diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzziwrj" "b/data_all_eng_slimpj/shuffled/split2/finalzziwrj" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzziwrj" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nIn this paper, we are concerned with numerically solving convection-diffusion equations of the form\n\\begin{equation}\\label{CDeqn}\n\\begin{cases}\n u_t + \\nabla\\cdot\\mathbf{F}(u) = \\epsilon\\Delta u + g(\\mathbf{x},t),&\\mathbf{x}\\in\\mathcal{D},\\quad t>0,\\\\\n u(\\mathbf{x},t=0)=u_0(\\mathbf{x}),&\\mathbf{x}\\in\\mathcal{D},\n\\end{cases}\n\\end{equation}\nwhere $\\epsilon\\geq 0$. We propose an Eulerian-Lagrangian Runge-Kutta finite volume (EL-RK-FV) scheme utilizing weighted essentially non-oscillatory (WENO) schemes with adaptive order (WENO-AO) for spatial reconstruction. The proposed method is designed for one-dimensional problems of the form \\eqref{CDeqn} and extended to higher-dimensional problems via dimensional splitting.\\\\\n\\indent Eulerian-Lagrangian (EL) and semi-Lagrangian (SL) schemes \\cite{Celia1990,Russell2002,Xiu2001} have proven to be computationally effective when solving hyperbolic problems because of their ability to employ high spatial resolution schemes while admitting very large CFL with numerical stability. Generally speaking, an EL or SL method involves working on a background grid so that high-order spatial resolutions can be used (the Eulerian part), and tracing characteristics of the cell boundaries backward\/forward in time to relax the CFL constraint (the Lagrangian part). Such methods have been developed in a wide variety of frameworks: discontinuous Galerkin \\cite{Cai2017,Ding2020,Qiu2011b,Rossmanith2011}, finite difference \\cite{Carrillo2007,Chen2021,Huot2003,Li2022,Qiu2010,Qiu2011,Xiong2019}, finite volume \\cite{Abreu2017,Benkhaldoun2015,Crouseilles2010,Filbet2001,Huang2012,Huang2017}. Although the SL and EL frameworks are similar in spirit, the SL framework assumes exact characteristic tracing and hence poses difficulties when considering nonlinear problems. Two other methods similar to EL and SL methods are the arbitrary Lagrangian-Eulerian (ALE) methods where an arbitrary mesh velocity not necessarily aligned with the fluid velocity is defined \\cite{Boscheri2014,Boscheri2013,Boscheri2015,Boscheri2017,Donea2004,Hirt1974,Peery2000}, and moving mesh methods where the PDE is first evolved in time and then followed by some mesh-redistribution procedure \\cite{Li2001,Li1997,Luo2019,Russell2002,Stockie2001,Tang2003}. The main difference between these two methods and the previously mentioned methods is that they move the mesh adaptively to focus resolving the solution around sharp transitions; whereas EL and SL methods evolve the equation by following characteristics.\\\\\n\\indent Our goal in this paper is to develop a new high-order EL method in the finite volume framework using method-of-lines (MOL) RK time discretizations. Finite volume methods are attractive since they are naturally mass conservative, easy to physically interpret, and modifiable for nonuniform grids. Similar to the recent developments made by Huang, Arbogast, and Qiu \\cite{Huang2012,Huang2017}, we use approximate characteristics to define a traceback space-time region, and then use WENO reconstructions to evaluate the modified flux. Huang and Arbogast developed a re-averaging technique that allows high-order reconstruction of the solution at arbitrary points by applying a standard WENO scheme \\cite{Cockburn1997,Shu2009} over a uniform reconstruction grid that is defined separately. They then used a natural continuous extension \\cite{Zennaro1986} of Runge-Kutta schemes, which requires the solution at several Gaussian nodes of the interval $[t^n,t^{n+1}]$, to evolve the solution along the approximate characteristics. \\\\\n\\indent The novelty of our proposed method is twofold: (1) the partition of space-time regions formed by linear approximations of the characteristic curves, and (2) integrating the differential equation over the partitioned space-time regions, followed by rewriting the space-time integral form of the equation into a spatial-integral time-differential form. In this way, a MOL RK type method can be directly applied for time discretization, thus avoiding the need to use a natural continuous extension of RK schemes. To be more precise, we construct linear approximate characteristics by using the Rankine-Hugoniot jump condition to define the traceback space-time regions. If the linear approximate characteristics are defined with zero velocity, then the proposed EL-RK-FV scheme reduces to the standard RK-FV method. Whereas, when linear space-time curves adequately approximate the exact characteristics, a large time stepping size is still permitted. We use WENO-AO to perform a solution remapping of the uniform cell averages onto the possibly nonuniform traceback cells. The recently developed WENO-AO schemes \\cite{Arbogast2020,Balsara2016,Balsara2020} are robust and guarantee the existence of the linear weights at arbitrary points. We note that Chen, et al. used WENO-AO schemes in the SL framework \\cite{Chen2021}, and Huang and Arbogast have recently used WENO-AO schemes in the Eulerian framework \\cite{Arbogast2019,Arbogast2020}. RK methods are used to evolve the MOL system along the approximate characteristics. Explicit RK methods, such as the strong stability-preserving (SSP) RK methods \\cite{Gottlieb2001}, are used for convection equations; and implicit-explicit (IMEX) RK methods \\cite{Ascher1997,Conde2017,Higueras2014} are used for convection-diffusion equations. In the latter case, the non-stiff convective term is treated explicitly and the stiff diffusive term is treated implicitly. Dimensional splitting is used to extend the one-dimensional algorithm to solve multi-dimensional problems. The proposed method is high-order accurate, capable of resolving discontinuities without oscillations, mass conservative, and stable with large time stepping sizes.\\\\\n\\indent The paper is organized as follows. We discuss the EL-RK-FV algorithm for pure convection problems in Section 2. In Section 3, we discuss the EL-RK-FV algorithm, coupled with IMEX RK schemes, for convection-diffusion equations. Numerical performance of the EL-RK-FV algorithm is shown in Section 4 by applying the algorithm to several linear and nonlinear test problems. Conclusions are made in Section 5. Appendices are listed after the references.\n\n\n\n\n\n\\section{The EL-RK-FV method for pure convection problems}\n\nThe spirit of the EL-RK-FV method is best demonstrated by starting with a pure convection problem in one dimension, i.e., equation \\eqref{CDeqn} with $\\epsilon=0$ and $g(x)=0$. Let the flux be denoted $f(x)$. We first discuss the formulation of the scheme in Section~\\ref{sec2.1}, followed by discussion of high-order spatial reconstruction in Section~\\ref{sec2.2} and time discretization in Section~\\ref{sec2.3}.\n\n\\subsection{\nScheme formulation\n}\n\\label{sec2.1}\n\nWe discretize the spatial domain $[a,b]$ into $N_x$ intervals with $N_x+1$ uniformly distributed nodes\n\\[a=x_{\\frac{1}{2}} 2$. We therefore conclude, that the DIFP can only be stable for $N_2 = N_3 = 1$.\n\nThere are two further exponents that we need to consider to establish the stability of the DIFP solution. Both follow from a scaling relation exploiting the decoupling of the $1$-sector: At the FP, the operators $\\mathcal{O}_1=\\phi_1^2 (\\phi_2^2+\\phi_3^2)$ and $\\mathcal{O}_2 = \\phi_1^2 \\left(\\phi_2^2 - \\frac{N_2}{N_2+N_3}(\\phi_2^2+\\phi_3^2) \\right)$ correspond to eigendirections of the stability matrix. As we know from the Wilson-Fisher FP that the scaling dimension $[\\phi_I^2] = -\\frac{1}{\\nu_I}+d$, we deduce that\n\\begin{equation}\n[\\mathcal{O}_1] = -\\frac{1}{\\nu_1} - \\frac{1}{\\nu_{2+3}} + 2d, \n\\end{equation}\nand accordingly the corresponding critical exponent is given by\n\\begin{equation}\n\\theta_{6} = \\frac{1}{\\nu_{1}} + \\frac{1}{\\nu_{2+3}} - d ~.\n\\end{equation}\nSimilarly, we deduce for the second operator that the corresponding critical exponent is given by\n\\begin{equation}\n\\theta_{4} = \\frac{1}{\\nu_{1}} + y_{2,2} - d ~,\n\\end{equation}\nwhere $y_{2,2}$ is the scaling dimension of the coupling belonging to $\\phi_2^2 - \\frac{N_2}{N_2+N_3}(\\phi_2^2+\\phi_3^2)$ in the two-field case, cf.\\ Ref.\\ \\cite{Eichhorn:2013zza}. The first relation is the one that arises for a two-field DFP, and gives a negative critical exponent (for values of $N_{1} > 1$).\n\nAt fixed $N_{2} = N_{3} = 1$, it is the critical exponent $\\theta_{4}$ that decides about the stability of the FP. Using results from the two-field case (LPA 12 including an anomalous dimension, cf.\\ Ref.\\ \\cite{Eichhorn:2013zza}) to obtain $\\theta_{1} = y_{2,2}$, $\\theta_{2} = \\frac{1}{\\nu_{1}}$, and $\\theta_{3} = \\frac{1}{\\nu_{2+3}}$, we arrive at the results shown in Tab.~\\ref{Tab:DIFP}. The DIFP is the stable FP for $N_{1} \\geq 6$ and $N_{2} = N_{3} = 1$.\n\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.4}\n\\renewcommand{\\tabcolsep}{4pt}\n\\begin{tabular}{c c c | c c c c c c}\n$N_1$ & $N_2$ & $N_3$& $\\theta_{1}$ & $\\theta_{2}$ & $\\theta_{3}$ & $\\theta_{4}$ & $\\theta_{5}$ & $\\theta_{6}$ \\\\\n\\hline\\hline\n1 & 1 & 1 & 1.765 & 1.571 & 1.459 & 0.336 & -0.042 & 0.030 \\\\ \n2 & 1 & 1 & 1.765 & 1.459 & 1.459 & 0.224 & -0.042 & -0.092 \\\\\n3 & 1 & 1 & 1.765 & 1.367 & 1.459 & 0.132 & -0.042 & -0.174 \\\\\n4 & 1 & 1 & 1.765 & 1.296 & 1.459 & 0.061 & -0.042 & -0.245 \\\\\n5 & 1 & 1 & 1.765 & 1.242& 1.459 & 0.007 & -0.042 & -0.299 \\\\\n6 & 1 & 1 & 1.765 & 1.203 & 1.459 & -0.032 & -0.042 & -0.338 \\\\ \n\\hline\\hline\n\\end{tabular}\n\\caption{\\label{Tab:DIFP}Critical exponents at the DIFP, using the LPA 12 including anomalous dimensions, and employing the above scaling relations.}\n\\end{table}\n\nNote that the derivation of the scaling relations is based on the assumption that the operators corresponding to these couplings are eigenoperators of the stability matrix. This property is, to the best of our knowledge, an assumption in $d = 3$ \\cite{Calabrese:2002bm}, and is usually not true within a truncation of the RG flow. Nevertheless, the stability properties are not incompatible with explicit numerical results within the LPA 8, where the transition to stability occurs already at $N_1=5$. { Explicitly, the critical exponents in LPA 8 at $N_1=5$ and $N_2=N_3=1$ read $\\theta_1=1.783, \\theta_2=1.193, \\theta_3=1.399, \\theta_4=-0.024,\\theta_5=-0.027$ and $\\theta_6=-0.395$.}\n\n\\vskip 5pt\n\n\n\\subsection{Decoupled biconical fixed point}\n\\label{SubSec:Decoupled biconical fixed point}\n\n\\begin{table}[!h]\n\\renewcommand{\\arraystretch}{1.4}\n\\renewcommand{\\tabcolsep}{2.7pt}\n\\begin{tabular}{c c c | c c c c c c}\n$N_1$ & $N_2$ &$N_3$& $\\theta_{1}$ & $\\theta_{2}$ & $\\theta_{3}$ & $\\theta_{4}$ & $\\theta_{5}$ & $\\theta_{6}$ \\\\\n\\hline\\hline\n1 & 1.2 & 1 & {\\bf 1.753} & {\\bf 1.381} & \\textit{1.537} & 0.285 & -0.075 & {\\bf -0.005} \\\\\n1 & 1.4 & 1 & {\\bf 1.535} & {\\bf 1.448} & \\textit{1.537} & 0.105 & -0.015 & {\\bf -0.010} \\\\\n1 & 1.5 & 1 & {\\bf 1.537} & {\\bf 1.462} & \\textit{1.537} & 0.068 & 0.001 & {\\bf -0.001} \\\\ \n1 & 1.2 & 2 & {\\bf 1.753} & {\\bf 1.381} & \\textit{1.399} & 0.161 & -0.200 & {\\bf -0.005} \\\\\n1 & 1.4 & 2 & {\\bf 1.535} & {\\bf 1.448} & \\textit{1.399} & -0.020 & -0.140 & {\\bf -0.010} \\\\\n1 & 1.5 & 2 & {\\bf 1.537} & {\\bf 1.462} & \\textit{1.399} & -0.057 & -0.124 & {\\bf -0.001} \\\\\n1.2 & 1 & 5 & {\\bf 1.753} & {\\bf 1.381} & \\textit{1.193} & -0.051 & -0.413 & {\\bf -0.005} \\\\\n1.5 & 1 & 5 & {\\bf 1.537} & {\\bf 1.462} & \\textit{1.193} & -0.270 & -0.338 & {\\bf -0.001} \\\\\n\\hline\\hline\n\\end{tabular}\n\\caption{\\label{Tab:BFP}Critical exponents at the DBFP in the LPA 8. Critical exponents emphasized in \\textbf{bold} font correspond to those of the two-field BFP, while those in \\textit{italic} arise in the sector with $O(N_3)$ group symmetry. Here, we list the DBFP for values of the field components $N_{I}$, where it is characterized by $\\Delta > 0$ (see text).}\n\\end{table}\n\nBeyond the isotropic and decoupled FP solutions, the two-field models feature another scaling solution, which is the biconical FP \\cite{PhysRevLett.33.813, Calabrese:2002bm, Folk:2008mi}. It is stable only in a restricted parameter region of these models, where $\\left. \\Delta^{\\textrm{2-field}} \\right|_{\\textrm{BFP}} = \\lambda_{2,0} \\lambda_{0,2} - \\lambda_{1,1}^2 > 0$, as it transfers the stability from the IFP to the DFP. Certainly, this FP should similarly manifest itself in the three-field case. While one of the three sectors decouples, the two remaining sectors should feature nondegenerate couplings, and we expect that in a given range of the parameter space such a decoupled BFP will be stable. \n\n{ To obtain as precise results as possible, we should make use of all methods available to us. In fact, results obtained using an $\\epsilon$-expansion around $d=4$ in the two-field case, allow us to infer the stability of the decoupled biconical fixed point in the three field case in one important instance: }\nFrom Refs.\\ \\cite{Folk:2008mi, Calabrese:2002bm}, we know that the biconical FP is stable for $N_1 = 1, N_2 = 2$ in two-field models (and similarly when the sectors are interchanged). Combined with the pattern in Tab.\\ \\ref{Tab:BFP} for the additional critical exponents in the three-field model, we conjecture that the DBFP is stable for $N_1 = 1, N_2 = 2, N_3\\geq 2$ (up to a permutation of the three sectors). To calculate the corresponding critical exponents directly in the three-field model, we expect that an extended truncation will be necessary, taking into account a field-dependent wavefunction renormalization.\n\n\n\\subsection{Search for further stable fixed points}\n\\label{SubSec:Search for further stable fixed points}\n\nWe summarize our results obtained so far in Fig.~\\ref{Fig:StabilityPlot}. The figure shows the stable FP solution for the corresponding values of field components, $(N_1, N_2, N_3)$. Apparently, no stable FP exists in the range $N_{1} < 6$, $N_{2} = N_{3}=1$ (up to a permutation of the sectors) that can be derived from the known scaling solutions in the one- and two-field models. This motivates an independent analysis of fully coupled FPs in the three-field model, which we describe in the following sections.\n\n\\begin{figure}[!t]\n\\includegraphics[width=0.8\\linewidth]{stability_cube_conjectured_modified.pdf}\n\\caption{\\label{Fig:StabilityPlot}(Color online) We show the stable DIFP (large blue dots), the stable DFP (small orange dots) and points without a stable FP (small gray dots) using the LPA 12$+\\eta$ results. We also include points where the DBFP is conjectured to be stable (middle-sized green dots).}\n\\end{figure}\n\n\n\\subsubsection{Stability trading between stable FPs}\n\\label{SubSec:Stability trading between stable FPs}\n\nGenerally, the $\\beta$ functions are non-polynomial functions of a large number of couplings in the multifield models. In the local potential type approximations the number of parameters and couplings increases from 9 to 34 if the order of the truncation is changed from 4 to 8. Thus, finding FPs in the multifield models becomes a highly nontrivial search for zeros of the $\\beta$ functions in a high-dimensional parameter space. In the following, we consider strategies to identify new FP solutions using a simple example. Consider the following $\\beta$ function which is expanded in terms of the coupling $g$ (assuming that higher than quadratic terms are zero):\n\\begin{equation}\n\\beta_g = g ( c + g ),\n\\end{equation}\nwhere $c$ is a function of the parameters of the model (e.g., dimensionality, number of field components, etc.) and possibly other couplings in a given truncation of the theory. Note that such a form captures the essential properties of typical fixed points, as it allows both for a trivial Gaussian FP and a nontrivial interacting FP, as a function of the parameter $c$. The critical exponent at a fixed point is given by\n\\begin{equation}\n\\theta = - \\left. \\frac{\\partial\\beta_g}{\\partial g} \\right|_{\\textrm{FP}} = \\left. - c - 2 g \\right|_{\\rm FP}. \n\\end{equation}\nAssuming that it is the exponent $\\theta$ that decides about the stability of the FP, we may distinguish the following scenarios: For $c < 0$, the interacting FP at $g = -c$ is infrared stable, whereas the Gaussian FP is unstable. As the parameter $c$ increases towards positive values (as a function of, e.g., $N_I$) the two FPs will approach each other. At $c = 0$, both FPs collide and exchange their stability properties. Moving apart again for $c > 0$, the interacting FP has become the unstable one, whereas the noninteracting FP is stable, cf.\\ Fig.\\ \\ref{Fig:StabilityTrading}. This simple example demonstrates that FPs will typically change their stability when they collide, as was also observed in Ref.\\ \\cite{Eichhorn:2013zza} in the two-field model with $O(N_{1})\\oplus O(N_{2})$ symmetry.\n\n\\begin{figure}[!h]\n\\includegraphics[width=0.8\\linewidth]{stability_trading3.pdf}\n\\caption{\\label{Fig:StabilityTrading}To illustrate the mechanism how stability properties are exchanged between FPs, we examine the $\\beta$-function $\\beta = g(c + g)$ for varying parameter $c$ describing the position of the nontrivial FP. As the FPs pass by one another, the local derivatives $\\theta = -\\beta'(g)$ exchange their sign indicating an interchange of their IR stability properties. }\n\\end{figure}\n\n\nInspired by the above stability-trading mechanism\nwe may devise a strategy to identify new FP solutions. Our search for FPs will concentrate on the vicinity of points in coupling space, where a known FP loses its stability.\n\n\n\\subsubsection{Fully coupled fixed points in the three-field model}\n\\label{SubSec:Fully coupled fixed points in the three-field model}\n\nIt turns out that the three-field model works in a different way from the mechanism described above, which applies in the two-field case: Within the LPA 4, we observe that the IFP, DFP, and DIFP have partially overlapping stability regions. No similar behavior occurs in two-field models, where stability regions of different FPs always touch, but never overlap, due to the above stability-trading mechanism. However, this changes dramatically as we include anomalous dimensions: While the IFP inhabits the same points, the DIFP is now only stable for $N_1 \\geq 4$, $N_2 = N_3=1$ etc.\\ Extensive numerical searches did not reveal a stable FP for $N_1 =3$, $N_2 = N_3=1$, and similarly for the cases where the sectors are interchanged. A similar result holds for higher orders of the LPA: As there are a larger number of independent operators that serve as a basis for the LPA, and thus potentially relevant directions in the three-field model, the stability exchange mechanism may not be captured by the simple model considered above. To elucidate the differences in the three-field model, we will focus on results obtained within the LPA to 8th order. \n\n\\begin{figure}[!t]\n\\includegraphics[width=\\linewidth]{BFPDFPcollision.pdf}\n\\caption{\\label{BFPDFPcollision}We plot $\\lambda_{1,0,1}, \\lambda_{0,1,1}$, and $\\lambda_{1,1,0}$ as a function of $N_I$ for the three DBFPs. At $N_I=1.244$ they each pass through the origin of that coordinate system, where the DFP sits. For $N_I>1.244$, the three BFPs move away from each other towards more negative values of the mixed couplings.}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\includegraphics[width=0.8\\linewidth]{BFPDFPthetas.pdf}\n\\includegraphics[width=0.8\\linewidth]{BFPDFPcoords.pdf}\n\\caption{\\label{BFPDFPthetas}We plot the fifth-largest critical exponent at the DBFPs (blue points of increasing value) and the corresponding critical exponent at the DFP (red points of decreasing value) as a function of $N_I$ (upper panel). Below, we show the coordinates of the couplings $\\lambda_{2,0,0}$ and $\\lambda_{0,2,0}$ at the DBFP (blue diamonds and black squares) and the couplings $\\lambda_{2,0,0}= \\lambda_{0,2,0} = \\lambda_{0,0,2}$ at the DFP (red dots) in the vicinity of the collision point.}\n\\end{figure}\n\nAs a first example, let us consider the point $N_1 = N_2 = N_3 \\approx 1.25$. Here, the DFP is stable, but it gains three additional relevant directions around $N_1 = N_2 = N_3 \\approx 1.244$. As within the two-field model, this point is marked by a collision with a decoupled biconical FP (DBFP). The main difference is that within a three-field model, three generalizations of the BFP exist, cf.\\ Sec.\\ \\ref{SubSec:Decoupled biconical fixed point}. For $N_I =1.25$, $I = 1,2,3$, all of these FPs feature five relevant directions. Toward smaller $N_I$, all three DBFPs approach the DFP, and simultaneously collide with it at $N_I \\approx 1.244$. At this point, the DFP gains one relevant direction from each of the three DBFPs, which subsequently feature only four relevant exponents, cf.\\ Fig.\\ \\ref{BFPDFPcollision} and Fig.~\\ref{BFPDFPthetas}. Thus this FP loses stability in a fixed-point collision. The central difference to the two-field case lies in the fact that the symmetry of the model which forces the DFP to collide with three other FPs simultaneously (for $N_1 = N_2 = N_3$ there is an exchange symmetry $N_1 \\leftrightarrow N_2 \\leftrightarrow N_3$). As each of them starts off with five relevant directions, the collision does not produce a stable FP for $N_I \\leq 1.24$, but instead leaves behind three FPs with four relevant directions each.\n\nAs a second example of new behavior in three-field models, we consider the point where the DBFP collides with the DIFP in the LPA 8. We fix $N_2 =1$ and $N_3=4$: Then the DBFP has five positive critical exponents at $N_1=1$. Going to larger values of $N_1$, it collides with the DIFP at $N_1 \\approx 1.6$. During this collision, the DIFP becomes unstable, and the DBFP gains one negative critical exponent, cf.\\ Fig.\\ \\ref{BFPDIFP}. Similar to the previous scenario, this FP collision is not sufficient to make the DBFP stable. Following the DBFP to even larger values of $N_1$, it undergoes another collision, this time being hit by two other FPs simultaneously. This is a novel feature that is not observed in simple two-field models. Starting in the region $N_1 < 1.2$, these FPs do not seem to exist for real values of the couplings -- at least no sign of them showed up in extensive numerical searches. They can be thought of as being created at the collision. Following this collision, they quickly move away from the collision point for increasing values of $N_{1}$. Each of the newly created FPs features four relevant critical exponents, while the DBFP is stable, cf.\\ Fig.\\ \\ref{BFPcollision}. Both new FPs are anisotropic, and define a new universality class that occurs for the first time if three fields are coupled, cf.\\ Tab.\\ \\ref{Tab:AFP}.\n\n\\begin{figure}[!h]\n\\includegraphics[width=0.9\\linewidth]{BFPcoords.pdf}\\\\\n\\includegraphics[width=0.9\\linewidth]{BFPthetas.pdf}\\\\\n\\includegraphics[width=0.9\\linewidth]{DIFPtheta.pdf}\n\\caption{\\label{BFPDIFP}(Color online) Here, we show the couplings $\\lambda_{2,0,0}$, $\\lambda_{0,2,0}$, and $\\lambda_{1,1,0}$ at the DBFP (blue circles, purple squares, cyan diamonds) and $\\lambda_{2,0,0} = \\lambda_{0,2,0} = \\lambda_{1,1,0}$ (red triangles) at the DIFP (uppermost panel). We show the fourth- and fifth-largest critical exponent (middle panel) at the DBFP as a function of $N_1$ for $N_2=1, N_3=4$. Around $N_1 \\approx 1.16$, the three couplings are clearly degenerate, as expected for a collision with the DIFP. At the same point, the fourth critical exponent crosses zero and becomes negative. Simultaneously, the fifth critical exponent at the DIFP (lower panel) crosses zero and becomes positive.}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\includegraphics[width=\\linewidth]{AFPcollision.pdf}\\\\\n\\includegraphics[width=\\linewidth]{AFPtheta.pdf}\\\\\n\\caption{\\label{BFPcollision}Here, we show the couplings $\\lambda_{1,0,1}$ and $\\lambda_{0,1,1}$ at the two anisotropic FPs. As a function of $N_1$ they converge towards zero, which is where the DBFP is sitting. The collision occurs at $N_1 \\approx 1.2$, which is where the fifth critical exponent of these two FPs (which numerically is nearly the same for both) approaches zero.}\n\\end{figure}\n\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.4}\n\\renewcommand{\\tabcolsep}{2.7pt}\n\\begin{tabular}{ccc|ccc|ccc|c}\n$N_1$ & $N_2$ & $N_3$ & $\\lambda_{2,0,0}$ & $\\lambda_{0,2,0}$ & $\\lambda_{0,0,2}$ & $\\lambda_{1,1,0}$ & $\\lambda_{1,0,1}$ & $\\lambda_{0,1,1}$ & $\\lambda_{1,1,1}$\\\\ \\hline \\hline\n2 & 2 & 2 & 6.2 & 6.4& 6.4 & 1.8 & 1.8 & -2.2 & -2.6\\\\\n2 & 1 & 2 & 6.5 & 7.5 & 6.7 & 1.2 & 0.5 & -2.1 & -1.4\\\\\n2 & 1 & 3 & 6.2 & 7.0 & 5.8 & 3.2 & 0.7 & -1.8 & -2.2\\\\\n2 & 1 & 4 & 6.0 & 6.5 & 5.1 & 4.6 & 0.6 & -1.3 & -1.6\\\\ \\hline \\hline\n\\end{tabular}\n\\vskip 4pt\n\\renewcommand{\\arraystretch}{1.4}\n\\renewcommand{\\tabcolsep}{6.8pt}\n\\begin{tabular}{ccc|ccc|ccc}\n{} & {} & {} & $\\theta_1$ & $\\theta_2$ & $\\theta_3$ & $\\theta_4$ & $\\theta_5$ & $\\theta_6$ \\\\ \\hline \\hline\n2 & 2 & 2 & 1.70 & 1.33 & 1.29 & 0.20 & -0.17 & -0.24\\\\\n2 & 1 & 2 & 1.62 & 1.40 & 1.35 & 0.08 & -0.05 & -0.23\\\\\n2 & 1 & 3 & 1.71 & 1.34 & 1.28 & 0.15 & -0.10 & -0.30\\\\\n2 & 1 & 4 & 1.77 & 1.32 & 1.23 & 0.16 & -0.07 & -0.35\\\\ \\hline \\hline\n\\end{tabular}\n\\caption{\\label{Tab:AFP}We list selected FP values and critical exponents of the first anisotropic FP in the LPA 8, restricting ourselves to couplings at fourth order in the fields, and additionally giving the value of the coupling associated with the three-field operator $\\sim \\phi_1^2 \\phi_2^2 \\phi_3^2$.}\n\\end{table}\n\nDue to the complicated dynamics of FPs in the parameter space of three-field models, it seems that these models do not necessarily feature a stable FP solution for all values of the model parameters. One might now wonder, why FPs of three-field models show such a disproportionate increase in the number of relevant directions for small values of $N_I$. While the Wilson-Fisher FP has one relevant direction, and two-field models always feature FPs with only two relevant directions, three-field models show FPs with more than three relevant directions. We conjecture that this is related to the fact that the transition from a two-field to a three-field model implies the existence of more than one additional new class of operators: While the transition from one to two fields only adds the mixed interactions $\\sim \\left( \\phi_1^{2} \\right)^{m} \\left( \\phi_2^{2} \\right)^{n}$, the transition to three-field models features three different classes of mixed interactions $\\sim \\left( \\phi_I^{2} \\right)^{m} \\left( \\phi_J^{2} \\right)^{n}$, $I,J = 1,2,3$, in addition to the new three-field couplings $\\sim \\left( \\phi_1^{2}\\right)^{l} \\left( \\phi_2^{2} \\right)^{m} \\left( \\phi_3^{2} \\right)^{n}$. These seem to play a particularly important role for small $N_I$ and imply that the IR scaling properties cannot be accounted for by only three free relevant parameters. Note that this does not imply that the corresponding additional couplings need to become relevant. Since these operators do not necessarily correspond to eigendirections of the RG, they may mix with other operators, and yield corrections to the scaling spectrum.\n\nApplying our results to determine the properties of possible multicritical points in phase diagrams for systems with three competing order parameters, we may conclude that models with small $N_I$ will typically feature a first order, rather than a second order (multicritical) transition. In particular, this applies to the phenomenologically relevant model of three interacting $Z_{2}$-Ising fields, i.e., $Z_{2} \\oplus Z_{2} \\oplus Z_{2}$ symmetry.\n\n\n\\section{Summary and conclusions}\n\\label{Sec:Summary and conclusions}\n\nHere, we present a renormalization group study of IR stable FP solutions in three-field models with $O(N_1) \\oplus O(N_2) \\oplus O(N_3)$ symmetry. Our main results regarding the existence of stable FPs are summarized in Fig.~\\ref{Fig:StabilityPlot}. Models in this class exhibit FPs that generalize the Wilson-Fisher FP, falling into three distinct categories, each characterized by the degree of symmetry-enhancement. We find a decoupled FP, a partially isotropic FP solution, and a fully isotropic FP solution. Their scaling spectrum can be deduced partially by considering perturbations around the single- and two-field models with $O(N)$ and $O(N_{1})\\oplus O(N_{2})$ symmetry, respectively. We proceed by deriving scaling relations between different critical exponents to discover the stability properties of nontrivial FPs in the three-field case. Apart from the generalized Wilson-Fisher scaling solutions, we identify a decoupled biconical FP whose scaling properties are partly inherited from the BFP in the $O(N_{1})\\oplus O(N_{2})$ symmetric model. As a main result of this work, we find that these FPs all show a significantly larger number of relevant critical exponents than in the two-field case, in the region of small $N_1$, $N_2$, and $N_3$. We tentatively connect this result to the existence of a large number of mixed interactions, and further conjecture, that similar results will hold for models with $n>3$ interacting fields.\n\nSummarizing our results, we find no IR stable FP for a small number of fields ($N_{1} < 6$, $N_{2} = N_{3} = 1$, up to permutations of the fields) in $d = 3$ dimensions. This result is certainly unexpected, as there is no evidence for similar behavior in coupled two-field models. While, in principle, we cannot exclude the possibility that stable FPs exist in that region of parameter space, we find no evidence for their existence in extensive numerical searches for FPs of the nonperturbative $\\beta$ functions. The identification of FPs in the three-field field models with $O(N_{1})\\oplus O(N_{2})\\oplus O(N_{3})$ is in general a difficult problem, since the search has to proceed through a high-dimensional coupling space. Nevertheless, the understanding of basic stability transitions between different FPs serves as a guiding principle to single out possible candidates for nontrivial FPs. Quite generally, in coupled-field models, stability seems to be inherited from single mergers or collisions of different FPs. Searches around such stability transition points have not yielded any FP that carries over the stability properties from the IFP (stable at small, noninteger values of $N_{I}<1$) to the decoupled FP. This indicates that the dynamics of FPs in $O(N_{1})\\oplus O(N_{2})\\oplus O(N_{3})$ symmetric three-field, or general $\\bigoplus_{I} O(N_{I})$ symmetric multifield models is very different from that encountered in the simpler $O(N_1)\\oplus O(N_2)$-type models.\n\nFrom these results, we may conclude that models with phenomenological relevance such as, e.g., the $Z_{2} \\oplus Z_{2} \\oplus Z_{2}$ symmetric model will not feature a multicritical point in its phase diagram. Certainly, it is challenging to find three parameters that are accessible experimentally, and may be tuned to the multicritical point. This would be necessary to quantify the scaling behavior close to the corresponding FP, or to show the absence of such a transition. Nevertheless, it is conceivable that in the context of ultracold atomic systems such a control of the system might be achievable \\cite{doi:10.1146\/annurev-conmatphys-070909-104059}. \n\nFinally, let us comment on the general applicability of our results to other systems of interest. The renormalization group flow equations are derived for general $d$ Euclidean dimensions and can be applied to $d = 2$, relevant for critical behavior of low-dimensional condensed-matter systems, and $d = 4$, for multifield models of inflation, as well as possible extensions of the Higgs sector of the standard model. We leave this for future work.\n\n\\begin{acknowledgements}\nWe thank P.~Calabrese for discussions.\nResearch at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. This work is supported by DOE grant No.\\ DE-FG0201ER41195. M.M.S. is supported by the grant ERC-AdG-290623 and DFG grant FOR 723. A.E. would like to thank the University of Heidelberg and the University of Illinois at Chicago for hospitality during the early stages of this work.\n\\end{acknowledgements}\n\n\\begin{appendix}\n\\label{Appendix}\n\n\\section{Functional renormalization group}\n\\label{Sec:Functional renormalization group}\n\nThe nonperturbative functional renormalization group \\cite{Wetterich:1992yh,Berges:2000ew,Polonyi:2001se,Pawlowski:2005xe,Gies:2006wv,Delamotte:2007pf} defines a functional flow for the scale dependent effective action $\\Gamma_{k}$, which interpolates between the microscopic action $S$ defined at some ultraviolet cutoff scale $k = \\Lambda$ and the full effective action $\\Gamma_{k\\rightarrow 0} = \\Gamma$, when the renormalization group parameter $k$ is removed. It is given by\n\\begin{equation}\n\\partial_t \\Gamma_{k} = \\frac{1}{2} \\Tr \\int\\! \\frac{d^{d}q}{(2\\pi)^{d}} \\left(\\Gamma_k^{(2)} (q) + R_k(q) \\right)^{-1} \\partial_t R_k(q) ~,\n\\label{Eq:WetterichEquation}\n\\end{equation}\nwhere the logarithmic scale derivative is written in terms of the parameter $t = \\ln (k\/\\Lambda)$, and the second functional derivative $\\Gamma_k^{(2)}(p,q) = \\frac{\\delta^2 \\Gamma_k}{\\delta \\chi(-p) \\delta \\chi(q)}$, and $\\Gamma^{(2)}(q) (2\\pi)^{d} \\delta^{(d)}(p-q) \\equiv \\Gamma^{(2)} (p,q)$. Here, $\\chi$ denotes the complete field content of our model and the trace $\\Tr$ denotes a summation over internal degrees of freedom, i.e., both fields and field components. The regulator function $R_{k}$ implements a mass-like cutoff and regulates the infrared divergences. We take $R_{I J} (q) = R_{I}(q) \\delta_{I J}$, $I, J = 1,2,3$, while the momentum dependence is given by $R_{I}(q) = Z_{I} (k^{2} - q^{2}) \\theta(k^{2} - q^{2})$, where the wavefunction renormalization is scale dependent. This choice is referred to as the optimized regulator \\cite{Litim:2000ci, Litim:2001up} which allows us to derive fully analytic expressions for the nonperturbative $\\beta$ functions, and it is thus a convenient choice to identify possible scaling solutions.\n\n\\onecolumngrid\n\\section{Scale dependent effective potential}\n\\label{Sec:Scale dependent effective potential}\n\nWe derive the renormalization group flow equation for the effective potential by plugging our \\textit{ansatz} Eq.\\ \\eqref{Eq:EffectivePotential} into the flow equation \\eqref{Eq:WetterichEquation} and projecting the right-hand-side onto a constant field configuration:\n\\begin{eqnarray}\n\\partial_t u_k &=& -d u_k +\\sum_I (d-2+ \\eta_I) \\rho_I \\partial_{\\rho_{I}} u_{k} + { 2 v_{d} \\sum_{I} \\Big\\{ (N_I-1) l_{0}^{(I)} \\left( \\partial_{\\rho_{I}} u_{k} \\right)} \\nonumber\\\\ && { +\\: l_{R}^{(I)} \\left( \\left\\{ \\partial_{\\rho_{J}} u_{k} + 2 \\rho_{J} \\partial_{\\rho_{J}}^{2} u_{k} \\right\\}, \\left\\{ 4 \\rho_{J} \\rho_{K} \\partial_{\\rho_{J}} \\partial_{\\rho_{K}} u_{k} \\right\\} \\right)} \\Big\\} ~. \n\\label{Eq:FlowEquationPotential}\n\\end{eqnarray}\nHere, $v_{d}^{-1} = 2^{d+1} \\pi^{\\frac{d}{2}} \\Gamma\\left(\\frac{d}{2}\\right)$ arises from the volume integration, and the anomalous dimensions are defined as $\\eta_I = - \\partial_t \\ln Z_I$. In the following the notation follows Sec.\\ \\ref{Sec:Model}. The threshold functions $l_{0}^{(I)}$ and $l_{R}^{(I)}$ (see, e.g., \\cite{Braun:2011pp}) define the diagrammatic contributions to the renormalization group flow of the scale dependent effective potential, where the upper index indicates the corresponding sector. Using an optimized regulator function \\cite{Litim:2000ci,*Litim:2001up} the threshold functions take the following form:\n\\begin{eqnarray}\nl_{0}^{(I)}(w_{I}) &=& \\frac{2}{d} \\left( 1 - \\frac{\\eta_{I}}{d+2}\\right)\\frac{1}{1+w_{I}} ~, \\\\\n\\hspace{-20pt} l_{R}^{(1)}(\\{ w_{I} \\}, \\{ \\delta_{I,J}^{2} \\}) &=& \\frac{2}{d} \\left( 1 - \\frac{\\eta_{1}}{d+2}\\right) \\frac{\\left( 1 + w_{2} \\right) \\left( 1 + w_{3} \\right) - \\delta_{2,3}^{2}}{2 \\delta_{1,2} \\delta_{1,3} \\delta_{2,3} \n- \\delta_{1,2}^{2} \\left( 1 + w_{3} \\right) - \\delta_{1,3}^{2} \\left( 1 + w_{2} \\right) - \\delta_{2,3}^{2} \\left( 1 + w_{1} \\right)\n+ \\prod_{J} \\left( 1 + w_{J} \\right)} ~, \\\\\n\\hspace{-20pt} l_{R}^{(2)}(\\{ w_{I} \\}, \\{ \\delta_{I,J}^{2} \\}) &=& \\frac{2}{d} \\left( 1 - \\frac{\\eta_{2}}{d+2}\\right) \n\\frac{\\left( 1 + w_{1} \\right) \\left( 1 + w_{3} \\right) - \\delta_{1,3}^{2}}{2 \\delta_{1,2} \\delta_{1,3} \\delta_{2,3} \n- \\delta_{1,2}^{2} \\left( 1 + w_{3} \\right) - \\delta_{1,3}^{2} \\left( 1 + w_{2} \\right) - \\delta_{2,3}^{2} \\left( 1 + w_{1} \\right)\n+ \\prod_{J} \\left( 1 + w_{J} \\right)} ~, \\\\\n\\hspace{-20pt} l_{R}^{(3)}(\\{ w_{I} \\}, \\{ \\delta_{I,J}^{2} \\}) &=& \\frac{2}{d} \\left( 1 - \\frac{\\eta_{3}}{d+2}\\right) \n\\frac{\\left( 1 + w_{1} \\right) \\left( 1 + w_{2} \\right) - \\delta_{1,2}^{2}}{2 \\delta_{1,2} \\delta_{1,3} \\delta_{2,3} \n- \\delta_{1,2}^{2} \\left( 1 + w_{3} \\right) - \\delta_{1,3}^{2} \\left( 1 + w_{2} \\right) - \\delta_{2,3}^{2} \\left( 1 + w_{1} \\right)\n+ \\prod_{J} \\left( 1 + w_{J} \\right)} ~.\n\\end{eqnarray}\nNote, that the nature of the interactions is such, that only the radial modes $l_{R}^{(I)}$ are affected by the couplings, e.g., $\\delta_{1,2}^{2} = 4 \\kappa_{1} \\kappa_{2} \\lambda_{1,1,0}^{2}$, and equivalent couplings between the remaining sectors. One may easily check that in the limit of vanishing couplings, $\\delta_{1,3} \\rightarrow 0$ and $\\delta_{2,3} \\rightarrow 0$, the radial contributions $l_{R}^{(I)}$ in the $(1,2)$ sectors reduce to the threshold functions that were already derived in the two-field model \\cite{Eichhorn:2013zza}:\n\\begin{eqnarray}\n\\hspace{-20pt} l_{0}^{(I)}( w_{I} ) &=& \\frac{2}{d} \\left( 1 - \\frac{\\eta_{I}}{d+2}\\right)\\frac{1}{1+w_{I}} ~, \n\\label{Eq:ThresholdFunctionL0} \\\\\n\\hspace{-20pt} l_{R}^{(1)}(\\{ w_{I} \\}, \\{ \\delta_{I,J}^{2} \\}) &=& \\frac{2}{d} \\left( 1 - \\frac{\\eta_{1}}{d+2}\\right) \\frac{1 + w_{2}}{\\left( 1 + w_{1} \\right) \\left( 1 + w_{2} \\right)- \\delta_{1,2}^{2}} ~, \\\\\n\\hspace{-20pt} l_{R}^{(2)}(\\{ w_{I} \\}, \\{ \\delta_{I,J}^{2} \\}) &=& \\frac{2}{d} \\left( 1 - \\frac{\\eta_{2}}{d+2}\\right) \\frac{1 + w_{1}}{\\left( 1 + w_{1} \\right) \\left( 1 + w_{2} \\right)- \\delta_{1,2}^{2}} ~.\n\\label{Eq:ThresholdFunctionLTildeR}\n\\end{eqnarray}\nFrom \\eqref{Eq:FlowEquationPotential} the flow equations for the couplings are derived by the differentiation with respect to the fields and successive projection onto a nonvanishing constant background field configuration $\\rho_{I} = \\kappa_{I}$, defined by the minimum of the effective potential. For some FPs it might be necessary to employ an expansion point where one or several of the $\\kappa_{I}$ are vanishing. In this case $\\beta_{\\kappa_{3}} \\equiv 0$. For a detailed discussion of this issue, we refer to Ref.\\ \\cite{Eichhorn:2013zza}. \n\nWe obtain the $\\beta$ functions for the couplings $\\lambda_{l,m,n}$, $l + m + n \\geq 2$:\n\\begin{equation}\n\\beta_{\\lambda_{l,m,n}} = \\left. \\left(\\frac{\\partial^{l+m+n}}{\\partial \\rho_1^{l} \\partial \\rho_2^{m} \\partial \\rho_3^{n}} \\partial_t u_k + \\beta_{\\kappa_1} \\frac{\\partial^{(l+1)+m+n} u_k}{\\partial \\rho_{1}^{l+1} \\partial \\rho_2^m \\partial \\rho_3^n} + \\beta_{\\kappa_2} \\frac{\\partial^{l+(m+1)+n} u_k}{\\partial \\rho_{1}^{l} \\partial \\rho_2^{m+1} \\partial \\rho_3^n} + \\beta_{\\kappa_3} \\frac{\\partial^{l+m+(n+1)} u_k}{\\partial \\rho_{1}^{l} \\partial \\rho_2^m \\partial \\rho_3^{n+1}} \\right) \\right|_{\\rho_I = \\kappa_I} ~,\n\\end{equation}\nwhere $\\beta_{\\lambda_{l,m,n}} \\equiv \\partial_{t} \\lambda_{l,m,n}$. The $\\beta$ functions for the scale dependent dimensionless field expectation values $\\kappa_{I}$, $I = 1,2,3$, are given by\n\\begin{eqnarray}\n\\beta_{\\kappa_1} &=& \\frac{1}{\\Delta} \\left\\{\n- \\Delta_{2,3} \\partial_{\\rho_1} \\partial_t u_k + \\left(\\lambda_{1,1,0} \\lambda_{0,0,2} - \\lambda_{1,0,1} \\lambda_{0,1,1}\\right) \\partial_{\\rho_2} \\partial_t u_k \\right. \\nonumber \\\\ && \\left.\\left. +\\: \\left(\\lambda_{1,0,1} \\lambda_{0,2,0} - \\lambda_{1,1,0} \\lambda_{0,1,1}\\right) \n\\partial_{\\rho_3} \\partial_t u_k \\right\\} \\right|_{\\rho_I = \\kappa_I} ~, \n\\end{eqnarray}\n\\begin{eqnarray}\n\\beta_{\\kappa_2} &=& \\frac{1}{\\Delta} \\left\\{ - \\Delta_{1,3} \\partial_{\\rho_2} \\partial_t u_k + \\left( \\lambda_{2,0,0} \\lambda_{0,1,1} - \\lambda_{1,1,0} \\lambda_{1,0,1} \\right) \\partial_{\\rho_3} \\partial_t u_k \\right. \\nonumber\\\\ \n&& \\left.\\left. +\\: \\left( \\lambda_{0,0,2} \\lambda_{1,1,0} - \\lambda_{1,0,1} \\lambda_{0,1,1} \\right) \\partial_{\\rho_1} \\partial_t u_k \\right\\} \\right|_{\\rho_I = \\kappa_I} ~, \\\\\n\\beta_{\\kappa_3} &=& \\frac{1}{\\Delta} \\left\\{\n- \\Delta_{1,2} \\partial_{\\rho_3} \\partial_t u_k + \\left( \\lambda_{2,0,0} \\lambda_{0,1,1} - \\lambda_{1,1,0} \\lambda_{1,0,1} \\right) \\partial_{\\rho_2} \\partial_t u_k \\right. \\nonumber\\\\ && \\left.\\left. +\\: \\left( \\lambda_{0,2,0} \\lambda_{1,0,1} - \\lambda_{1,1,0} \\lambda_{0,1,1}\\right) \\partial_{\\rho_1} \\partial_t u_k \\right\\} \\right|_{\\rho_I = \\kappa_I} ~. \n\\end{eqnarray}\nHere, we have defined the coupling parameter\n\\begin{equation}\n\\Delta_{1,2} = \\lambda_{2,0,0} \\lambda_{0,2,0} - \\lambda_{1,1,0}^{2} ~,\n\\end{equation}\nand equivalently $\\Delta_{1,3}$ and $\\Delta_{2,3}$ (defined from the remaining quartic couplings in the three-field model), as well as the parameter\n\\begin{eqnarray}\n\\Delta &=& - 2 \\left( \\lambda_{2,0,0} \\lambda_{0,2,0} \\lambda_{0,0,2} - \\lambda_{1,1,0} \\lambda_{1,0,1} \\lambda_{0,1,1} \\right) + \\lambda_{0,0,2} \\Delta_{1,2} + \\lambda_{0,2,0} \\Delta_{1,3} + \\lambda_{2,0,0} \\Delta_{2,3} ~.\n\\end{eqnarray}\nThese parameters quantify the symmetry enhancement properties of the system. In particular, for certain symmetry enhanced FPs, these quantities vanish exactly.\n\n\\section{Wavefunction renormalization and anomalous dimensions}\n\\label{Sec:Wavefunction renormalization and anomalous dimensions}\n\nTo determine the scale dependence of the field independent renormalization factor $Z_{I}$ from the functional flow equation \\eqref{Eq:WetterichEquation} we perform a projection of the flow onto operators of the type $\\sim \\left( \\partial_{\\mu} \\phi_{I} \\right)^{2} $. This yields the scale dependence of the coefficient for the corresponding operator in the effective action. We have\n\\begin{equation}\n\\partial_{t} Z_{I} = \\frac{(2\\pi)^{d}}{\\delta^{(d)}(0)} \\lim_{Q\\rightarrow 0} \\frac{\\partial}{\\partial Q^{2}} \\frac{\\delta^{2}}{\\delta \\varphi_{I}(-Q) \\delta \\varphi_{I}(Q)} \\partial_{t} \\Gamma_{k} ~,\n\\label{Eq:RenormalizationFactorProjection}\n\\end{equation}\nwhere the functional derivatives are taken with respect to the Nambu-Goldstone (NG) degrees of freedom $\\varphi_{I}$ in the $I$-sector. Note, there is no summation implied over the $I$-index. For details of the derivation in the context of the $O(N)$ vector model, we refer to \\cite{Tetradis:1993ts,Berges:2000ew}. \n\nThe anomalous dimensions are defined via the scaling contribution to the wavefunction renormalization, i.e., $\\eta_{I} = - \\partial_{t} \\ln Z_{I}$, and take the following form in the three-field model:\n\\begin{eqnarray}\n\\eta_{1} &=& \\frac{16 v_{d}}{d} \\Xi^{-1} \\left\\{ \\kappa_{2} \\lambda_{1,1,0}^2 + \\kappa_{3} \\lambda_{1,0,1}^2 + \\kappa_{1} \\left( \\lambda_{2,0,0} + 2 \\kappa_{2} \\Delta_{1,2} + 2 \\kappa_{3} \\Delta_{1,3} + 4 \\kappa_{2} \\kappa_{3} \\Delta \\right)^2 \\right. \\nonumber\\\\ && \\left. +\\: 4 \\kappa_{2} \\kappa_{3} \\left( \\Delta_{2,3} \\left( \\lambda_{2,0,0} + \\kappa_{2} \\Delta_{1,2} + \\kappa_{3} \\Delta_{1,3} \\right) - \\Delta \\left(1 + \\kappa_{2} \\lambda_{0,2,0} + \\kappa_{3} \\lambda_{0,0,2} \\right) \\right) \\right\\} ~, \n\\label{Eq:Eta1} \\\\\n\\eta_{2} &=& \\frac{16 v_{d}}{d} \\Xi^{-1} \\left\\{ \\kappa_{1} \\lambda_{1,1,0}^{2} + \\kappa_{3} \\lambda_{0,1,1}^2 + \\kappa_{2} \\left( \\lambda_{0,2,0} + 2 \\kappa_{1} \\Delta_{1,2} + 2 \\kappa_{3} \\Delta_{2,3} + 4 \\kappa_{1} \\kappa_{3} \\Delta \\right)^2 \\right. \\nonumber\\\\ \n&& \\left. +\\: 4 \\kappa_{1} \\kappa_{3} \\left( \\Delta_{1,3} \\left( \\lambda_{0,2,0} + \\kappa_{1} \\Delta_{1,2} + \\kappa_{3} \\Delta_{2,3} \\right) - \\Delta \\left(1 + \\lambda_{2,0,0} \\kappa_{1} + \\lambda_{0,0,2} \\kappa_{3} \\right) \\right) \\right\\} ~, \n\\label{Eq:Eta2} \\\\\n\\eta_{3} &=& \\frac{16 v_{d}}{d} \\Xi^{-1} \\left\\{ \\kappa_{1} \\lambda_{1,0,1}^2 + \\kappa_{2} \\lambda_{0,1,1}^2 + \\kappa_{3} \\left( \\lambda_{0,0,2} + 2 \\kappa_{1} \\Delta_{1,3} + 2 \\kappa_{2} \\Delta_{2,3} + 4 \\kappa_{1} \\kappa_{2} \\Delta \\right)^2 \\right. \\nonumber\\\\ \n&& \\left. +\\: 4 \\kappa_{1} \\kappa_{2} \\left( \\Delta_{1,2} \\left( \\lambda_{0,0,2} + \\kappa_{1} \\Delta_{1,3} + \\kappa_{2} \\Delta_{2,3} \\right) - \\Delta \\left(1 + \\lambda_{2,0,0} \\kappa_{1} + \\lambda_{0,2,0} \\kappa_{2} \\right) \\right) \\right\\} ~,\n\\label{Eq:Eta3}\n\\end{eqnarray} \nwhere the prefactor in the above expressions is defined as:\n\\begin{eqnarray}\n\\Xi &=& \\Big(1 + 2 \\sum_{I} \\lambda_{I} \\kappa_{I} + 4 \\sum_{I < J} \\kappa_{I} \\kappa_{J} \\Delta_{I,J} + 8 \\kappa_{1} \\kappa_{2} \\kappa_{3} \\Delta \\Big)^2 ~.\n\\end{eqnarray}\nNote, that Eqs.\\ \\eqref{Eq:Eta1} -- \\eqref{Eq:Eta3} reduce to the two-field results in the limit $\\delta_{1,3}^{2} = \\delta_{2,3}^{2} = 0$ (cf.\\ Appendix \\ref{Sec:Scale dependent effective potential})\n\\begin{eqnarray}\n\\hspace{-20pt} \\eta_{1} &=& \\frac{16 v_{d}}{d} \\frac{ \\kappa_{2} \\lambda_{1,1}^2 + \\kappa_{1} \\left(\\lambda_{2,0} + 2 \\kappa_{2} \\Delta_{1,2} \\right)^2}{\\left(1 + 2 \\kappa_{1} \\lambda_{2,0} + 2 \\kappa_{2} \\lambda_{0,2} + 4 \\kappa_{1} \\kappa_{2} \\Delta_{1,2} \\right)^2} ~, \\\\\n\\hspace{-20pt} \\eta_{2} &=& \\frac{16 v_{d}}{d} \\frac{ \\kappa_{1} \\lambda_{1,1}^2 + \\kappa_{2} \\left(\\lambda_{0,2} + 2 \\kappa_{1} \\Delta_{1,2} \\right)^2}{\\left(1 + 2 \\kappa_{1} \\lambda_{2,0} + 2 \\kappa_{2} \\lambda_{0,2} + 4 \\kappa_{1} \\kappa_{2} \\Delta_{1,2} \\right)^2} ~,\n\\end{eqnarray}\nwhere we have written $\\lambda_{1,1} \\equiv \\lambda_{1,1,0}$, etc., as the $3$-sector effectively decouples in this case. From here, it is easy to identify the decoupling and symmetry enhancement scenarios for the spectrum of anomalous dimensions, by considering the proper limits of the couplings between different sectors. These results for the two-field case are equivalent to the anomalous dimensions given in Refs.\\ \\cite{Friederich:2010hr,Wetzel:2013}.\n\\twocolumngrid\n\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Approach}\n\\label{sec:approach}\nDetermining whether the bottleneck router in a network path is employing a drop-tail or AQM queuing scheme is fundamentally a classification problem, which has been rigorously studied in Machine Learning literature. To this end, we employ a data-driven approach\nthat entails generating random network configurations of varying complexity, simulating the generated scenarios using Mininet \\cite{mininet}, and training a classification algorithm on the obtained data. Our approach contains three main components: (i) randomized network topology generation and simulation, (ii) feature engineering to encode relevant information, and (iii) training of the classification algorithm. \n\n\\begin{figure}[!htb]\n \\centering\n \\includegraphics[width=\\linewidth]{figures\/topology}\n \\centering\n \\includegraphics[width=\\linewidth]{figures\/legend}\n\\caption{A random network topology generated by our framework. Edge weights denote link capacities.}\n\\label{fig:topology}\n\\end{figure}\n\n\\subsection{Random Topology Generation}\nProperly training a machine learning algorithm to simultaneously achieve high prediction accuracy and generalization to different kinds of network configurations requires a large and diverse set of training data. Hence, we opted to leverage randomization in order to generate a wide array of sufficiently diverse network topologies, which we subsequently simulated using Mininet. In particular, we developed a framework for constructing varied network configurations. \n\nAn example of a random network topology $G = (V, E)$ generated by our framework is depicted in Fig.~\\ref{fig:topology} and is constructed as follows. We initialize a \\textit{path graph}, i.e., a graph defined by a path from the source node $s$ to the sink node $g$, $\\pi = (s, v_1, \\ldots, v_n, g)$, consisting of $n \\sim \\mathcal{U}(3,5)$ \\footnote{$\\mathrm{U}(a,b)$ denotes a continuous uniform distribution, whereas $\\mathcal{U}(a,b)$ denotes a discrete uniform distribution.} switches, i.e., routers. For each switch along the path $v \\in \\{v_1,\\ldots,v_n\\}$, a random Erd\\\"{o}s-R\\'{e}nyi graph $G_\\text{rand}(n_p, p)$ \\cite{erdos} characterized by the uniformly drawn number of nodes $n_p \\sim \\mathcal{U}(1,5)$ and connectivity probability $p \\sim \\mathrm{U}(0,1)$ is generated such that each node is a switch or a host with equal probability; the largest connected component $G_\\text{conn} \\subseteq G_\\text{rand}$ is identified and connected to switch $v$ by an edge (i.e., network link). Additional $h \\sim \\mathcal{U}(1,5)$ hosts are generated and connected to vertex $v$.\n\nRandomly chosen values for link delays and capacities are generated to diversify the network configuration. For each link $e \\in E$, we set a random link delay $d \\in \\mathcal{U}(10,100)$ and link capacity $c \\in \\mathcal{U}(10,1000)$. To increase network complexity and simulate scenarios with bottlenecks caused by competing flows, we add auxiliary flows, i.e., paths, from randomly chosen host nodes to the sink node. The randomly generated link capacities are adjusted accordingly so that there exists a single bottleneck router along the path $\\pi$ from source to sink, which is subsequently labeled as either Tail Drop or AQM (denoting the queuing scheme to be simulated).\\footnote{The ranges used for the uniform distributions can be easily altered to generate random networks of desired complexity.}\n\n\\subsection{Feature Engineering}\n\n\\begin{figure}[t!]\n\\centering\n\\subfigure[Tail Drop]{%\n \\includegraphics[width=\\linewidth]{figures\/Basic-2.pdf}%\n\\label{fig:tail-drop}}\n\\subfigure[PIE]{%\n \\includegraphics[width=\\linewidth]{figures\/PIE-2.pdf}%\n\\label{fig:pie}}\n\\caption{RTT data obtained from two distinct Mininet simulations of the same network topology with the exception of the queuing scheme used by the bottleneck router: (a) Tail Drop and (b) PIE (AQM).}\n\\label{fig:data}\n\\end{figure}\nThe randomly generated network topologies are simulated using Mininet to obtain training data in the form of RTT and CWND information over time, as further detailed in Sect.~\\ref{subsect:training}. Figs.~\\ref{fig:tail-drop} and \\ref{fig:pie} depict the RTT data gathered from a 20 second Mininet simulation of the topology shown in Fig.~\\ref{fig:topology}. In order to construct a vector of distinguishing features, we examined a multitude of diversely-ranged RTT and CWND figures, similar to the figures shown above. Our observations indicated that high amount of noise was present in the data, therefore, we used the Exponentially Weighted Moving Average (EWMA) filter with parameter $\\alpha=0.3$ to smoothen the resulting RTT and CWND data.\n\nOur observations reaffirmed the inherent differences in the methodology of Tail Drop and AQM schemes: Tail Drop does not drop packets until the queue is full, which leads to a pronounced sawtooth pattern in the RTT graph (Fig.~\\ref{fig:tail-drop}), i.e., lengthy periods of linear increase in the RTT plot followed by a sharp drop. PIE, on the other hand, preemptively (and probabilistically) drops packets, which results in relatively short bursts of increasing and decreasing RTT values. \n\nWe combined our insights from experimental observations and the inherent differences in Tail Drop and AQM schemes to construct a vector of distinguishing features. Our feature vector was 72 dimensional (36 features each for RTT and CWND) that were computed for each simulated time series. Examples of particularly distinguishing features that constituted our feature vector include: statistics regarding the first and second order gradient (e.g., mean, variance, maximum), the number of local minima \\& maxima, the $L_2$-norm squared total variation, the maximum sum subsequence, and the mean and variances of the local maxima. Fig.~\\ref{fig:features} shows a complete list of the features that were used, ranked by descending importance.\n\n\\subsection{Training}\n\\label{subsect:training}\nTo train our classifier, we generated 1,100 different network topologies of varying complexity and heterogeneity so that balanced, labeled data relevant to both Tail Drop and AQM schemes were obtained. For each random network topology, we instantiated two consecutive, distinct Mininet simulations by varying the queue management technique on the bottleneck between Tail Drop and PIE. We simulated a long running TCP flow on each network from our known source to the sink and auxiliary flows from other randomly generated hosts to their respective sinks for 20 seconds. From the simulation, we gathered the resulting congestion window and RTT over time using \\verb|tcpprobe| and \\verb|ping| respectively. This data, as exemplified by the RTT plot in Fig.~\\ref{fig:data}, were then used as labeled training data in our classifier.\n\\section{Conclusion}\n\\label{sec:conclusion}\nWe presented a data-driven approach that leveraged a Neural Network for reliably determining whether a bottleneck router along a given path employs a Tail Drop or AQM scheme. We developed a framework for generating random network topologies and used Mininet to simulate a wide range of scenarios and obtain diverse training data. We identified distinguishing features by analyzing the intrinsic differences between AQM and Tail Drop and quantified the respective importance of each feature. Our feature importance analysis indicated that the most salient features were statistics pertaining to the gradient and RTT time-series data.\n\nOur algorithm attained an accuracy of 97\\% when evaluated against test data and generalized and performed well against data from highly complex topologies that it had not been exposed to before. We conjecture that our algorithm has potential to generalize to and provide reliable and accurate predictions in complex, real-world networks. Future work includes further training and evaluation on even more complex topologies and real-world data. We also plan to extend our method for classification of different AQM interfaces such as RED and PIE, in addition to the Tail Drop scheme.\n\\section{Contributions}\n\\emph{Cenk Baykal} developed the algorithm used to generate the random network configurations and helped develop the features that were used to classify the queueing scheme. \n\\\\\n\n\\emph{Wilko Schwarting} worked on the classification algorithm and evaluated state-of-the-art classification methods and conducted hyper-parameter optimization.\n\\\\\n\n\\emph{Alex Wallar} developed the Mininet codebase that was used to simulate the randomly generated network configurations and obtain data.\n\n\\section{Evaluation}\n\\label{sec:evaluation}\n\nUsing the the data obtained from our simulations, we trained and benchmarked state-of-the-art classification algorithms such as Support Vector Machines (SVMs) \\cite{svm}, Neural Networks (NNs) \\cite{hornik1989multilayer}, Logistic Regression \\cite{logistic}, and Random Forests, among others. We conducted k-fold cross validation (with $k = 10$) for each classifier on multiple data sets and found Neural Networks to be consistently outperforming all other tested classification techniques.\n\nHaving converged on using NNs as our classification algorithm, we employed a randomized search to optimize hyper-parameters by drawing from a previously specified parameter space distribution.\nThe parameter space included 1) NN architecture, i.e., the number of layers (1-4) and neurons (2-20) per layer,\n2) $L_2$-Regularization parameter $\\alpha$ penalizing large connection weights, and\n3) solvers, including Stochastic Gradient Descent (SGD), ADAM, and Limited-memory Broyden Fletcher Goldfarb Shanno (LBFGS) algorithms.\nThe randomized search evaluates each parameter tuple with a score consisting of the average accuracy over 100 stratified cross-validations with $90\\%$ training data and $10\\%$ test data, to avoid biased parameter selection from overfitting. After 10,000 evaluations, the optimal hyper-parameters were found to be a NN topology with 1 hidden layer consisting of 14 neurons, a regularization of $\\alpha = 0.5e-10$, and the LBFGS solver.\n\n\\subsection{Training Results}\nFrom the 2,200 labeled data points that were generated, 200 were left out of the training to be used later for testing. After training the classifier and optimizing its hyper-parameters, we tested how well the classification generalized by evaluating its performance on the left-out data set. Fig.~\\ref{fig:cm-left-out} shows a confusion matrix summarizing the results of this classification test. On a per-category basis, we achieve 94\\% and 99\\% prediction accuracy for correctly classifying the bottleneck router as Tail Drop or PIE respectively. Overall, we were able to accurately classify whether the bottleneck router was utilizing a Tail Drop or PIE as its queuing scheme 97\\% of the time. This can be seen as an improvement over current state-of-the-art AQM detection algorithms, such as the recently proposed method by Bideh et al. \\cite{tada}, which achieved an overall classification accuracy of 73\\% on a serial bottleneck within a relatively simple network topology. In addition to achieving a higher classification accuracy than that of previous work, we also note that our algorithm was evaluated on significantly more complex and diverse network configurations, whereas the algorithm presented in \\cite{tada} was evaluated against a single topology consisting of solely 3 hosts and 3 routers in total.\n\nTo thoroughly assess the dexterity of our algorithm in generating reliable predictions, we evaluated the classification accuracy of our algorithm on a data-set obtained from simulations of highly complex topologies with significantly more hosts, routers, and flows than the topologies that were used to train our algorithm. We emphasize the fact that our classifier was \\emph{not} trained on any data from these more complicated scenarios.\n\nDespite the challenges of this classification task, our classifier produced accurate labels for Tail Drop and PIE 80\\% and 65\\% of the time respectively, for an average classification accuracy of 73\\% overall. The confusion matrix for these tests is shown in Fig.~\\ref{fig:cm-complex}. This evaluation is evidence for our classifier's ability in generalizing well even to data obtained from highly complex network configurations that it was not explicitly trained to handle beforehand. Our favorable results demonstrate our algorithm's potential to be applied to real-world settings and achieve high prediction accuracy even against highly complex network configurations.\n\n\\begin{figure}[t!]\n\\centering\n\\subfigure[Left Out Data]{%\n\\includegraphics[width=0.49\\columnwidth]{figures\/cm_left_out}%\n\\label{fig:cm-left-out}}\n\\subfigure[Complex Topologies]{%\n\\includegraphics[width=0.49\\columnwidth]{figures\/cm_complex}%\n\\label{fig:cm-complex}}\n\\caption{Confusion matrices for the two testing scenarios, (a) data left out of the training with the same topology complexity and (b) data generated using more complex topologies than used for training}\n\\label{fig:cm}\n\\end{figure}\n\n\\subsection{Feature Importance}\nAs introduced in Sec.~\\ref{sec:approach}, our feature vector for a single data point was 72 dimensional, with 36 features each for RTT and CWND. The 72 distinguishing features for each data point mentioned in Sec.~\\ref{sec:approach} were motivated by our knowledge of the profound differences between Tail Drop and AQM schemes, as we covered in class \\cite{class}. Fig.~\\ref{fig:features} depicts a quantification of feature importance for each of the 72 features. By inspecting the highest ranking features, we obtain the insight that features pertaining to gradient-like statistics were the most distinguishing features and that data about the RTT over time were more helpful than the CWND data.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=\\columnwidth]{figures\/feature_importances}\n\\caption{Plot showing each feature's importance for classification. Note the high importance scores of features pertaining to RTT information and gradients' statistics.}\n\\label{fig:features}\n\\end{figure}\n\n\n\\section{Problem Definition}\n\\label{sec:problem}\nActive Queue Management (AQM) schemes, such as PIE \\cite{pie} and RED \\cite{red}, have been particularly effective in combating bufferbloat and attenuating network congestion. Despite their widespread success and improved performance over traditional Tail Drop queuing schemes, \nthe extent of AQM deployment in contemporary networks is not known \\cite{tada}. Detecting the presence of AQMs in a network has potential to not only provide insight into a network's internal characteristics (i.e., network tomography), but also facilitate network optimization \\footnote{e.g., in the case that a transport protocol is known to perform poorly over an AQM or certain subset of AQMs \\cite{tada}.}.\n\nIn this paper, we address the problem of determining whether a bottleneck router on a given network path is using an AQM or a drop-tail scheme. We assume that we are given a source-to-sink path of interest -along which a bottleneck router exists- and data regarding the Round-Trip Times (RTT) and Congestion Window (CWND) sizes with respect to this flow. We develop a reliable classification algorithm that solely uses RTT and CWND information pertaining to a single flow to classify the queuing scheme, Tail Drop or AQM, used by the bottleneck router. We evaluate our method and present results that demonstrate our algorithm's highly accurate classification ability across a wide array of complex network topologies and configurations.","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\t\\label{S:11}\n\t\n\tChemometric classification methods are often used to discriminate high-dimensional chemical signatures of unknown samples to determine their most likely class label \\cite{patternrec}. These methods have proven to be valuable for the fields of archeaometry and forensics where the origin, or provenance, of a manufactured item is often of interest\\cite{patternrec, obsidian, crypto, shipwreck, butter, heroin} because the chemical data, especially those obtained from neutron activation analysis (NAA) \\cite{sayre,NAArev}, tend to be highly multivariate. The high dimensionality of data poses a challenge for understanding sample relationships because it cannot be easily visualized or interpreted directly. While, these methods can succeed at classifying samples, they do so by including information from the class labels into the model, and they provide little insight about trends or patterns found in the data with the absence of labels. Information pertaining to which samples are most similar to one another between classes, or which are different from the rest of the samples within a given source class, without the influence of label discrimination is not available from classification methods. \n\t\n\tExploratory methods are often used to reveal trends and other patterns hidden in data without the use of class labels \\cite{patternrec}. Exploratory data analysis (EDA) is a form of data analysis which encompasses a number of visual exploration methods. Success in an EDA study depends on the creativity of the analyst as much as on the technique. Although there is no strict definition of EDA, it has been stated that, ``Exploratory data analysis isolates patterns and features of the data and reveals these forcefully to the analyst\" \\cite{tukeybook}, and also that ``Exploratory data analysis' is an attitude, a state of flexibility, a willingness to look for those things that we believe are not there, as well as those we believe to be there\"\\cite{tukeybooktwo}. The coupling of EDA methods with preprocessing and\/or feature selection can aid an analyst in the discovery of patterns, or the lack thereof, in data. The information gained from EDA may be used with feature selection, preprocessing and modeling methods to iteratively improve a data analysis pipeline.\n\t\n\tThe two predominant types of EDA focus on dimension reduction and clustering. These methods have been extensively used for the EDA of archeometric data \\cite{crypto, shipwreck, butter, cluster} because high dimensional data are not readily visualized. Dimension reduction techniques such as principal component analysis (PCA), discriminant analysis, or exploratory projection pursuit are used to change the basis of the data to one based on lower-dimensional projections \\cite{patternrec}. Unfortunately, these methods do not offer a guarantee that the resulting 2-D or 3-D projections will offer meaningful information about class relationships or sample-class relationships, due to the presence of class overlap or class heterogeneity\\cite{patternrec}. Unsupervised classification as implemented in clustering algorithms suffers from the opposite problem clustering methods readily create groupings of high-dimensional data but tend to offer little information about the cluster assignments themselves. Clustering methods have been known to incorrectly associate archeological samples depending on the method used and on the chemistry of the samples \\cite{cluster}. Information about sample or class relationships may be inferred by employing many techniques, but the resulting information is often hidden, and typically key relationships in the data cannot be seen by an analyst \\cite{cluster}.\n\t\n\tThis work reports a new approach, which we call the uncharted forest, to the visualization and measuring of relationships within and between classes of data. This approach has elements of a clustering algorithm in that it groups similar samples with one another, but is also similar to dimension reduction methods in that it outputs a single heat map which can be interpreted to reveal information about the samples. Uncharted forest analysis uses a partitioning method that is related to the sample partitioning approach used in decision trees but, it does not use class labels like most tree methods do \\cite{CARTBook}. Instead, the uncharted forest analysis explores how samples relate to one another under the context of univariate variance partitions. Although the method is unsupervised, we show that when the results are overlaid with external class labels, the method can be used to investigate sample relationships as they pertain to class labels. We demonstrate that this technique can be used as a tool for exploratory data analysis to visualize class or cluster associations, sample-sample associations, class heterogeneity, and uninformative classes. The utility of uncharted forest analysis is demonstrated on two classification datasets and two provenance datasets. Additionally, two empirical clustering metrics are compared with two of the metrics obtained by uncharted forest analysis on another provenance dataset.\n\t\n\t\n\t\\section{Theory}\n\tTo motivate the use of an unsupervised tree ensemble, a brief review of supervised trees and bagged tree ensembles as used in classification is provided here. A complete review of these methods can be found elsewhere \\cite{CARTBook,BreimanRF}. \n\t\n\t\\subsection{Supervised Classification Trees}\n\tBefore describing the mechanisms by which supervised decision trees are used to partition data, a survey of the vocabulary common to these methods is presented. Classification trees are a type of supervised classifier, which means that the assignment of new data to a class label requires that each sample the tree was trained on also has a label. The aim of developing a classification tree is the establishment of a set of sequential rules that can be applied to label a sample based on its features. Supervised decision trees are collections of many binary decisions, where each decision is made at one of three locations: roots, branches, or terminal nodes\/leaves. These three locations are displayed in their hierarchical ordering in Figure 1. A root is the first decision made in the tree. Branch nodes indicate later, non-terminal decisions. Terminal nodes indicate where a branch has been terminated and where final decisions that assign class labels are made. \n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=0.40\\linewidth]{Figure1}\n\t\t\\caption{A block diagram showing the relationships between roots, branches and terminal nodes via arrows.}\n\t\\end{figure}\n\t\n\tEvery sample from the training set is partitioned at a branch or node based on whether a selected variable for the sample has a value that is greater than or less than a specified value. The combination of a variable and value that partitions a set of samples is often referred to as a decision boundary. Decision boundaries are obtained by exhaustively searching each variable and finding the threshold value which affords the highest gain \\cite{CARTBook}. Here, gain is a metric-dependent measure of how well a decision separates the available samples according to their class labels. Metrics that can be used to minimize class label impurities at terminal nodes, such as the Gini impurity, informational entropy, and classification accuracy are commonly used to train supervised classification trees by finding a set of decision boundaries \\cite{CARTBook}. \n\t\n\tFor example, if the samples available at a branch node are better separated into their respective classes by application of some decision, two new branches are made at that node. If not, the partitioning terminates, the node becomes a terminal node, and the decision tree ceases growth using those samples. In this way, each observation is sorted from the root of a tree through branches and eventually to terminal nodes, based on whether creating a new decision boundary better separates samples by known class labels\\cite{CARTBook}. \t\n\t\n\tOne advantage to the use of decision trees is that the classifier that results can be easily understood by examining the decision boundaries that partition the data and the relation of these boundaries to class labels. However, the predictive performance of classification trees on new samples is generally poor relative to other models because the partitions are sensitive to noise in the variables \\cite{CARTez}. Tree-based classifiers also tend to overfit the data, and some sort of branch removal, or pruning step, is usually employed to reduce the complexity of the tree \\cite{CARTBook}. \n\t\n\t\\subsection{Random Forest Classification}\n\tThe random forest classifer is a multiclass classification technique that utilizes the results from an ensemble, or collection of supervised decision trees \\cite{BreimanRF} to assign class labels. In this technique, a user-specified number of supervised decision trees is created, each overfit to a selection of bootstrapped samples and random selections of variables. Random forests make use of the complexity of many, biased decision trees to make robust classification assignments by using the average class designation assigned by every tree to predict overall class membership for new data. This approach to ensemble model building is referred to as bootstrap aggregating, or bagging \\cite{bagging}. The average of many, complex trees tends to perform well in practice \\cite{BreimanRF}.\n\t\n\tRandom forest classification has some distinct advantages relative to many other classifiers. It is robust to outliers because sample selection at branches is bootstrapped, and the effect of a few outlying samples does not affect its cost function significantly \\cite{BreimanRF}. Random forest models are also robust to noise in variables because many classifiers, each with randomly selected variables, are used to decide on class memberships. Another notable advantage to random forest classification is that there is no requirement for linearity in class boundaries. Random forest models, unlike methods such as linear discriminant analysis or support vector machines, are not based on the hypothesis that classes are linearly separable \\cite{linearmethods,kernel}. Random forest classification also requires relatively few hyperparameters and works without the need for significant tuning of the classifier on many kinds of data \\cite{CARTez}. \n\t\n\t\\subsection{Uncharted Forest Tree}\n\tThe concept of unsupervised decision tree modeling is relatively new to chemometric applications. Only a few approaches using unsupervised trees have been reported. An approach reported by Khudanpur, et al. is based on an algorithm similar to the one reported here, but their approach uses the Kullback-Leibler distance to find sets of unsupervised partitions that are optimal with respect to an information theoretic measure\n\t\\cite{InformationApp}. Other methods for developing unsupervised decision trees are based on the use of an assumed class label for each observation; the trees are trained in a manner that is similar to that used for training a random forest classifier \\cite{UnsupRF, CLTrees}. \n\t\n\tOur approach to unsupervised decision trees, the uncharted forest tree algorithm, focuses on intuitive concepts from statistics or machine learning rather than on information theory, to allow for easy interpretation. The trees can be created without the need for class labels because the gain function that is optimized in the construction of each tree relies only on information in the data matrix, not on the labels. This algorithm does not utilize the usual supervised metrics for optimization such as the Gini importance or entropy \\cite{BreimanRF}. Instead, the tree hierarchies used in uncharted forest are constructed from decision boundaries that reduce measures of spread in a given variable. The reduction of spread is common in pattern recognition and chemometrics; it is the mechanism underlying techniques such as K-means clustering, and principal component analysis for dimension reduction \\cite{patternrec}. Metrics based on changes in the variance, median absolute deviation (MAD), or mean absolute deviation (AD) can also be used for creating the decision boundaries used in an uncharted forest tree. If variance is used, however, univariate Gaussian probabilities may be calculated from the samples on either side of a decision boundary. We prefer the variance metric because it is related to several established figures of merit for clustering, as we describe later. The gain metric based on changes in variance that we use is\n\t\n\n\t\\begin{equation}\n\tgain = 1 - \\frac{Var(Branch_1) - Var(Branch_2)} {Var(Parent Node)}\n\t\\end{equation}\n\t\n\t\\noindent where the sample data in either partition are denoted as $Branch_1$ and $Branch_2$, and the sample data prior to introducing the partition is contained in the previous - or parent - node, which has been denoted as, $Parent Node$. The gain in the unsupervised case can then be seen as the loss of variance ($Var$) in the data resulting from the construction of a given decision boundary. The gain in the unsupervised case can then be seen as the loss of variance in the data resulting from the construction of a given decision boundary, relative to that in the previous - or parent - node. Similar gain equations can be written for measures of spread based on AD and MAD. These are not considered in this work because they do not have direct comparisons to clustering metrics. \n\t\n\tThe spread in data can almost always be reduced by partitioning data around an available decision boundary. Thus, uncharted forest trees require a user-defined parameter, the tree depth, to limit the number of possible partitioning decisions made. Tree depth can be defined as the maximum number of branches that are counted from the root node to a given terminal node. For exploratory analysis of classification or cluster data, a reasonable depth may be obtained from the rounded base 2 logarithm of either the number of known classes or the number of clusters selected. The number of classes\/clusters available is often known information; for example, in most provenance studies the number of classes is decided upon by the number of sampling sites used. In our implementation the minimum node size, or when branch growth is ceased after the number of samples becomes fewer than a given number, has been set to five samples because five samples allows for a calculation of variance that is not undersized but still accommodates a small cluster\/class size. Pseudocode that depicts the recursive algorithm for constructing an uncharted decision tree is shown in Algorithm 1.\n\t\n\t\n\n\t\n\t\\begin{algorithm}\n\t\t\\caption{Uncharted Forest Decision Tree}\\label{unchartedtree}\n\t\t\\begin{algorithmic}[1]\n\t\t\t\\Procedure{ }{}\n\t\t\t\\State for each variable in variables\n\t\t\t\\State \\ \\ \\ \\ Calculate gain. (Eqn. 1)\n\t\t\t\\State \\ \\ \\ \\ Store highest gain.\n\t\t\t\\State end for\n\t\t\t\\State Partition samples at largest gain.\n\t\t\t\\If {branch depth $\\geq$ number of branches} \n\t\t\t\\State Stop growth.\n\t\t\t\\Else\n\t\t\t\\State For each partition \\textbf{goto} 2\n\t\t\t\\EndIf\n\t\t\t\\State \\textbf{end if}\n\t\t\t\\EndProcedure\n\t\t\\end{algorithmic}\n\t\\end{algorithm}\n\t\n\tThe uncharted forest decision tree partitions a set of data by finding decision boundaries that reduce the spread in the data. It functions as a kind of clustering, where the clusters result from divisive partitioning of spread in the data. Unfortunately, unsupervised trees of the form described above often have marginal utility in practice. These trees can be used for clustering but, like a single supervised tree, the individual uncharted forest decision trees are sensitive to noise effects in the data. The variables in a set of data have many contributions to their spreads, and as a consequence, the partitions that result from a single unsupervised tree tend to be uninformative.\n\t\n\t\\subsection{Uncharted Forest}\n\tOne solution to the uninformative partitions created by a single uncharted forest decision tree is to create an ensemble of unsupervised trees, an ``uncharted forest\". The approach is similar to that taken in the random forest classifier. For each tree in this uncharted forest, the variables used are randomly selected, so that potentially relevant information within an individual variable is not obscured by other contributions to its spread or by another variable with much larger gain in the reduction of spread. The algorithm for the uncharted forest method follows that of the random forest algorithm \\cite{BreimanRF}, except that the supervised classification and regression tree models are replaced with the uncharted forest decision tree and the samples are not bootstrapped.\n\t\n\tBecause construction of the trees is unsupervised, the output of the uncharted ensemble algorithm differs from that of the random forest. An uncharted forest returns a square matrix whose dimensionality is that of the number of observations, so that each row or column vector stores information about a single observation and how it relates to the other samples. The matrix is populated in the following manner: when samples are partitioned into a terminal node together, the row and column representative of each of these samples each gain a +1 count to account for their associations with one another. Similarly the row and column of any samples that were not present at a terminal node with those samples each receive a -1 count. The associations formed by random chance become less pronounced by including the -1 penalty on observations that were not grouped with one another at their respective terminal nodes. After row-wise normalization, the matrix possesses diagonal symmetry. In the case of variance-only partitions, a Gaussian probability for node membership may be used instead of $\\pm$ 1. \n\t\n\n\t\n\t\\subsection{Figures of Merit}\n\tWe refer to the matrix generated by the tree ensemble as a probabilistic affinity matrix, because of its resemblance to the distance-based affinity matrices utilized by clustering methods \\cite{spectral}. Each element in the matrix represents the row-wise, likelihood that a given row (observation) is associated with a given column (another or the same observation) at a terminal node in an uncharted forest tree. In order to maintain the above interpretation, a ramp function is applied to every matrix element (p), max(0, p), to remove any negative entries. In this way, the diagonal elements have a sampled probability of association of 1.0, because there is a 100\\% likelihood that a given sample will be associated with itself at each terminal node where it was assigned. \n\t\n\tWe introduce two schemes that allow for the probability values in the matrix to be quantified for exploratory analysis. Two figures of merit can be calculated to measure how often samples between- or within- clusters\/classes are partitioned together. The submatrix of the probability affinity matrix (P) that contains the associations between two clusters\/classes i and j, can be quantified by what we define as the interaction quotient (IQ),\n\t\n\n\t\\begin{equation}\n\tIQ_{i,j} = \\frac{\\sum_{c=j_{min}}^{j_{max}} \\sum_{r=i_{min}}^{i_{max}} P_{r,c}} {(j_{max} - j_{min}))(i_{max} - i_{min}) }\\end{equation}\n\t\n\t\\noindent where the min and max subscripts denote the contigious row\/column beginning and end for cluster\/class indices within the matrix. The IQ is simply the sum of all probabilities inside of the submatrix that is defined by the rows shared from cluster\/class $i$, and the columns shared by cluster\/classes $j$ normalized by the maximum possible probabilities for association within that submatrix. Similarly, we define the self association quotient (SAQ) as a measure of within- class or cluster sample associations. The SAQ metric is calculated from the probabilistic affinity matrix for the special case where $i = j$. For purposes of visualization, the matrix blocks from which these metrics are calculated are displayed below,\n\t\n\t\\[\n\t\\begin{bmatrix}\n\tSAQ_{1} & IQ_{1,2} & IQ_{1,3} & \\dots & IQ_{1,j} \\\\\n\tIQ_{2,1} & SAQ_{2} & IQ_{2,3} & \\dots & IQ_{2,j} \\\\\n\t\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n\tIQ_{i,1} & IQ_{i,2} & IQ_{i,3} & \\dots & SAQ_{i}\n\t\\end{bmatrix}\n\t\\]\n\t\n\tThe total interaction quotient (TIQ) which describes the extent of the total interactions between the samples of all class\/cluster assignments can be expressed by the sum of probabilities within each IQ block divided by the number of matrix entries in those blocks(n),\n\t\n\n\t{\\centering\n\t\t\\begin{equation} TIQ = \\sum_{i} \\sum_{j} IQ_{i,j} \/ n \\ \\ \\ \\ \\ provided \\ \\ i \\neq j \\end{equation} \\par\n\t}\n\t\n\tIn an analogous manner, the total self association quotient (TSAQ) can be used to describe how often the samples in all class\/clusters are partitioned with samples in their same respective class\/cluster by uncharted forest,\n\t\n\n\t\\begin{equation}TSAQ = \\sum_{i} SAQ_{i} \/ n \\end{equation}\n\t\n\tThe metrics SAQ and IQ provide a means to assess the likelihoods that samples are within the same terminal nodes after variance partitioning with reference to class\/cluster ownership. SAQ can provide a measure of class\/cluster heterogeneity by indicating how likely it is that the samples inside of a given label are partitioned together. The IQ provides a measure of how often data were associated to the samples that belong to a different class\/cluster label. The IQ measure can therefore be used as an aid to identify classes in the data that do not associate with samples from other classes of data, or classes which are similar to one another. These measures can also be made on a sample-wise basis to discover outliers which may either be between class\/cluster labels or are not associated with any class\/cluster label at all.\n\t\n\t\n\t\\section{Scope of the Uncharted Forest Method}\n\tThere is one restriction to the use of uncharted forest analysis that should be noted. The associations that a uncharted forest analysis can describe are often not meaningful without data ordered by class or cluster labels, despite that it is an unsupervised technique. The hypothesized membership of samples allow interpretations of the data as it pertains to class\/cluster labels. In some cases, data have serial correlations, such as in time series, or variables from spectroscopic instruments, and the probabilistic affinity matrix appears well-ordered. However, with uncorrelated, labelless data, the probabilistic affinity matrix most often appears similar to random noise. One interesting feature of the probabilistic affinity matrix is that it is invariant to the ordering of data. This is a useful attribute when the matrix could display seemingly random disjoint probabilities, because the order of the rows and columns of the matrix can be changed to permit a more meaningful interpretation, like a cluster for example, without running the algorithm again. The studies explored in this work address the case where class or cluster labels were available for provenance and classification data.\n\t\n\t\\section{Selected Examples}\n\tIn this work, five datasets were investigated by uncharted forest analysis. The first, the well-known iris dataset, is presented as an instructional example to demonstrate how the technique can be used to examine data with class labels, and how it compares to the widely used principal components analysis method. The well-known ARCH obsidian data is presented to compare the sample-sample associations obtained by uncharted forest analysis with two supervised classification methods. The Nan'ao One shipwreck and cryptocrystalline sillicate datasets are presented to compare the results from uncharted forest analysis to those of published works that used other data analysis methods on the same data. The fifth dataset, the butter data, is used to compare the new method to established variance-based partitioning methods and to show that the TIQ and TSAQ figures of merit are similar to well-established, variance-based clustering metrics.\n\t\n\t\\subsection{Iris Data EDA}\n\tThe iris dataset contains measurements of 150 iris flower samples. Both the widths and lengths of the sepals and petals of each flower were measured. Each sample belongs to either the virginica, versicolor, or setosa species \\cite{iris1, iris2}. The iris data is widely regarded as a standard dataset for introducing classification problems and algorithms because it features two elements common to classification problems: it possesses a linearly separable class (setosa) and two classes with overlapping boundaries (virginica, versicolor).\n\t\n\tWhen the uncharted forest algorithm was applied to the iris data using 100 trees, a depth of 4, and using the variance metric, the probabilistic affinity matrix indicated associations between several samples of virginica and versicolor. The row-wise total interaction quotients for the samples in the setosa class were $\\ll$ 5\\%; this suggested that the setosa class was well separated from the others. Four samples that were greater than 3$\\sigma$ from the mean row-wise IQ were found in both the virginica and versicolor classes. This result implies that, by making consecutive decision boundaries to reduce variance, the virginica and versicolor classes had several samples that appeared to share the same class label space. \n\t\n\tPrincipal components analysis was employed to reduce the dimensions of the iris data and to visually assess which samples uncharted forest analysis indicated as associated with other classes. When the iris data were projected onto the first two principal component axes, five samples which had significant row-wise IQ's were located between the virginica and versicolor classes, as can be seen in Figure 2. A conventional random forest model with 500 trees was then trained on the entire dataset. The random forest model misclassified 5 samples. Of those 5 samples, 3 were the samples with row-wise IQ values that exceeded 3$\\sigma$. The heat map from uncharted forest algorithm highlighted many of the samples that the random forest classifier had difficulty classifying due to class overlap.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=0.9\\linewidth]{Figure2}\n\t\t\\label{fig:Ng1} \n\t\t\n\t\t\\caption{A heat map of the probabilistic affinity matrix of the Iris data with class labels (top left). The row-wise IQ values for the corresponding heat-map with the 3$\\sigma$ vertical line (top right). Plot of the PCA scores for the iris data. Samples denoted with an X, are the samples with row-wise IQ values above the 3$\\sigma$ line (bottom).}\n\t\\end{figure}\n\t\n\t\\subsection{Obsidian Data Pattern Recognition}\n\tThis dataset contains X-ray fluorescence measurements made on 12 obsidian artifacts and 63 obsidian samples that were collected from four locations in Northern California. The four source locations were from quarries located near: Mount Konocti, Borax Lake, Glass Mountain (Napa County), and Anadel (Sonoma County). We used this dataset demonstrate that the sample-sample associations afforded by uncharted forest analysis could be used to establish the likely origin of the artifacts from measured quantities of the following trace elements: iron, titanium, barium, calcium, potassium, manganese, rubidium, strontium, yttrium, and zirconium \\cite{Obsidian}. This dataset has previously been demonstrated as a test-bed for unsupervised and supervised learning methods in chemometrics \\cite{kowalski}.\n\t\n\tThe unknown samples were assigned source labels based on an uncharted forest model made with 200 trees and a tree depth of 2, as well as a nearest neighbor classification (k = 1) after PCA projection onto the first two components of the unknown samples The assignments of the artifact samples to their predicted source classes can be seen in Figure 3. The PCA was created from standard normal variate-scaled obsidian source data that had the X-ray fluorescence measurements of iron, strontium, and manganese removed due to correlation with other variables. The uncharted forest analysis assignments were made from the row-wise IQ values of the rows in the probabalistic affinity matrix, which corresponded to the twelve unknown samples via a maximum voting scheme. The voting scheme was performed to remove analyst bias and to automate the assignment of samples to suspected classes, but the results were equivalent to our visual assessment. \n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=0.95\\linewidth]{Figure3}\n\t\t\n\t\t\\caption{The probability affinity matrix (left) and the PCA scores plot obtained from the first two principal components of the standard normal variate-scaled ARCH data (right). The unknown artifact samples are denoted with a `?', and the four source classes of the obsidian data are labeled as follows: Mount Konocti (1), Borax Lake (2), Glass Mountain (3), and Anadel (4). }\n\t\\end{figure}\n\t\n\tThe source assignments of the artifacts from the uncharted forest analysis with the voting scheme differed from those made from the nearest neighbor (1-NN) PCA classification model by only one sample. For a sense of comparison, an ordinary random forest classification model with 200 trees was also trained on the source data. Again, the assignments made by the maximum vote uncharted forest analysis differed by only one sample from those of the random forest model. However, the classification produced by the nearest neighbor classifier differed from the assignments made by the random forest model by two samples. None of the models assigned any of the artifact samples to the Anadel source class. This result is consistent with a classification analysis performed by a linear learning machine \\cite{kowalski}.\n\t\n\tSimilar assignments between unknown samples and assigned source classes were obtained by the uncharted forest method and the two supervised classification models. This finding suggested that the unsupervised partitioning method used in the uncharted forest method with a maximum voting scheme can offer sample-sample associations that are comparable to that found using supervised methods.\n\t\n\t\n\t\\subsection{Nan'ao One Shipwreck Data EDA}\n\tThis dataset was selected because it was relatively high in dimension, and because it possessed complex class labels and sample membership. Twelve porcelain samples were collected from a shipwreck and compared, by using NAA measurements of 27 elements, with samples from several nearby kilns. The data were scaled by applying a base ten logarithm prior to analysis as suggested by the authors \\cite{shipwreck}. The goal of the study was to determine which kilns were most likely used to create the blue and white porcelain artifacts that accounted for 95\\% of the samples found on the wreck \\cite{shipwreck}.\n\t\n\tAn uncharted forest analysis using 30 trees and a tree depth of 2 was used to investigate the source and artifact data, as shown in Figure 4. It was found that two of the artifact samples had high associations with source samples obtained from the Jingdezhen kiln, and no associations with other source samples. This result implied that those samples could be assigned to the Jingdezhen kiln. Zhu et al.'s PCA analysis agreed with the assignment of those two samples to the Jingdezhen kiln \\cite{shipwreck}. \n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=0.65\\linewidth]{Figure4}\n\t\t\n\t\t\\caption{Probability affinity matrix of the three source classes and the shipwreck samples. }\n\t\\end{figure}\n\t\n\tThe probabilistic affinity matrix also indicated that samples from the Jingdezhen kiln may be composed of two subclasses. Our PCA analysis did not readily show that the Jingdezhen kiln could be represented as two subclasses. It was found that the scores of the subclassess that were identified by uncharted forest analysis were linearly separable after projection from the fourth and fifth principal components. Manual assessment of the 729 pairs plots showed that the two subclasses identified by uncharted forest were linearly separable in the Lutetium measurements. The base-10 logarithmically scaled Lutetium variable was was used in 16\\% of all decision boundaries made from the random selections of variables and was the third most used variable for constructing decision boundaries in all of the uncharted forest trees. In every instance the decision boundaries were created at -0.569, and this boundary was never imposed at a root node. A plot of normalized kernel density estimates of Lutetium, displayed in Figure 5, showed that the samples from these two possible subclasses were well-separated by the Lutetium decision boundary. This information was readily accessible by uncharted forest analysis, whereas, by the other EDA methods we did not know that this division of samples in the Jingdezhen class may be of interest, or that it existed. The reason why the Jingdezhen class has two well-separated distributions of Lu measurements is not known, but this finding could be of value for future investigations or to provide information about appropriate sample selection that accounts for the sources of variation, as is necessary in the training of classification models. \n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=0.75\\linewidth]{Figure5}\n\t\t\n\t\t\\caption{ Kernel density estimates for the base-10 logarithmically scaled Lutetium concentrations in the speculated Jingdezhen 1 and 2 subclasses as well as in the other samples. The kernel density estimates were created using a bandwidth of 0.04 and were normalized relative to the kernel estimates of the entirety of the data. The decision boundary created by the uncharted forest algorithm is shown as a vertical line. }\n\t\\end{figure}\n\t\n\tUncharted forest analysis revealed that the remaining artifact samples were found to be most associated with those from the Zhangzou province. However, there was notable interaction between samples from the Zhangzou province and three source samples from the Dapu kiln. Similarly, Zhu et al. found that the remaining samples appeared to be from the Zhangzou province, and that several samples were outside of the 90\\% confidence bounds of their principal component analysis \\cite{shipwreck}. \n\t\n\tTo make the partitions of uncharted forest more specific to the samples believed to be in the Zhangzou province, both the source and artifact samples attributed to the Jingdezhen kiln were removed. The source samples collected from the Dapu kiln were also removed to focus the partitioning only on the kilns in the Zhangzou province. The goal of this analysis was to assess the samples obtained from specific kilns within the Zhangzou province as they related to the shipwreck, rather than assuming the Zhangzou province was a homogeneous collection of source kilns. The probabilistic affinity matrix, shown in Figure 6, produced by uncharted forest analysis with 75 trees and a tree depth of 4 displayed no associations (IQ = 0) between the samples from the shipwreck and the Hauzilou and Tiankeng kilns. This finding suggests that both kilns were unlikely sources for the artifacts, a finding that was not observed in the original study\\cite{shipwreck}.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=0.65\\linewidth]{Figure6}\n\t\t\n\t\t\\caption{The probabilistic affinity matrix after the fragmentation of the Zhangzou province into its respective source kilns and the removal of the two artifact samples associated with the Jingdezhen kiln, and the sources from the Jingdezhen kiln and the Dapu kiln. }\n\t\\end{figure}\n\t\n\t\n\tAlthough the additional analysis did not clearly display which individual kilns in the Zhangzou province were most likely associated with the artifact samples, it did discount two unlikely locations. In a provenance study it is also of interest to know where samples likely did not originate from, because that information can be used for the design of future studies.\n\t\n\t\\subsection{Cryptocrystalline Sillicate Data EDA}\n\tHoard et al. investigated the geological source of several cryptocrystalline sillicate artifacts that appeared identical by eye and by microscope \\cite{crypto}. Artifacts were collected in Eckles, Kansas and at three sites in Nebraska (Signal Butte I\/II\/III). Eleven trace elements in the artifacts were quantified by NAA and compared to source materials collected from Flattop Butte, Gering formation, Nelson Butte, Table Mountain, and from West Horse Creek. \n\t\n\tThe Gering formation TIQ was calculated to be $\\ll$ 5\\% by an uncharted forest model that had 100 trees and a tree depth of 4. This result suggested that the Gering formation was not commonly partitioned with samples outside of its assigned class label, in agreement with the authors' key finding that the Gering formation was not a likely source for the artifacts\\cite{crypto}. This finding is a clear example of the identification of an uninformative class by uncharted forest analysis. This finding was corroborated from the scores obtained by a PCA constructed using only the source classes. However, the PCA scores plots were uninformative with regards to obtaining other information Hoard et al. had described using canonical discriminant analysis (CDA), or that observed via the probabilistic affinity matrix obtained from uncharted forest analysis. The score plots of the first four principal components showed nondistinct overlap of almost all of the samples belonging to the classes, even after the CDA-separable Gering formation class was removed. \n\t\n\tThe probabilistic affinity matrix obtained by uncharted forest analysis using 100 trees and a tree depth of 4 with only the Eckles artifact class is displayed in Figure 7. This Figure clearly shows that the most likely source for the samples collected at the Eckles site was the Flattop Butte. This finding by uncharted forest concurred with the CDA analysis performed by Hoard et al \\cite{crypto}. \n\t\n\n\n\n\n\t\n\n\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=0.70\\linewidth]{Figure7}\n\t\t\n\t\t\\caption{The probabilistic affinity matrix from uncharted forest clustering of the cryptocrystalline sillicate data with only the Eckles Kansas artifact and the source classes included. Class boundaries are shown by the overlays of vertical and horizontal lines. }\n\t\\end{figure}\n\t\n\tHowever, unlike the CDA analysis performed by Hoard et al. or the PCA analysis explored in this study, an uncharted forest analysis displayed class heterogeneity. A very clear example of a class with heterogeneous structure, shown in Figure 8, was observed via uncharted forest analysis using 100 trees and a tree depth of 4, by comparing associations between artifacts from the Signal Butte III site and the source locations. Seven of the twelve samples collected from the Signal Butte III site were most frequently partitioned with two samples collected from Table Mountain, not the Signal Butte III samples. Those two samples that were collected from Table Mountain did not appear to be associated with the rest of the data that was labeled as Table Mountain. This instance of class heterogeneity may be indicative of a missing source class, the introduction of a sampling error, or of another anomaly in the Table Mountain data. Hoard et al. was unable to determine the likely origin for several of the samples that were collected from the Signal Butte III site; CDA indicated that the samples from Signal Butte III were tightly grouped \\cite{crypto}. Uncharted forest analysis, on the other hand, indicated that the samples may actually be from two groups, one not well-represented in the NAA data, and the other most like the samples obtained from Table Mountain.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=0.70\\linewidth]{Figure8}\n\t\t\n\t\t\\caption{The probabilistic affinity matrix from uncharted forest clustering of the cryptocrystalline sillicate data, with overlaid class boundaries. Heterogeneity within the first 7 samples collected from the Signal Butte III site can be observed in that they only associate with two samples collected from Table Mountain. These samples do not associate with the rest of the data from Table Mountain. }\n\t\\end{figure}\n\t\n\t\n\t\\subsection{Geographical Butter Data Clustering Analysis}\n\tThe butter dataset consists of 78 butter samples collected from 17 countries. The goal of this study was to assess the viability of determining the origin of butter samples from their chemical signatures \\cite{butter}. Butter samples were analyzed by inductively coupled plasma mass spectrometry (ICP-MS) and isotope-ratio mass spectrometry using isotopic ratios of $^{13}$C, $^{15}$N, $^{18}$O, and $^{34}$S \\cite{butter}. Of the 17 countries included in the data, 8 were represented by two samples or less; these data were removed. A total of 65 samples were retained for analysis, of which 44.6\\% were of German origin. A significant proportion of the measurements of $^{34}$S in butter proteins were lower than the detection limit \\cite{butter}. To make use of the information in this variable, the samples with $^{34}$S values lower than the detection limit were imputed with zero concentrations. \n\t\n\tIn this study, the butter dataset was used to compare the variance partitions made by the uncharted forest algorithm with two variance-based clustering algorithms. K-means clustering and hierarchical clustering with a Ward's criterion were applied to the butter data, using a range of clusters (2-7), with 15 replicates. An uncharted forest analysis with 200 trees and a tree depth of 5 was used to assess each replicate of the clustering methods. K-means clustering optimizes the location of cluster partitions by minimizing within-cluster variance in an iterative fashion \\cite{kmeans}. Ward's clustering was employed with a loss function that minimizes within-cluster variance by forming clusters using an agglomerative approach\\cite{ward}. The goal of this study was to assess how the TIQ and TSAQ metrics compared to within- and between-cluster variance metrics, respectively. \n\t\n\tConfidence intervals of the correlation coefficients between uncharted forest metrics and the empirical cluster variances for both clustering methods were created by Fisher transforming the correlation coefficients, calculating the confidence intervals as usual, and transforming back \\cite{Fisher}. All intervals reported here were computed at 99\\% confidence. TSAQ and within-cluster variance were found to be strongly correlated: K-means clustering had a correlation of 0.90 $\\pm$ 0.12 with TSAQ, and Ward's clustering had a correlation of 0.94 $\\pm$ 0.05. This result strongly suggested that the relationship between TSAQ and the within-cluster variance metric were well approximated by a linear relationship. The intervals for the correlation coefficients between TIQ and between-cluster variances were -0.50 $\\pm$ 0.07 for K-means clustering and -0.60 $\\pm$ 0.05 for Ward's clustering. Overall, the correlation between TIQ and between-cluster variance of either clustering method was moderate; this finding indicated that TIQ provides information that is only somewhat similar in direction to between-cluster variance.\n\t\n\tThe negative correlation between TIQ and between cluster variances was expected, because as the number of clusters increases, the cluster centroid distances also increase, and so do the between cluster variances. However, as cluster distances increase, the associations found by uncharted forest analysis between the samples inside of those clusters would be expected to become less prevalent, because the clusters are becoming more specific to the groups in the data; thus, TIQ values would be expected to decrease. The magnitude of the correlations were not near-perfect (-1.0) because TIQ and TSAQ metrics result from randomly created partitions and the uncharted forest algorithm does not partition data the same way that either K-means or Ward clustering does. However, the fact that the correlation was correct in sign suggests that the partitioning mechanism is overall variance-based, as intended.\n\t\n\t\n\t\\section{Conclusion}\n\tWe have presented a novel adaptation to the random forest paradigm which extends it to unsupervised exploratory data analysis. The method allows for high- and low-dimensional data to be represented as a matrix where every entry is a sampled probability-like value that represents the likelihood that a given sample (row) resides in the same terminal node as the other samples (column). This exploratory data analysis tool and its associated metrics are shown to be a informative aid for the analysis of classification and provenance data. Uncharted forest analysis was able to provide information related to uninformative class labels, class heterogeneity, and sample-wise relationships. \n\t\n\tThe uncharted forest analysis method using variance partitions was explored by comparing its measurable results with both between- and within- cluster variances of two known variance based clustering algorithms. It was shown that the uncharted forest algorithm can provide metrics (TIQ, TSAQ) that are correlated to global clustering statistics. However, the true utility of the method is found in the visualization of sample-wise information, and how it relates to cluster\/class-wise associations. As a proof-of-concept though, when sample-wise variance partitions were used in uncharted forest analyses, the probabilistic affinity matrix generated metrics that were correlated to rigorously defined statistical measures of spread for clustered data. \n\t\n\tVery similar findings to the classification based provenance studies performed by Zhu et al. and Hoard et al. were obtained from exploratory uncharted forest analysis. This is of interest because the uncharted forest algorithm is unsupervised, but with an overlay of class labels, it was possible to visually infer class labels for unknown samples in high-dimensional, multiclass datasets. Most importantly, uncharted forest analysis was able to display information about the data that was not readily apparent from the classification methods used in the aforementioned studies and other exploratory data analysis methods. The uncharted forest method was shown to be an EDA technique that can provide information that is relevant to the domains of exploratory data analysis, clustering, and classification.\n\t\n\t\\section{Acknowledgments}\n\tThis work was supported by the United States National Science Foundation grant 1506853.\n\t\n\t\\section{References}\n\n\n\n\t\n\n\n\t\n\n\n\n\n\n\n\n\t\n\n\t\n\t\\bibliographystyle{model1-num-names}\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\\label{sec:intro}\nIn this work, we study local problems on regular trees from three different perspectives. \n\n\\textit{First,} we consider the perspective of distributed algorithms. In distributed computing, the studied setup is a network of computers where each computer can only communicate with its neighbors. Roughly speaking, the question of interest in this area is which problems can be solved with only a few rounds of communication in the underlying network. \n\n\\textit{Second}, we consider the perspective of (finitary) factors of iid processes.\nIn probability, random processes model systems that appear to vary in a random manner.\nThese include Bernoulli processes, Random walks etc.\nA particular, well-studied, example is the Ising model.\n\n\\textit{Third}, we investigate the perspective of descriptive combinatorics. The goal of this area is to understand which constructions on infinite graphs can be performed without using the so-called axiom of choice.\n\nAlthough many of the questions of interest asked in these three areas are quite similar to each other, no systematic connections were known until an insightful paper of Bernshteyn~\\cite{Bernshteyn2021LLL} who showed that results from distributed computing can automatically imply results in descriptive combinatorics.\nIn this work, we show that the connections between the three areas run much deeper than previously known, both in terms of techniques and in terms of complexity classes.\nIn fact, our work suggests that it is quite useful to consider all three perspectives as part of a common theory, and we will attempt to present our results accordingly.\nWe refer the reader to \\cref{fig:big_picture_trees} for a partial overview of the rich connections between the three perspectives, some of which are proven in this paper. \n\nIn this work, we focus on the case where the graph under consideration is a regular tree.\nDespite its simplistic appearance, regular trees play an important role in each of the three areas, as we will substantiate at the end of this section.\nTo already provide an example, in the area of distributed algorithms, almost all locality lower bounds are achieved on regular trees.\nMoreover, when regarding lower bounds, the property that they already apply on regular trees actually \\emph{strengthens} the result---a fact that is quite relevant for our work as our main contribution regarding the transfer of techniques between the areas is a new lower bound technique in the area of distributed computation that is an adaptation and generalization of a technique from descriptive combinatorics.\nRegarding our results about the relations between complexity classes from the three areas, we note that such connections are also studied in the context of paths and grids in two parallel papers \\cite{grebik_rozhon2021LCL_on_paths,grebik_rozhon2021toasts_and_tails}.\n\nIn the remainder of this section, we give a high-level overview of the three areas that we study.\nThe purpose of these overviews is to provide the reader with a comprehensive picture of the studied settings that can also serve as a starting point for delving deeper into selected topics in those areas---in order to follow our paper, it is not necessary to obtain a detailed understanding of the results and connections presented in the overviews. Necessary technical details will be provided in \\cref{sec:preliminaries}.\nMoreover, in \\cref{sec:contribution}, we present our contributions in detail.\n\n\n\n\n\n\\paragraph{Distributed Computing}\nThe definition of the $\\mathsf{LOCAL}$ model of distributed computing by Linial~\\cite{linial92LOCAL} was motivated by the desire to understand distributed algorithms in huge networks. \nAs an example, consider a huge network of wifi routers. Let us think of two routers as connected by an edge if they are close enough to exchange messages. \nIt is desirable that such close-by routers communicate with user devices on different channels to avoid interference.\nIn graph-theoretic language, we want to properly color the underlying network. \nEven if we are allowed a color palette with $\\Delta+1$ colors where $\\Delta$ denotes the maximum degree of the graph (which would admit a simple greedy algorithm in a sequential setting), the problem remains highly interesting in the distributed setting, as, ideally, each vertex decides on its output color after only a few rounds of communication with its neighbors, which does not allow for a simple greedy solution. \n\nThe $\\mathsf{LOCAL}$ model of distributed computing formalizes this setup: we have a large network, where each vertex knows the network's size, $n$, and perhaps some other parameters like the maximum degree $\\Delta$. In the case of randomized algorithms, each vertex has access to a private random bit string, while in the case of deterministic algorithms, each vertex is equipped with a unique identifier from a range polynomial in the size $n$ of the network. \nIn one round, each vertex can exchange any message with its neighbors and can perform an arbitrary computation. The goal is to find a solution to a given problem in as few communication rounds as possible. \nAs the allowed message size is unbounded, a $t$-round $\\mathsf{LOCAL}$ algorithm can be equivalently described as a function that maps $t$-hop neighborhoods to outputs---the output of a vertex is then simply the output its $t$-hop neighborhood\nmapped to by this function.\nAn algorithm is correct if and only if the collection of outputs at all vertices constitutes a correct solution to the problem. \n\nThere is a rich theory of distributed algorithms and the local complexity of many problems is understood. \nThe case of trees is a highlight of the theory: it is known that any local problem (a class of natural problems we will define later) belongs to one of only very few complexity classes.\nMore precisely, for any local problem, its randomized local complexity is either $O(1), \\Theta(\\log^* n), \\Theta(\\log\\log n), \\Theta(\\log n)$, or $\\Theta(n^{1\/k})$ for some $k \\in \\mathbb{N}$. Moreover, the deterministic complexity is always the same as the randomized one, except for the case $\\Theta(\\log\\log n)$, for which the corresponding deterministic complexity is $\\Theta(\\log n)$ (see \\cref{fig:big_picture_trees}). \n\n\n\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{fig\/big_picture_trees.pdf}\n \\caption{Complexity classes on regular trees considered in the three areas of distributed computing, factors of iid processes\/uniform algorithms, and descriptive combinatorics. The left part shows complexity classes of distributed computing. We use the shorthand $\\mathsf{LOCAL}$ if it does not matter whether we talk about the deterministic or randomized complexity. These two notions differ only for the class of problems of randomized local complexity $O(\\log\\log n)$, which have deterministic complexity $O(\\log n)$. \\\\\n The uniform complexity classes of sublogarithmic complexity are in correspondence to appropriate classes in the randomized local complexity model, as proven in \\cref{sec:separating_examples}. On the other hand, the class fiid is very similar to the class $\\mathsf{MEASURE}$ from descriptive combinatorics. The equivalence of the class $\\mathsf{CONTINUOUS}$ and $\\mathsf{LOCAL}(O(\\log^* n)) = \\mathsf{OLOCAL}(O(\\log^* 1\/\\varepsilon))$ is marked with a dashed arrow as it was proven in case the tree is generated by a group action (think of the tree being equipped with an additional $\\Delta$-edge coloring). The inclusion $\\mathsf{LOCAL}(O(\\log^* n)) \\subseteq \\mathsf{BOREL}$ however clearly holds also in our setting. The class $\\mathsf{BOREL}$ is incomparable with $\\mathsf{RLOCAL}(O(\\log\\log n))$, as proven in \\cref{sec:separating_examples}. }\n \\label{fig:big_picture_trees}\n\\end{figure}\n\n\n\\todo{thm 5.1 - 5.3}\n\n\n\n\n\\paragraph{(Finitary) Factors of iid Processes and Uniform Algorithms}\n\nIn recent years, \\emph{factors of iid (fiid) processes} on trees attracted a lot of attention in combinatorics, probability, ergodic theory and statistical physics \\cite{ABGGMP,ArankaFeriLaci,BackSzeg,BackVir,backhausz,backhausz2,BGHV,Ball,Bowen1,Bowen2,CPRR,csoka2015invariant,GabLyo,GamarnikSudan,GGP,HarangiVirag,HolroydPoisson,HLS,HPPS,HolroydPeres,HoppenWormald,Kun,RahmanPercolation,RahVir,LyonsTrees,Mester,schmidt2016circular,Timar1,Timar2}.\nIntuitively, factors of iid processes are randomized algorithms on, e.g., infinite $\\Delta$-regular trees, where each vertex outputs a solution to a problem after it explores random strings on vertices of the whole tree.\nAs an example, consider the \\emph{perfect matching} problem.\nAn easy parity argument shows that perfect matching cannot be solved by any local randomized algorithm on finite trees.\nHowever, if we allow a small fraction of vertices not to be matched, then, by a result of Nguyen and Onak \\cite{nguyen_onak2008matching} (see also \\cite{ElekLippner}), there is a constant-round randomized algorithm that produces such a matching on high-girth graphs (where the constant depends on the fraction of unmatched vertices that we allow).\nThis result can also be deduced from a result of Lyons and Nazarov \\cite{lyons2011perfect}, who showed that perfect matching can be described as a factor of iid process on an infinite $\\Delta$-regular tree.\nThe high-level idea behind this connection is that high-girth graphs approximate the infinite $\\Delta$-regular tree and constant-round local algorithms approximate factors of iid processes.\nThis correspondence is formalized in the notion of \\emph{Benjamini-Schramm} or \\emph{local-global} convergence \\cite{benjamini2011recurrence, hatamilovaszszegedy}.\nWe note that getting only ``approximate'' solutions, that is, solutions where a small fraction of vertices does not have to satisfy the constraints of a given problem, is intrinsic in this correspondence.\nRegardless, there are many techniques, such as entropy inequality \\cite{backhausz} or correlation decay \\cite{backhausz2}, and particular results such as the aforementioned perfect matching problem \\cite{lyons2011perfect} that provide lower and upper bounds, respectively, in our setting as well.\nWe refer the reader to \\cite{LyonsTrees,BGHV} for a comprehensive summary of the field. \n\n\nIn this paper, we mostly consider a stronger condition than fiid, namely so-called \\emph{finitary} factors of iid (ffiid) processes that are studied in the same context as fiid \\cite{holroyd2017one_dependent_coloring,HolroydSchrammWilson2017FinitaryColoring,Spinka}. \nPerhaps surprisingly, the notion of ffiid is identical to the notion of so-called uniform distributed randomized algorithms \\cite{Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring,grebik_rozhon2021toasts_and_tails} that we now describe. \nWe define an uniform local algorithm as a randomized local algorithm that does not know the size of the graph $n$ -- this enables us to run such an algorithm on infinite graphs, where there is no $n$. \nMore precisely, we require that each vertex eventually outputs a solution that is compatible with the output in its neighborhood, but the time until the vertex finishes is a potentially unbounded random variable. \nAs in the case of classical randomized algorithms, we can now measure the \\emph{uniform complexity} of an uniform local algorithm (known as the tail decay of ffiid \\cite{HolroydSchrammWilson2017FinitaryColoring}). \nThe uniform complexity of an algorithm is defined as the function $t(\\varepsilon)$ such that the probability that the algorithm run on a specific vertex needs to see outside its $t(\\varepsilon)$-hop neighborhood is at most $\\varepsilon$. \nAs in the case of classical local complexity, there is a whole hierarchy of possible uniform complexities (see \\cref{fig:big_picture_trees}). \n\nWe remark that uniform distributed local algorithms can be regarded as Las Vegas algorithms. The output will always be correct; there is however no fixed guarantee at what point all vertices have computed their final output. On the other hand, a randomized distributed local algorithm can be viewed as a Monte Carlo algorithm as it needs to produce an output after a fixed number of rounds, though the produced output might be incorrect. \n\n\n\n\n\n\n\n\n\\paragraph{Descriptive Combinatorics}\n \n \n\t\n\tThe Banach-Tarski paradox states that a three-dimensional ball of unit volume can be decomposed into finitely many pieces that can be moved by isometries (distance preserving transformations such as rotations and translations) to form two three-dimensional balls each of them with unit volume(!).\n The graph theoretic problem lurking behind this paradox is the following: fix finitely many isometries of $\\mathbb{R}^3$ and then consider a graph where $x$ and $y$ are connected if there is an isometry that sends $x$ to $y$.\n\tThen our task becomes to find a perfect matching in the appropriate subgraph of this graph -- namely, the bipartite subgraph where one partition contains points of the first ball and the other contains points of the other two balls. \n\tBanach and Tarski have shown that, with a suitably chosen set of isometries, the axiom of choice implies the existence of such a matching. \n\tIn contrast, since isometries preserve the Lebesgue measure, the pieces in the decomposition cannot be Lebesgue measurable. \n\tSurprisingly, Dougherty and Foreman \\cite{doughertyforeman} proved that the pieces in the Banach-Tarski paradox can have the \\emph{Baire property}. \n\tThe Baire property is a topological analogue of being Lebesgue measurable; a subset of $\\mathbb{R}^3$ is said to have the Baire property if its difference from some open set is topologically negligible.\n\t\n \n\t \n\t\n\t\n Recently, results similar to the Banach-Tarski paradox that lie on the border of combinatorics, logic, group theory, and ergodic theory led to an emergence of a new field often called \\textit{descriptive} or \\textit{measurable combinatorics}. \n The field focuses on the connection between the discrete and continuous and is largely concerned with the investigation of graph-theoretic concepts. \n The usual setup in descriptive combinatorics is that we have a graph with uncountably many connected components, each being a countable graph of bounded degree. \n For example, in case of the Banach-Tarski paradox, the vertices of the underlying graph are the points of the three balls, edges correspond to isometries, and the degree of each vertex is bounded by the number of chosen isometries. \n Some of the most beautiful results related to the field include \\cite{laczk, marksunger,measurablesquare,doughertyforeman,marks2016baire,gaboriau,KST,DetMarks,millerreducibility,conley2020borel,csokagrabowski,Bernshteyn2021LLL}, see \\cite{kechris_marks2016descriptive_comb_survey,pikhurko2021descriptive_comb_survey} for recent surveys. \n \n Importantly, in many of these results, including the Banach-Tarski paradox, graphs where each component is an infinite $\\Delta$-regular tree appear naturally. \n Oftentimes, questions considered in descriptive combinatorics lead to constructing a solution to a local problem in the underlying uncountable graph (in the case of Banach-Tarski, the local problem is perfect matching). \n The construction needs to be such that the solution of the problem has some additional regularity properties. For example in the case of Banach-Tarski, a solution is possible when the regularity condition is the Baire property, but not if it is Lebesgue measurability.\n In fact, together with Borel measurability these are the most prominent regularity conditions studied in descriptive combinatorics.\n \n The corresponding complexity classes of local problems that always admit a solution with the respective regularity property are $\\mathsf{BOREL},\\mathsf{MEASURE},\\mathsf{BAIRE}$ (See \\cref{fig:big_picture_trees}). \n In this paper, we moreover consider the setting where each connected component of the underlying graph is a $\\Delta$-regular tree. \n \n \n \n \n \n \n\t\n\t\n\t\t\n\t\n\t\t\n\t\t\n\n\t The connection between distributed computing and descriptive combinatorics arises from the fact that in descriptive combinatorics we care about constructions that do not use the axiom of choice. \n\t In the distributed language, the axiom of choice corresponds to leader election, that is, the constructions in descriptive combinatorics do not allow picking exactly one point in every component. \n\t \n\t To get some intuition about the power of the complexity class $\\mathsf{BOREL}$, we note that Borel constructions allow us to alternate countably many times the following two operations. \n\t First, any local algorithm with constant local complexity can be run. \n\t Second, we have an oracle that provides a maximal independent set (MIS) on any graph that can be constructed locally from the information computed so far \\cite{KST}. \n Note that from the speedup result of \\cite{chang_kopelowitz_pettie2019exp_separation} we get that every local problem with local complexity $O(\\log^* n)$ can be solved by constant local constructions and \\emph{one} call to such an MIS oracle. This implies the inclusion $\\mathsf{LOCAL}(O(\\log^* n)) \\subseteq \\mathsf{BOREL}$ in \\cref{fig:big_picture_trees} proven in the insightful paper of Bernshteyn \\cite{Bernshteyn2021LLL}. \n The relationship of the class $\\mathsf{MEASURE}$ (and of the class $\\mathsf{fiid}$ from the discussion of factors) to the class $\\mathsf{BOREL}$ is analogous to the relationship of randomized distributed algorithms to deterministic distributed algorithms. \n\n \n \n\n\n\n\n\n\n\t\n\n\t\n\t\n\t\n\n\n\n\n\n\n\\paragraph{Local Problems on Regular Trees}\n\nAfter introducing the three areas of interest in this work, we conclude the section by briefly discussing the kinds of problems we focus on, which are local problems on regular trees.\nMore precisely, we study \\emph{locally checkable labeling (LCL)} problems, which are a class of problems, where the correctness of the solution can be checked locally.\nExamples include classical problems from combinatorics such as proper vertex coloring, proper edge coloring, perfect matching, and maximal independent set.\nOne main goal of this paper is to understand possible complexity classes of LCLs without inputs on infinite $\\Delta$-regular trees and their finite analogues.\nWe refer the reader to \\cref{sec:preliminaries} for a precise definition of a finite $\\Delta$-regular tree.\n\nThe motivation for studying regular trees in this work stems from different sources: (a) infinite $\\Delta$-regular trees are studied in the area of ergodic theory \\cite{Bowen1,Bowen2}, random processes \\cite{backhausz,backhausz2,lyons2011perfect} and descriptive combinatorics \\cite{DetMarks,BrooksMeas}, (b) many lower bounds in distributed computing are proven in regular trees \\cite{balliu_et_al:binary_lcl,balliu2019LB,brandt_etal2016LLL,brandt19automatic,Brandt20tightLBmatching,ChangHLPU20,goeoes14_PODC}, and (c) connecting and comparing the techniques of the three areas in this simple setting reveals already deep connections, see \\cref{sec:contribution}.\n\n\n\n\n\n\n\n\n\n\\section{Our Contributions}\n\\label{sec:contribution}\n\nWe believe that our main contribution is presenting all three perspectives as part of a common theory. Our technical contribution is split into three main parts. \n\n\n\n\n\\subsection{Generalization of Marks' Technique}\\label{sec:intromarks}\nIn Section \\ref{sec:marks} we extend the Borel determinacy technique of Marks \\cite{DetMarks}, which was used to prove the nonexistence of Borel $\\Delta$-colorings and perfect matchings, to a broader class of problems, and \nadapt the extended technique to the distributed setting, thereby obtaining a simple method for proving distributed lower bounds.\nThis method is the first lower bound technique for distributed computing using ideas coming from descriptive combinatorics (see \\cite{BernshteynVizing} for a distributed computing upper bound motivated by descriptive combinatorics).\nMoreover, we show how to use the developed techniques to obtain both $\\mathsf{BOREL}$ and $\\mathsf{LOCAL}$ lower bounds for local problems from a natural class, called homomorphism problems.\nOur key technical ingredient for obtaining the mentioned techniques and results is a novel technique based on the notion of an \\emph{ID graph}.\nWe note that a very similar concept to the ID graph was independently discovered by \\cite{id_graph}. \n\n\\paragraph{Marks' technique}\nIn the following we give an introduction to Marks' technique by going through a variant of his proof~\\cite{DetMarks,Marks_Coloring} that shows that $\\Delta$-coloring has deterministic local complexity $\\Omega(\\log n)$. The proof already works in the case where the considered regular tree comes with an input $\\Delta$-edge coloring. In this case, the output color of a given vertex $u$ can be interpreted as that $u$ ``grabs'' the incident edge of that color. The problem is hence equivalent to the \\emph{edge grabbing} problem where every vertex is required to grab an incident edge such that no two vertices grab the same edge. \n\nWe first show the lower bound in the case that vertices do not have unique identifiers but instead are properly colored with $L > \\Delta$ colors. \nSuppose there is an algorithm $\\mathcal{A}$ solving the edge grabbing problem with local complexity $t(n) = o(\\log n)$, and consider a tree rooted at vertex $u$ of depth $t(n)$; such a tree has less than $n$ vertices, for large enough $n$.\nAssume that $u$ has input color $\\sigma \\in [L]$, and consider, for some fixed edge color $\\alpha$, the edge $e$ that is incident to $u$ and has color $\\alpha$.\nTwo players, Alice and Bob, are playing the following game. In the $i$-th round, Alice colors the vertices at distance $i$ from $u$ in the subtree reachable via edge $e$ with colors from $[L]$. Then, Bob colors all other vertices at distance $i$ from $u$ with colors from $[L]$ (see \\cref{fig:game}).\nConsider the output of $u$ when executing $\\mathcal{A}$ on the obtained colored tree.\nBob wins the game if $u$ grabs the edge $e$, and Alice wins otherwise. \n\nNote that either Alice or Bob has a winning strategy. \nGiven the color $\\sigma$ of $u$, if, for each edge color $\\alpha$, Alice has a winning strategy in the game corresponding to the pair $(\\sigma, \\alpha)$, then we can create $\\Delta$ copies of Alice and let them play their strategy on each subtree of $u$, telling them that the colors chosen by the other Alices are what Bob played. The result is a coloring of the input tree such that $u$, by definition, does not pick any edge, contradicting the fact that $\\mathcal{A}$ provides a valid solution!\nSo for every $\\sigma$ there is at least one $\\alpha$ such that Bob has a winning strategy for the game corresponding to $(\\sigma, \\alpha)$. By the pigeonhole principle, there are two colors $\\sigma_1, \\sigma_2$, such that Bob has a winning strategy for both pairs $(\\sigma_1, \\alpha)$ and $(\\sigma_2, \\alpha)$. \nBut now we can imagine a tree rooted in an edge between vertices $u_1, u_2$ that are colored with colors $\\sigma_1, \\sigma_2$. We can now take two copies of Bob, one playing at $u_1$ and the other playing at $u_2$ and let them battle it out, telling each copy that the other color from $\\{\\sigma_1, \\sigma_2\\}$ and whatever the other copy plays are the moves of Alice. The resulting coloring has the property that both $u_1$ and $u_2$, when executing $\\mathcal{A}$ on the obtained colored tree, grab the edge between them, a contradiction that finishes the proof!\n\n\\paragraph{The ID graph}\nThe downside of the proof is that it does not work in the model with unique identifiers (where the players' moves consist in assigning identifiers instead of colors), since gluing copies of the same player could result in an identifier assignment where the identifiers are not unique. \nOne possible remedy is to conduct the whole proof in the context of Borel graphs as was done by Marks. This proves an even stronger statement, namely that $\\Delta$-coloring is not in the class $\\mathsf{BOREL}$, but requires additional ad-hoc tricks and a pretty heavy set theoretic tool---Martin's celebrated Borel determinacy theorem~\\cite{martin} stating that even for infinite two-player games one of the players has to have a winning strategy if the payoff set is Borel. \nThe ID graph enables us to adapt the proof (and its generalization that we develop in \\cref{sec:marks}) to the distributed setting, where the fact that one of the players has a winning strategy is obvious. \nMoreover, we use an infinite version of the ID graph to generalize Marks' technique also in the Borel setting. \n\nHere is how it works: \nThe ID graph is a specific graph whose vertices are the possible unique input identifiers (for input graphs of size $n$), that is, numbers from $[n^{O(1)}]$. Its edges are colored with colors from $[\\Delta]$ and its girth is $\\Omega(\\log n)$. When we define the game between Alice and Bob, we require them to label vertices with identifiers in such a way that whenever a new vertex is labeled with identifier $i$, and its already labeled neighbor has identifier $j$, then $ij$ is an edge in the ID graph.\nMoreover, the color of edge $ij$ in the ID graph is required to be the same as the color of the edge between the vertices labeled $i$ and $j$ in the tree where the game is played. \nIt is straightforward to check that these conditions enforce that even if we let several copies of the same player play, the resulting tree is labeled with unique identifiers. Hence, the same argument as above now finally proves that the deterministic local complexity of $\\Delta$-coloring is $\\Omega (\\log n)$. \n\nWe note that our ID graph technique is of independent interest and may have applications in many different contexts. \nTo give an example from distributed computing, consider the proof of the deterministic $\\Omega(\\log n)$-round lower bound for $\\Delta$-coloring developed by the distributed community~\\cite{brandt_etal2016LLL,chang_kopelowitz_pettie2019exp_separation}, which is based on the celebrated round elimination technique. \nEven though the result is deterministic, the proof is quite technical due to the fact that it relies on examining randomized algorithms, for reasons similar to the reasons why Marks' proof does not apply directly to the setting with unique identifiers. \nFortunately, it can be again streamlined with the use of the ID graph technique. \nMoreover, in a follow-up work, the ID graph technique has already led to a new lower bound for the Lov\u00e1sz local lemma~\\cite{brandt_grunau_rozhon2021LLL_in_LCA} in the area of Local Computation Algorithms (which is part of the realm of sequential computation), thereby giving further evidence for the versatility of the technique. \n\n\n\\paragraph{Marks vs. Round Elimination}\nIt is quite insightful to compare Marks' technique (and our generalization of it) with the powerful round elimination technique~\\cite{brandt19automatic}, which has been responsible for all locality lower bounds of the last years~\\cite{brandt_etal2016LLL, balliu2019LB, brandt19automatic, balliu2019hardness_homogeneous, balliu_et_al:binary_lcl, Brandt20tightLBmatching, balliu20rulingset, ChangHLPU20, balliu21dominatingset}. \nWhile, on the surface, Marks' approach developed for the Borel world may seem quite different from the round elimination technique, there are actually striking similarities between the two methods.\nOn a high level, in the round elimination technique, the following argument is used to prove lower bounds in the LOCAL model:\nIf a $T$-round algorithm exists for a problem $\\Pi_0$ of interest, then there exists a $(T-1)$-round algorithm for some problem $\\Pi_1$ that can be obtained from $\\Pi_0$ in a mechanical manner.\nBy applying this step iteratively, we obtain a problem $\\Pi_t$ that can be solved in $0$ rounds; by showing that there is no $0$-algorithm for $\\Pi_t$ (which is easy to do if $\\Pi_t$ is known), a $(T+1)$-round lower bound for $\\Pi_0$ is obtained.\n\nThe interesting part regarding the relation to Marks' technique is how the ($T-i-1$)-round algorithms $\\mathcal{A}'$ are obtained from the ($T-i$)-round algorithms $\\mathcal{A}$ in the round elimination framework: in order to execute $\\mathcal{A}'$, each vertex $v$, being aware of its ($T-i-1$)-hop neighborhood, essentially asks whether, for all possible extensions of its view by one hop along a chosen incident edge, there exists some extension of its view by one hop along all other incident edges such that $\\mathcal{A}$, executed on the obtained ($T-i$)-hop neighborhood, returns a certain output at $v$, and then bases its output on the obtained answer.\nIt turns out that the vertex sets corresponding to these two extensions correspond precisely to two moves of the two players in the game(s) played in Marks' approach: more precisely, in round $T-i$ of a game corresponding to the considered vertex $v$ and the chosen incident edge, the move of Alice consists in labeling the vertices corresponding to the first extension, and the move of Bob consists in labeling the vertices corresponding to the second extension.\n\nHowever, despite the similarities, the two techniques (at least in their current forms) have their own strengths and weaknesses and are interestingly different in that there are local problems that we know how to obtain good lower bounds for with one technique but not the other, and vice versa.\nFinding provable connections between the two techniques is an exciting research direction that we expect to lead to a better understanding of the possibilities and limitations of both techniques.\n\nIn \\cref{sec:marks} we use our generalized and adapted version of Marks' technique to prove new lower bounds for so-called homomorphism problems. \nHomomorphism problems are a class of local problems that generalizes coloring problems---each vertex is to be colored with some color and there are constraints on which colors are allowed to be adjacent. \nThe constraints can be viewed as a graph---in the case of coloring this graph is a clique. \nIn general, whenever the underlying graph of the homomorphism problem is $\\Delta$-colorable, its deterministic local complexity is $\\Omega(\\log n)$, because solving the problem would imply that we can solve $\\Delta$-coloring too (in the same runtime). \nIt seems plausible that homomorphism problems of this kind are the only hard, i.e., $\\Omega(\\log n)$, homomorphism problems. However, our generalization of Marks' technique asserts that this is not true. \n\n\\begin{theorem}\nThere are homomorphism problems whose deterministic local complexity on trees is $\\Omega(\\log n)$ such that the chromatic number of the underlying graph is $2\\Delta-2$.\n\\end{theorem}\n\nIt is not known how to prove the same lower bounds using round elimination\\footnote{Indeed, the descriptions of the problems have comparably large numbers of labels and do not behave like so-called ``fixed points'' (i.e., nicely) under round elimination, which suggests that it is hard to find a round elimination proof with the currently known approaches.}; in fact, as far as we know, these problems are the only known examples of problems on $\\Delta$-regular trees for which a lower bound is known to hold but currently not achievable by round elimination. \nProving the same lower bounds via round elimination is an exciting open problem. \n\n\n\n\n\n\n\n\n\n\n\\subsection{Separation of Various Complexity Classes}\n\\label{sec:introseparation}\n\\paragraph{Uniform Complexity Landscape}\nWe investigate the connection between randomized and uniform distributed local algorithms, where uniform algorithms are equivalent to the studied notion of finitary factors of iid. \nFirst, it is simple to observe that local problems with uniform complexity $t(\\varepsilon)$ have randomized complexity $t(1 \/ n^{O(1)})$ -- by definition, every vertex knows its local output after that many rounds with probability $1 - 1\/n^{O(1)}$. The result thus follows by a union bound over the $n$ vertices of the input graph.\n\nOn the other hand, we observe that on $\\Delta$-regular trees the implication also goes in the opposite direction in the following sense. Every problem that has a randomized complexity of $t(n) = o(\\log n)$ has an uniform complexity of $O(t(1\/\\varepsilon))$. \n\nOne could naively assume that this equivalence also holds for higher complexities, but this is not the case. Consider for example the $3$-coloring problem.\nIt is well-known in the distributed community that $3$-coloring a tree can be solved deterministically in $O(\\log n)$ rounds using the rake-and-compress decomposition~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup,MillerR89}.\nOn the other hand, there is no uniform algorithm for $3$-coloring a tree. \nIf there were such an uniform algorithm, we could run it on any graph with large enough girth and color $99\\%$ of its vertices with three colors. This in turn would imply that the high-girth graph has an independent set of size at least $0.99 \\cdot n\/3$. This is a contradiction with the fact that there exists high-girth graphs with a much smaller independence number~\\cite{bollobas}. \n\nInterestingly, the characterization of Bernshteyn \\cite{Bernshteyn_work_in_progress} implies that \\emph{any} uniform distributed algorithm can be ``sped up'' to a deterministic local $O(\\log n)$ complexity, as we prove in~\\cref{thm:MainBaireLog}. \n\nWe show that there are local problems that can be solved by an uniform algorithm but only with a complexity of $\\Omega(\\log 1\/\\varepsilon)$. Namely, the problem of constructing a $2$-hop perfect matching on infinite $\\Delta$-regular trees for $\\Delta \\ge 3$ has an uniform local complexity between $\\Omega(\\log 1\/\\varepsilon)$ and $O(\\operatorname{poly}\\log 1\/\\varepsilon)$. \nFormally, this proves the following theorem. \n\n\\begin{theorem}\n$\\mathsf{OLOCAL}(O(\\log\\log 1\/\\varepsilon)) \\subsetneq \\mathsf{OLOCAL}(O(\\operatorname{poly}\\log 1\/\\varepsilon))$. \n\\end{theorem}\n\nThe uniform algorithm for this problem is based on a so-called one-ended forest decomposition introduced in \\cite{BrooksMeas} in the descriptive combinatorics context. In a one-ended forest decomposition, each vertex selects exactly one of its neighbors as its parent by orienting the corresponding edge outwards. This defines a decomposition of the vertices into infinite trees. We refer to such a decomposition as a one-ended forest decomposition if the subtree rooted at each vertex only contains finitely many vertices. \nHaving computed such a decomposition, $2$-hop perfect matching can be solved inductively starting from the leaf vertices of each tree.\n\nWe leave the understanding of the uniform complexity landscape in the regime $\\Omega(\\log 1\/\\varepsilon)$ as an exciting open problem. In particular, does there exist a function $g(\\varepsilon)$ such that each local problem that can be solved by an uniform algorithm has an uniform complexity of $O(g(\\varepsilon))$?\n\n\n\n\\paragraph{Relationship of Distributed Classes with Descriptive Combinatorics}\n\nBernshteyn recently proved that $\\mathsf{LOCAL}(\\log^* n) \\subseteq \\mathsf{BOREL}$ \\cite{Bernshteyn2021LLL}. That is, each local problem with a deterministic $\\mathsf{LOCAL}$ complexity of $O(\\log^* n)$ also admits a Borel-measurable solution. A natural question to ask is whether the converse also holds. \nIndeed, it is known that $\\mathsf{LOCAL}(\\log^* n) = \\mathsf{BOREL}$ on paths with no additional input \\cite{grebik_rozhon2021LCL_on_paths}. \nWe show that on regular trees the situation is different. \nOn one hand, a characterization of Bernshteyn \\cite{Bernshteyn_work_in_progress} implies that $\\mathsf{BOREL} \\subseteq \\mathsf{BAIRE} \\subseteq \\mathsf{LOCAL}(O(\\log n))$. \nOn the other hand, we show that this result cannot be strengthened by proving the following result. \n\n\\begin{theorem}\n$\\mathsf{BOREL} \\subsetneq \\mathsf{RLOCAL}(o(\\log n))$.\n\\end{theorem}\n\nThat is, there exists a local problem that admits a Borel-measurable solution but cannot be solved with a (randomized) $\\mathsf{LOCAL}$ algorithm running in a sublogarithmic number of rounds. \n\nLet us sketch a weaker separation, namely that $\\mathsf{BOREL} \\setminus \\mathsf{LOCAL}(O(\\log^* n)) \\neq \\emptyset$. Consider a version of $\\Delta$-coloring where a subset of vertices can be left uncolored. However, the subgraph induced by the uncolored vertices needs to be a collection of doubly-infinite paths (in finite trees, this means each path needs to end in a leaf vertex). \nThe nonexistence of a fast distributed algorithm for this problem essentially follows from the celebrated $\\Omega(\\log n)$ deterministic lower bound for $\\Delta$-coloring of~\\cite{brandt_etal2016LLL}. \nOn the other hand, the problem allows a Borel solution. First, sequentially compute $\\Delta-2$ maximal independent sets, each time coloring all vertices in the MIS with the same color, followed by removing all the\ncolored vertices from the graph. In that way, a total of $\\Delta-2$ colors are used. Moreover, each\nuncolored vertex has at most $2$ uncolored neighbors. \nThis implies that the set of uncolored vertices forms a disjoint union of finite paths, one ended\ninfinite paths and doubly infinite paths. The first two classes can be colored inductively with two\nadditional colors, starting at one endpoint of each path in a Borel way (namely it can be done by making use of the countably many MISes in larger and larger powers of the input graph). \nHence, in the end only doubly infinite paths are left uncolored, as desired. \n\nTo show the stronger separation between the classes $\\mathsf{BOREL}$ and $\\mathsf{RLOCAL}(o(\\log n))$ we use a variation of the\n$2$-hop perfect matching problem. In this variation, some of the vertices can be left unmatched, but similar as in the variant of the $\\Delta$-coloring problem described above, the graph induced by all the unmatched vertices needs to satisfy some additional constraints.\n\nWe conclude the paragraph by noting that the separation between the classes $\\mathsf{BOREL}$ and $\\mathsf{LOCAL}(O(\\log^* n))$ is not as simple as it may look in the following sense. \nThis is because problems typically studied in the $\\mathsf{LOCAL}$ model with a $\\mathsf{LOCAL}$ complexity of $\\omega(\\log^* n)$ like $\\Delta$-coloring and perfect matching also do not admit a Borel-measurable solution due to the technique of Marks \\cite{DetMarks} that we discussed in \\cref{sec:intromarks}.\n\n\n\n\n\n\\subsection{$\\mathsf{LOCAL}(O(\\log n)) = \\mathsf{BAIRE}$} \n\nWe already discussed that one of complexity classes studied in descriptive combinatorics is the class $\\mathsf{BAIRE}$.\nRecently, Bernshteyn proved \\cite{Bernshteyn_work_in_progress} that all local problems that are in the complexity class $\\mathsf{BAIRE}, \\mathsf{MEASURE}$ or $\\mathsf{fiid}$ have to satisfy a simple combinatorial condition which we call being $\\ell$-full. \nOn the other hand, all $\\ell$-full problems allow a $\\mathsf{BAIRE}$ solution \\cite{Bernshteyn_work_in_progress}.\nThis implies a complete combinatorial characterization of the class $\\mathsf{BAIRE}$. \nWe defer the formal definition of $\\ell$-fullness to Section \\ref{sec:baire} as it requires a formal definition of a local problem. \nInformally speaking, in the context of vertex labeling problems, a problem is $\\ell$-full if we can choose a subset $S$ of the labels with the following property.\nWhenever we label two endpoints of a path of at least $\\ell$ vertices with two labels from $S$, we can extend the labeling with labels from $S$ to the whole path such that the overall labeling is valid.\nFor example, proper $3$-coloring is $3$-full with $S=\\{1,2,3\\}$ because for any path of three vertices such that its both endpoints are colored arbitrarily, we can color the middle vertex so that the overall coloring is proper. \nOn the other hand, proper $2$-coloring is not $\\ell$-full for any $\\ell$. \n\nWe complement this result as follows. \nFirst, we prove that any $\\ell$-full problem has local complexity $O(\\log n)$, thus proving that all complexity classes considered in the areas of factors of iids and descriptive combinatorics from \\cref{fig:big_picture_trees} are contained in $\\mathsf{LOCAL}(O(\\log n))$. \nIn particular, this implies that the existence of \\emph{any} uniform algorithm implies a local distributed algorithm for the same problem of local complexity $O(\\log n)$. \nWe obtain this result via the well-known rake-and-compress decomposition~\\cite{MillerR89}. \n\n\nOn the other hand, we prove that any problem in the class $\\mathsf{LOCAL}(O(\\log n))$ satisfies the $\\ell$-full condition. \nThe proof combines a machinery developed by Chang and Pettie~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}\nwith additional nontrivial ideas.\nIn this proof we construct recursively a sequence of sets of rooted, layered, and partially labeled trees, where the partial labeling is computed by simulating any given $O(\\log n)$-round distributed algorithm, \nand then the set $S$ meeting the $\\ell$-full condition is constructed by considering all possible extensions of the partial labeling to complete correct labeling of these trees.\n\nThis result implies the following equality:\n\\begin{theorem}\n$\\mathsf{LOCAL}(O(\\log n)) = \\mathsf{BAIRE}$. \n\\end{theorem}\n\nThis equality is surprising in that the definitions of the two classes do not seem to have much in common on the first glance!\nMoreover, the proof of the equality relies on nontrivial results in both distributed algorithms (the technique of Chang and Pettie~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}) and descriptive combinatorics (the fact that a hierarchical decomposition, so-called toast, can be constructed in $\\mathsf{BAIRE}$, \\cite{conleymillerbound}, see Proposition \\ref{pr:BaireToast}). \n\n\n\n\n\n\n\n\n\n\n\n\nThe combinatorial characterization of the local complexity class $\\mathsf{LOCAL}(O(\\log n))$ on $\\Delta$-regular trees is interesting from the perspective of distributed computing alone. \nThis result can be seen as a part of a large research program aiming at classification of possible local complexities on various graph classes~\\cite{balliu2020almost_global_problems,brandt_etal2016LLL,chang_kopelowitz_pettie2019exp_separation,chang_pettie2019time_hierarchy_trees_rand_speedup,chang2020n1k_speedups,Chang_paths_cycles,balliu2021_rooted_trees,balliu_et_al:binary_lcl,balliu2018new_classes-loglog*-log*}. \nThat is, we wish not only to understand possible complexity classes (see the left part of \\cref{fig:big_picture_trees} for possible local complexity classes on regular trees), but also to find combinatorial characterizations of problems in those classes that allow us to efficiently decide for a given problem which class it belongs to. \nUnfortunately, even for grids with input labels, it is \\emph{undecidable} whether a given local problem can be solved in $O(1)$ rounds~\\cite{naorstockmeyer,brandt_grids}, since\nlocal problems on grids can be used to simulate a Turing machine.\nThis undecidability result does not apply to paths and trees, hence for these graph classes it is still hopeful that we can find simple and useful characterizations for different classes of distributed problems. \n\nIn particular, on paths it is decidable what classes a given local problem belongs to, for all classes coming from the three areas considered here, and this holds even if we allow inputs~\\cite{Chang_paths_cycles,grebik_rozhon2021LCL_on_paths}. \nThe situation becomes much more complicated when we consider trees. \nRecently, classification results on trees were obtained for so-called binary-labeling problems \\cite{balliu_et_al:binary_lcl}. \nMore recently, a complete classification was obtained in the case of \\emph{rooted regular trees}~\\cite{balliu2021_rooted_trees}. \nAlthough their algorithm takes exponential time in the worst case, the authors provided a practical implementation fast enough to classify many typical problems of interest. \n\nMuch less is known for general, \\emph{unoriented} trees, with an arbitrary number of labels. \nIn general, deciding the optimal distributed complexity for a local problem on bounded-degree trees is $\\mathsf{EXPTIME}$-hard~\\cite{chang2020n1k_speedups}, such a hardness result does not rule out the possibility for having a simple and polynomial-time characterization for the case of \\emph{regular trees}, where there is no input and the constraints are placed only on degree-$\\Delta$ vertices. \nIndeed, it was stated in~\\cite{balliu2021_rooted_trees} as an open question to find such a characterization. Our characterization of $\\mathsf{LOCAL}(O(\\log n)) = \\mathsf{BAIRE}$ by $\\ell$-full problems makes progress in better understanding the distributed complexity classes on trees and towards answering this open question. \n\n\n\n\n\n\n\n\\paragraph{Roadmap}\n\\hypertarget{par:roadmap}{}\nIn \\cref{sec:preliminaries}, we define formally all the three setups we consider in the paper. \nIn \\cref{sec:marks} we discuss the lower bound technique of Marks and the new concept of an ID graph.\nNext, in \\cref{sec:separating_examples} we prove some basic results about the uniform complexity classes and give examples of problems separating some classes from \\cref{fig:big_picture_trees}. \nFinally, in \\cref{sec:baire} we prove that a problem admits a Baire measurable solution if and only if it admits a distributed algorithm of local complexity $O(\\log n)$.\n\nThe individual sections can be read largely independently of each other.\nMoreover, most of our results that are specific to only one of the three areas can be understood without reading the parts of the paper that concern the other areas.\nWe encourage the reader interested mainly in one of the areas to skip the respective parts.\n\n\n\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\n\nIn this section, we explain the setup we work with, the main definitions and results. \nThe class of graphs that we consider in this work are either infinite $\\Delta$-regular trees, or their finite analogue that we define formally in \\cref{subsec:local_problems}.\n\nWe sometimes explicitly assume $\\Delta > 2$. \nThe case $\\Delta = 2$, that is, studying paths, behaves differently and seems much easier to understand \\cite{grebik_rozhon2021LCL_on_paths}. \nUnless stated otherwise, we do not consider any additional structure on the graphs, but sometimes it is natural to work with trees with an input $\\Delta$-edge-coloring. \n\n\n\\subsection{Local Problems on $\\Delta$-regular trees}\n\\label{subsec:local_problems}\n\nThe problems we study in this work are locally checkable labeling (LCL) problems, which, roughly speaking, are problems that can be described via local constraints that have to be satisfied in a suitable neighborhood of each vertex.\nIn the context of distributed algorithms, these problems were introduced in the seminal work by Naor and Stockmeyer~\\cite{naorstockmeyer}, and have been studied extensively since.\nIn the modern formulation introduced in~\\cite{brandt19automatic}, instead of labeling vertices or edges, LCL problems are described by labeling half-edges, i.e., pairs of a vertex and an incident edge.\nThis formulation is very general in that it not only captures vertex and edge labeling problems, but also others such as orientation problems, or combinations of all of these types.\nBefore we can provide this general definition of an LCL, we need to introduce some definitions.\nWe start by formalizing the notion of a half-edge.\n\n\\begin{definition}[Half-edge]\n\\label{def:half-edge}\n A \\emph{half-edge} is a pair $(v, e)$ where $v$ is a vertex, and $e$ an edge incident to $v$.\n We say that a half-edge $(v, e)$ is \\emph{incident} to a vertex $w$ if $w = v$, we say that a vertex $w$ is \\emph{contained} in a half edge $(v,e)$ if $w=v$, and we say that $(v, e)$ \\emph{belongs} to an edge $e'$ if $e' = e$.\n We denote the set of all half-edges of a graph $G$ by $H(G)$.\n A \\emph{half-edge labeling} is a function $c \\colon H(G) \\to \\Sigma$ that assigns to each half-edge an element from some label set $\\Sigma$.\n\\end{definition}\n\nIn order to speak about finite $\\Delta$-regular trees, we need to consider slightly modified definition of a graph.\nWe think of each vertex to be contained in $\\Delta$-many half-edges, however, not every half edge belongs to an actual edge of the graph.\nThat is half-edges are pairs $(v,e)$, but $e$ is formally not a pair of vertices.\nSometimes we refer to these half-edges as \\emph{virtual} half-edges.\nWe include a formal definition to avoid confusions. See also \\cref{fig:Delta_reg_tree}. \n\n\n\\begin{definition}[$\\Delta$-regular trees]\n A tree $T$, finite or infinite is a \\emph{$\\Delta$-regular tree} if either it is infinite and $T=T_\\Delta$, where $T_\\Delta$ is the unique infinite $\\Delta$-regular tree, that is each vertex has exactly $\\Delta$-many neighbors, or it is finite of maximum degree $\\Delta$ and each vertex $v\\in T$ of degree $d\\le \\Delta$ is contained in $(\\Delta-d)$-many virtual half-edges.\n \n \n Formally, we can view $T$ as a triplet $(V(T), E(T), H(T))$, where $(V(T),E(T))$ is a tree of maximum degree $\\Delta$ and $H(T)$ consists of real half-edges, that is pairs $(v,e)$, where $v\\in V(T)$, $e\\in E(T)$ and $e$ is incident to $v$, together with some virtual edges, in the case when $T$ is finite, such that each vertex $v\\in V(T)$ is contained in exactly $\\Delta$-many half-edges (real or virtual).\n \n \n\\end{definition}\n\n\\begin{figure}\n \\centering\n \\includegraphics{fig\/Delta_reg_tree.pdf}\n \\caption{A $3$-regular tree on 15 vertices. Every vertex has $3$ half-edges but not all half-edges lead to a different vertex. }\n \\label{fig:Delta_reg_tree}\n\\end{figure}\n\nAs we are considering trees in this work, each LCL problem can be described in a specific form that provides two lists, one describing all label combinations that are allowed on the half-edges incident to a vertex, and the other describing all label combinations that are allowed on the two half-edges belonging to an edge.\\footnote{Every problem that can be described in the form given by Naor and Stockmeyer~\\cite{naorstockmeyer} can be equivalently described as an LCL problem in this list form, by simply requiring each output label on some half-edge $h$ to encode all output labels in a suitably large (constant) neighborhood of $h$ in the form given in~\\cite{naorstockmeyer}.}\nWe arrive at the following definition for LCLs on $\\Delta$-regular trees.\\footnote{Note that the defined LCL problems do not allow the correctness of the output to depend on input labels.}\n\n\\begin{definition}[LCLs on $\\Delta$-regular trees]\\label{def:LCL_trees}\nA \\emph{locally checkable labeling problem}, or \\emph{LCL} for short, is a triple $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$, where $\\Sigma$ is a finite set of labels, $\\mathcal{V}$ is a subset of unordered cardinality-$\\Delta$ multisets\\footnote{Recall that a multiset is a modification of the concept of sets, where repetition is allowed.} of labels from $\\Sigma$ , and $\\mathcal{E}$ is a subset of unordered cardinality-$2$ multisets of labels from $\\Sigma$.\n\nWe call $\\mathcal{V}$ and $\\mathcal{E}$ the \\emph{vertex constraint} and \\emph{edge constraint} of $\\Pi$, respectively.\nMoreover, we call each multiset contained in $\\mathcal{V}$ a \\emph{vertex configuration} of $\\Pi$, and each multiset contained in $\\mathcal{E}$ an \\emph{edge configuration} of $\\Pi$.\n\nLet $T$ be a $\\Delta$-regular tree and $c:H(T)\\to \\Sigma$ a half-edge labeling of $T$ with labels from $\\Sigma$.\nWe say that $c$ is a \\emph{$\\Pi$-coloring}, or, equivalently, a correct solution for $\\Pi$, if, for each vertex $v$ of $T$, the multiset of labels assigned to the half-edges incident to $v$ is contained in $\\mathcal{V}$, and, for each edge $e$ of $T$, the cardinality-$2$ multiset of labels assigned to the half-edges belonging to $e$ is an element of $\\mathcal{E}$.\n\\end{definition}\n\nAn equivalent way to define our setting would be to consider $\\Delta$-regular trees as commonly defined, that is, there are vertices of degree $\\Delta$ and vertices of degree $1$, i.e., leaves.\nIn the corresponding definition of LCL one would consider leaves as unconstrained w.r.t. the vertex constraint, i.e., in the above definition of a correct solution the condition ``for each vertex $v$'' is replaced by ``for each non-leaf vertex $v$''.\nEquivalently, we could allow arbitrary trees of maximum degree $\\Delta$ as input graphs, but, for vertices of degree $< \\Delta$, we require the multiset of labels assigned to the half-edges to be extendable to some cardinality-$\\Delta$ multiset in $\\mathcal{V}$.\nWhen it helps the exposition of our ideas and is clear from the context, we may make use of these different but equivalent perspectives.\n\nWe illustrate the difference between our setting and ``standard setting'' without virtual half-edges on the perfect matching problem.\nA standard definition of the perfect matching problem is that some edges are picked in such a way that each vertex is covered by exactly one edge.\nIt is easy to see that there is no local algorithm to solve this problem on the class of finite trees (without virtual half-edges), this is a simple parity argument.\nHowever, in our setting, every vertex needs to pick exactly one half-edge (real or virtual) in such a way that both endpoints of each edge are either picked or not picked.\nWe remark that in our setting it is not difficult to see that (if $\\Delta>2$), then this problem can be solved by a local deterministic algorithm of local complexity $O(\\log(n))$.\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The $\\mathsf{LOCAL}$ model}\n\nIn this section, we define local algorithms and local complexity.\nWe discuss the general relation between the classical local complexity and the uniform local complexity. Recall that when we talk about distributed algorithm on $\\Delta$-regular trees, the algorithm has access to $n$, the size of the tree. The measure of complexity is the classical local complexity. On the other hand, when we talk about uniform distributed algorithms on $\\Delta$-regular trees, we talk about an infinite $\\Delta$-regular tree and the measure of complexity is the uniform local complexity. \nWe start with the classical notions.\nRecall that $B(v,t)$ is the $t$-hop neighborhood of a vertex $v$ in an underlying graph $G$.\n\n\\begin{definition}[Local algorithm]\n\\label{def:local_algorithm}\nA \\emph{distributed local algorithm} $\\mathcal{A}$ of local complexity $t(n)$ is a function defined on all possible $t(n)$-hop neighborhoods of a vertex. Applying an algorithm $\\mathcal{A}$ on an input graph $G$ means that the function is applied to a $t(n)$-hop neighborhood $B(u, t(n))$ of each vertex $u$ of $G$. The output of the function is a labeling of the half-edges around the given vertex. The algorithm also takes as input the size of the input graph $n$. \n\\end{definition}\n\n\\begin{definition}[Local complexity]\n\\label{def:local_complexity}\nWe say that an LCL problem $\\Pi$ has a deterministic local complexity $t(n)$ if there is a local algorithm $\\mathcal{A}$ of local complexity $t(n)$ such that when run on the input graph $G$, with each of its vertices having a unique identifier from $[n^{O(1)}]$, $\\mathcal{A}$ always returns a valid solution to $\\Pi$. \nWe also say $\\Pi \\in \\mathsf{DLOCAL}( O(t(n))$. \n\nThe randomized complexity is define analogously, but instead of unique identifiers, each vertex of $G$ has an infinite random string. The solution of $\\mathcal{A}$ needs to be a valid solution with probability $1 - 1\/n$. \nWe also say $\\Pi \\in \\mathsf{RLOCAL}( O(t(n))$. \n\\end{definition}\n\nWhenever we talk about \\emph{local} complexity on $\\Delta$-regular trees, we always tacitly think about the class of finite $\\Delta$-regular trees. We use the notation $\\mathsf{LOCAL}(O(t(n)))$ whenever the deterministic and the randomized complexity of the problems are the same up to constant factor (cf. \\cref{thm:basicLOCAL}). \n\n\\paragraph{Classification of Local Problems on $\\Delta$-regular Trees}\n\nThere is a lot of work aiming to classify all possible local complexities of LCL problems on bounded degree trees. This classification was recently finished.\nEven though the results in the literature are stated for the classical notion of finite trees, we note that it is easy to check that the same picture emerges if we restrict ourselves to $\\Delta$-regular trees according to our definition.\n\n\\begin{theorem}[Classification of local complexities of LCLs on $\\Delta$-regular trees \\cite{naorstockmeyer,chang_kopelowitz_pettie2019exp_separation,chang_pettie2019time_hierarchy_trees_rand_speedup,chang2020n1k_speedups,balliu2020almost_global_problems,balliu2019hardness_homogeneous,brandt_grunau_rozhon2021classification_of_lcls_trees_and_grids}]\n\\label{thm:basicLOCAL}\nLet $\\Pi$ be an LCL problem. Then the deterministic\/randomized local complexity of $\\Pi$ on $\\Delta$-regular trees is one of the following:\n\\begin{enumerate}\n \\item $O(1)$,\n \\item $\\Theta(\\log^*n)$,\n \\item $\\Theta(\\log n)$ deterministic and $\\Theta(\\log\\log n)$ randomized,\n \\item $\\Theta(\\log n)$,\n \\item $\\Theta(n^{1\/k})$ for $k \\in \\mathbb{N}$.\n\\end{enumerate}\n\\end{theorem}\n\nWe do not know whether the complexity classes $\\Theta(n^{1\/k})$ are non-empty on \\emph{$\\Delta$-regular} trees. The $2 \\frac{1}{2}$ coloring problems $\\mathcal{P}_k$ are known to have complexity $\\Theta(n^{1\/k})$ on \\emph{bounded-degree} trees~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}. However, as the correctness criterion of $\\mathcal{P}_k$ depends on the degree of vertices, it does not fit into the class of LCL problems that we consider in \\cref{def:LCL_trees}. \nNevertheless, as can be seen in \\cref{fig:big_picture_trees}, all classes of factors of iid and from descriptive combinatorics are already contained in the class $\\mathsf{LOCAL}(O(\\log n))$. \n\n\n\n\\paragraph{Uniform Algorithms}\nAs we mentioned before, when talking about local complexities, we always have in mind that the underlying graph is finite.\nIn particular, the corresponding algorithm knows the size of the graph.\nOn infinite $\\Delta$-regular trees, or infinite graphs in general, we use the following notion \\cite{HolroydSchrammWilson2017FinitaryColoring,Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring}. \n\n\\begin{definition}\nAn \\emph{uniform local algorithm} $\\mathcal{A}$ is a function that is defined on all possible (finite) neighborhoods of a vertex.\nFor some neighborhoods it outputs a special symbol $\\emptyset$ instead of a labeling of the half-edges around the central vertex.\nApplying $\\mathcal{A}$ on a graph $G$ means that for each vertex $u$ of $G$ the function is applied to $B(u,t)$, where $t$ is the minimal number such that $\\mathcal{A}(u,t)\\not=\\emptyset$.\nWe call $t$ the coding radius of $\\mathcal{A}$, and denote it, as a function on vertices, as $R_\\mathcal{A}$.\n\\end{definition}\n\nWe define the corresponding notion of \\emph{uniform local complexity} for infinite $\\Delta$-regular trees where each vertex is assigned an infinite random string.\n\n\\begin{definition}[Uniform local complexity \\cite{HolroydSchrammWilson2017FinitaryColoring}]\n\\label{def:uniform_complexity}\nWe say that the \\emph{uniform local (randomized) complexity} of an LCL problem $\\Pi$ is $t(\\varepsilon)$ if there is an uniform local algorithm $\\mathcal{A}$ such that the following hold on the infinite $\\Delta$-regular tree.\nRecall that $R_\\mathcal{A}$ is the random variable measuring the coding radius of a vertex $u$, that is, the distance $\\mathcal{A}$ needs to look at to decide the answer for $u$. Then, for any $0 < \\varepsilon < 1$:\n\\[\n\\P(R_\\mathcal{A} \\ge t(\\varepsilon)) \\leq \\varepsilon.\n\\]\nWe also say $\\Pi \\in \\mathsf{OLOCAL}(O(t(\\varepsilon))$.\n\\end{definition}\n\nWe finish by stating the following lemma that bounds the uniform complexity of concatenation of two uniform algorithm (we need to be little bit more careful and cannot just add the complexites up). \n\n\\begin{lemma}[Sequential Composition]\n\\label{lem:sequential_composition}\nLet $A_1$ and $A_2$ be two distributed uniform algorithms with an uniform local complexity of $t_1(\\varepsilon)$ and $t_2(\\varepsilon)$, respectively. \nLet $A$ be the sequential composition of $A_1$ and $A_2$. That is, $A$ needs to know the output that $A_2$ computes when the local input at every vertex is equal to the output of the algorithm $A_1$ at that vertex. Then, $t_A(\\varepsilon) \\leq t_{A_1} \\left( \\frac{\\varepsilon\/2}{\\Delta^{t_{A_2}(\\varepsilon\/2) + 1}}\\right) + t_{A_2}(\\varepsilon\/2)$.\n\\end{lemma}\n\\begin{proof}\nConsider some arbitrary vertex $u$. Let $E_1$ denote the event that the coding radius of $\\mathcal{A}_1$ is at most $t_{A_1} \\left( \\frac{\\varepsilon\/2}{\\Delta^{t_{A_2}(\\varepsilon\/2) + 1}}\\right)$ for all vertices in the $t_{A_2}(\\varepsilon\/2)$-hop neighborhood around $u$. As the $t_{A_2}(\\varepsilon\/2)$-hop neighborhood around $u$ contains at most $\\Delta^{t_{A_2}(\\varepsilon\/2) + 1}$ vertices, a union bound implies that $P(E_1) \\geq 1 - \\varepsilon\/2$. Moreover, let $E_2$ denote the event that the coding radius of algorithm $\\mathcal{A}_2$ at vertex $u$ is at most $t_{A_2}(\\varepsilon\/2)$. By definition, $P(E_2) \\geq 1 - \\varepsilon\/2$. Moreover, if both events $E_1$ and $E_2$ occur, which happens with probability at least $1 - \\varepsilon$, then the coding radius of algorithm $\\mathcal{A}$ is at most $t_{A_1} \\left( \\frac{\\varepsilon\/2}{\\Delta^{t_{A_2}(\\varepsilon\/2) + 1}}\\right) + t_{A_2}(\\varepsilon\/2)$, thus finishing the proof.\n \\end{proof}\n\n\n\\subsection{Descriptive combinatorics}\n\\label{subsec:descriptivecombinatorics}\n\nBefore we define formally the descriptive combinatorics complexity classes, we give a high-level overview on their connection to distributing computing for the readers more familiar with the latter.\n\nThe complexity class that partially captures deterministic local complexity classes is called $\\mathsf{BOREL}$ (see also Remark \\ref{rem:cont}).\nFirst note that by a result of Kechris, Solecki and Todor\\v{c}evi\\'c \\cite{KST} the \\emph{maximal independent set problem} (with any parameter $r\\in \\mathbb{N}$) is in this class for any bounded degree graph.\\footnote{That is, it is possible to find a Borel maximal independent set, i.e., a maximal independent set which is, moreover, a Borel subset of the vertex set.}\nIn particular, this yields that $\\mathsf{BOREL}$ contains the class $\\mathsf{LOCAL}(O(\\log^* n))$ by the characterization of \\cite{chang_kopelowitz_pettie2019exp_separation}, see \\cite{Bernshteyn2021LLL}. Moreover, as mentioned before, $\\mathsf{BOREL}$ is closed under countably many iterations of the operations of finding maximal independent set (for some parameter that might grow) and of applying a constant local rule that takes into account what has been constructed previously.\\footnote{It is in fact an open problem, whether this captures fully the class $\\mathsf{BOREL}$. However, note that an affirmative answer to this question would yield that problems can be solved in an ``effective\" manner in the Borel context, which is known not to be the case in unbounded degree graphs \\cite{todorvcevic2021complexity}.}\n\nTo get a better grasp of what this means, consider for example the proper vertex $2$-coloring problem on half-lines. It is clear that no local algorithm can solve this problem.\nHowever, as it is possible to determine the starting vertex after countably many iterations of the maximal independent set operation, we conclude that this problem is in the class $\\mathsf{BOREL}$.\nThe idea that $\\mathsf{BOREL}$ can compute some unbounded, global, information will be implicitly used in all the construction in \\cref{sec:separating_examples} that separate $\\mathsf{BOREL}$ from local classes.\n\nThe intuition behind the class $\\mathsf{MEASURE}$ is that it relates in the same way to the class $\\mathsf{BOREL}$, as randomized local algorithms relate to deterministic ones.\nIn particular, the operations that are allowed in the class $\\mathsf{MEASURE}$ are the same as in the class $\\mathsf{BOREL}$ but the solution of a given LCL can be incorrect on a measure zero set.\n\nThe class $\\mathsf{BAIRE}$ can be considered as a topological equivalent of the measure theoretic class $\\mathsf{MEASURE}$, that is, a solution can be incorrect on a topologically negligible set.\nThe main difference between the classes $\\mathsf{MEASURE}$ and $\\mathsf{BAIRE}$ is that in the later there is a hierarchical decomposition that is called \\emph{toast}.\n(Note that this phenomenon is present in the case of $\\mathsf{MEASURE}$ exactly on so-called amenable graphs. It is also tightly connected with the notion of hyperfiniteness \\cite{conleymillerbound,gao2015forcing}.)\nThe independence of colorings on a tree together with this structure allows for a combinatorial characterization of the class $\\mathsf{BAIRE}$, which was proven by Bernshteyn \\cite{Bernshteyn_work_in_progress}, see also \\cref{sec:baire}.\n\n\n\n\nNext we formulate the precise definitions.\nWe refer the reader to \\cite{pikhurko2021descriptive_comb_survey,kechris_marks2016descriptive_comb_survey,Bernshteyn2021LLL,kechrisclassical}, or to \\cite[Introduction,~Section~4.1]{grebik_rozhon2021LCL_on_paths} and \\cite[Section~7.1,~7.2]{grebik_rozhon2021toasts_and_tails} for intuitive examples and standard facts of descriptive set theory.\nIn particular, we do not define here the notion standard Borel\/probability space, a Polish topology, a Borel probability measure, Baire property etc.\n\nLet $\\mathcal{G}$ be a Borel graph of bounded maximum degree on a standard Borel space $X$.\nIn this paper we consider exclusively acyclic $\\Delta$-regular Borel graphs and we refer to them as \\emph{$\\Delta$-regular Borel forests}.\nIt is easy to see that the set of half-edges (see Definition \\ref{def:half-edge}) is naturally a standard Borel space, we denote this set by $H(\\mathcal{G})$.\nThus, it makes sense to consider Borel labelings of $H(\\mathcal{G})$.\nMoreover, if $\\mathcal{G}$ is a $\\Delta$-regular Borel forest and $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ is an LCL, we can also decide whether a coloring $f:\\mathcal{H}(\\mathcal{G})\\to \\Sigma$ is a solution to $\\Pi$ as in Definition \\ref{def:LCL_trees}. \nSimilarly, we say that the coloring $f$ solves $\\Pi$, e.g., on a $\\mu$-conull set for some Borel probability measure $\\mu$ on $X$ if there is a Borel set $C\\subseteq X$ such that $\\mu(C)=1$, the vertex constraints are satisfied around every $x\\in C$ and the edge constraints are satisfied for every $x,y\\in C$ that form an edge in $\\mathcal{G}$.\n\n\\begin{definition}[Descriptive classes]\n\\label{def:descriptive}\nLet $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ be an LCL.\nWe say that $\\Pi$ is in the class $\\mathsf{BOREL}$ if for every acyclic $\\Delta$-regular Borel graph $\\mathcal{G}$ on a standard Borel space $X$, there is a Borel function $f:H(\\mathcal{G})\\to \\Sigma$ that is a $\\Pi$-coloring of $\\mathcal{G}$.\n\nWe say that $\\Pi$ is in the class $\\mathsf{BAIRE}$ if for every acyclic $\\Delta$-regular Borel graph $\\mathcal{G}$ on a standard Borel space $X$ and every compatible Polish topology $\\tau$ on $X$, there is a Borel function $f:H(\\mathcal{G})\\to \\Sigma$ that is a $\\Pi$-coloring of $\\mathcal{G}$ on a $\\tau$-comeager set.\n\nWe say that $\\Pi$ is in the class $\\mathsf{MEASURE}$ if for every acyclic $\\Delta$-regular Borel graph $\\mathcal{G}$ on a standard Borel space $X$ and every Borel probability measure $\\mu$ on $X$, there is a Borel function $f:H(\\mathcal{G})\\to \\Sigma$ that is a $\\Pi$-coloring of $\\mathcal{G}$ on a $\\mu$-conull set.\n\\end{definition}\n\nThe following claim follows directly from the definition.\n\n\\begin{claim}\\label{cl:basicDC}\nWe have $\\mathsf{BOREL}\\subseteq \\mathsf{MEASURE}, \\mathsf{BAIRE}$.\n\\end{claim}\n\nRecently Bernshteyn \\cite{Bernshteyn2021LLL,Bernshteyn2021local=cont} proved several results that connect distributed computing with descriptive combinatorics.\nUsing the language of complexity classes we can formulate some of the result as inclusions in \\cref{fig:big_picture_trees}.\n\n\\begin{theorem}[\\cite{Bernshteyn2021LLL}]\nWe have\n\\begin{itemize}\n \\item $\\mathsf{LOCAL}(O(\\log^*(n)))\\subseteq \\mathsf{BOREL}$,\n \\item $\\mathsf{RLOCAL}(o(\\log(n))) \\subseteq \\mathsf{MEASURE}, \\mathsf{BAIRE}$.\n\\end{itemize}\nIn fact, these inclusions hold for any reasonable class of bounded degree graphs.\n\\end{theorem}\n\n\\begin{remark}\n\\label{rem:cont}\nThe ``truly'' local class in descriptive combinatorics is the class $\\mathsf{CONTINUOUS}$.\nEven though we do not define this class here\\footnote{To define the class $\\mathsf{CONTINUOUS}$, rather than asking for a continuous solution on all possible $\\Delta$-regular Borel graphs, one has to restrict to a smaller family of graphs, otherwise the class trivializes. To define precisely this family is somewhat inconvenient, and not necessary for our arguments.}, and we refer the reader to \\cite{Bernshteyn2021local=cont,GJKS,grebik_rozhon2021toasts_and_tails} for the definition and discussions in various cases, we mention that the inclusion \\ \\[\\mathsf{LOCAL}(O(\\log^* n))\\subseteq \\mathsf{CONTINUOUS} \\tag{*}\\] holds in most reasonable classes of bounded degree graphs, see \\cite{Bernshteyn2021LLL}.\nThis also applies to our situation\nRecently, it was shown by Bernshteyn \\cite{Bernshteyn2021local=cont}, and independently by Seward \\cite{Seward_personal}, that $(*)$ can be reversed for Cayley graphs of finitely generated groups.\nThis includes, e.g., natural $\\Delta$-regular trees with proper edge $\\Delta$-coloring, as this is a Cayely graph of free product of $\\Delta$-many $\\mathbb{Z}_2$ induced by the standard generating set.\nIt is, however, not clear whether it $(*)$ can be reversed in our situation, i.e., $\\Delta$-regular trees without any additional labels.\n\n\n\n\\end{remark}\n\n\n\n\n\n\n\\subsection{Random processes}\n\nWe start with an intutitve description of fiid processes. Let $T_\\Delta$ be the infinite $\\Delta$-regular tree.\nInformally, \\emph{factors of iid processes (fiid)} on $T_\\Delta$ are infinite analogues of local randomized algorithms in the following way.\nLet $\\Pi$ be an LCL and $u\\in T_\\Delta$.\nIn order to solve $\\Pi$, we are allowed to explore the whole graph, and the random strings that are assigned to vertices, and then output a labeling of half-edges around $u$.\nIf such an assignment is a measurable function and produces a $\\Pi$-coloring almost surely, then we say that $\\Pi$ is in the class $\\mathsf{fiid}$.\nMoreover, if every vertex needs to explore only finite neighborhood to output a solution, then we say that $\\Pi$ is in the class $\\mathsf{ffiid}$.\nSuch processes are called \\emph{finitary} fiid (\\emph{ffiid}). There is also an intimate connection between ffiid and uniform algorithms.\nThis is explained in \\cite[Section 2.2]{grebik_rozhon2021toasts_and_tails}.\nInformally, an ffiid process that almost surely solves $\\Pi$ is, in the language of distributed computing, an uniform local algorithm that solves $\\Pi$.\nThis allows us to talk about uniform local complexity of an ffiid.\nIn the rest of the paper we interchange both notions freely with slight preference for the distributed computing language.\n\nNow we define formally these classes, using the language of probability.\nWe denote by $\\operatorname{Aut}(T_\\Delta)$ the automorphism group of $T_\\Delta$.\nAn \\emph{iid process} on $T_\\Delta$ is a collection of iid random variables $Y=\\{Y_v\\}_{v\\in V(T_\\Delta)}$ indexed by vertices, or edges, half-edges etc, of $T_\\Delta$ such that their joint distribution is invariant under $\\operatorname{Aut}(T_\\Delta)$.\nWe say that $X$ is a \\emph{factor of iid process (fiid)} if $X=F(Y)$, where $F$ is a measurable $\\operatorname{Aut}(T_\\Delta)$-invariant map and $Y$ is an iid process on $T_\\Delta$.\\footnote{We always assume $Y=(2^\\mathbb{N})^{T_\\Delta}$ endowed with the product Lebesgue measure.}\nMoreover, we say that $X$ is a \\emph{finitary factor if iid process (ffiid)} if $F$ depends with probability $1$ on a finite (but random) radius around each vertex.\nWe denote as $R_F$ the random variable that assigns minimal such radius to a given vertex, and call it the \\emph{coding radius} of $F$.\nWe denote the space of all $\\Pi$-colorings of $T_\\Delta$ as $X_\\Pi$.\nThis is a subspace of $\\Sigma^{H(T_\\Delta)}$ that is invariant under $\\operatorname{Aut}(T_\\Delta)$.\n\n\\begin{definition}\nWe say that an LCL $\\Pi$ is in the class $\\mathsf{fiid}$ ($\\mathsf{ffiid}$) if there is an fiid (ffiid) process $X$ that almost surely produces elements of $X_\\Pi$.\n\nEquivalently, we can define $\\mathsf{ffiid} = \\bigcup_f \\mathsf{OLOCAL}(f(\\varepsilon))$ where $f$ ranges over all functions. That is, $\\mathsf{ffiid}$ is the class of problems solvable by any uniform distributed algorithm. \n\n\\end{definition}\n\n\nIt is obvious that $\\mathsf{ffiid}\\subseteq \\mathsf{fiid}$.\nThe natural connection between descriptive combinatorics and random processes is formulated by the inclusion $\\mathsf{MEASURE}\\subseteq \\mathsf{fiid}$.\nWhile this inclusion is trivially satisfied, e.g., in the case of $\\Delta$-regular trees with proper edge $\\Delta$-coloring, in our situation we need a routine argument that we include for the sake of completeness in \\cref{app:ktulu}.\n\n\\begin{restatable}{lemma}{FIIDinMEASURE}\\label{thm:FIIDinMEASURE}\nLet $\\Pi$ be an LCL such that $\\Pi\\in \\mathsf{MEASURE}$.\nThen $\\Pi\\in \\mathsf{fiid}$.\n\\end{restatable}\n\n\n\n\n\\subsection{Specific Local Problems}\n\nHere we list some well-known local problems that were studied in all three considered areas. We describe the results known about these problems with respect to classes from \\cref{fig:big_picture_trees}. \n\n\\paragraph{Edge Grabbing}\nWe start by recalling the well-known problem of edge grabbing (a close variant of the problem is known as sinkless orientation\\cite{brandt_etal2016LLL}). \nIn this problem, every vertex should mark one of its half-edges (that is, grab an edge) in such a way that no edge can be marked by two vertices. \nIt is known that $\\Pi_{\\text{edgegrab}} \\not\\in \\mathsf{LOCAL}(O(\\log^* n))$ \\cite{brandt_etal2016LLL} but $\\Pi_{\\text{edgegrab}} \\in \\mathsf{RLOCAL}(O(\\log\\log n))$.\n\nSimilarly, $\\Pi_{\\text{edgegrab}} \\not \\in \\mathsf{BOREL}$ by \\cite{DetMarks}, but $\\Pi_{\\text{edgegrab}}\\in \\mathsf{MEASURE}$ by \\cite{BrooksMeas}: to see the former, just note that if a $\\Delta$-regular tree admits proper $\\Delta$-colorings of both edges and vertices, every vertex can grab an edge of the same color as the color of the vertex. Thus, $\\Pi_{\\text{edgegrab}}\\in \\mathsf{BOREL}$ would yield that $\\Delta$-regular Borel forests with Borel proper edge-colorings admit a Borel proper $\\Delta$-coloring, contradicting \\cite{DetMarks}.\n\nThis completes the complexity characterization of $\\Pi_{\\text{edgegrab}}$ as well as the proper vertex $\\Delta$-coloring with respect to classes in \\cref{fig:big_picture_trees}. \n\n\n\n\n\n\\paragraph{Perfect Matching}\nAnother notorious LCL problem, whose position in \\cref{fig:big_picture_trees} is, however, not completely understood, is the perfect matching problem $\\Pi_{\\text{pm}}$.\nRecall that the perfect matching problem $\\Pi_{\\text{pm}}$ asks for a matching that covers all vertices of the input tree.\\footnote{In our formalism this means that around each vertex exactly one half-edge is picked.}\nIt is known that $\\Pi_{\\text{pm}} \\in \\mathsf{fiid}$ \\cite{lyons2011perfect}, and it is easy to see that $\\Pi_{\\text{pm}} \\not\\in \\mathsf{RLOCAL}(O(\\log\\log(n)))$ (we will prove a stronger result in \\cref{pr:AddingLineLLL}).\nMarks proved \\cite{DetMarks} that it is not in $\\mathsf{BOREL}$, even when the underlying tree admits a Borel bipartition.\nIt is not clear if $\\Pi_{\\text{pm}}$ is in $\\mathsf{ffiid}$, nor whether it is in $\\mathsf{MEASURE}$.\n\n\n\n\n\n\\paragraph{Graph Homomorphism}\nWe end our discussion with LCLs that correspond to graph homomorphisms (see also the discussion in Section \\ref{sec:marks}).\nThese are also known as edge constraint LCLs.\nLet $G$ be a finite graph.\nThen we define $\\Pi_G$ to be the LCL that asks for a homomorphism from the input tree to $G$, that is, vertices are labeled with vertices of $G$ in such a way that edge relations are preserved.\nThere are not many positive results except for the class $\\mathsf{BAIRE}$.\nIt follows from the result of Bernshteyn \\cite{Bernshteyn_work_in_progress} (see \\cref{sec:baire}) that $\\Pi_G\\in \\mathsf{BAIRE}$ if and only if $G$ is not bipartite.\nThe main negative results can be summarized as follows.\nAn easy consequence of the result of Marks \\cite{DetMarks} is that if $\\chi(G)\\le \\Delta$, then $\\Pi_G\\not \\in \\mathsf{BOREL}$.\nIn this paper, we describe examples of graphs of chromatic number up to $2\\Delta-2$ such that the corresponding homomorphism problem is not in $\\mathsf{BOREL}$, see \\cref{sec:marks}.\nThe theory of \\emph{entropy inequalities} see \\cite{backhausz} implies that if $G$ is a cycle on more than $9$ vertices, then $\\Pi_G\\not\\in \\mathsf{fiid}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Generalization of Marks' technique}\n\\label{sec:marks}\n\nIn this section, we first develop a new way of proving lower bounds in the $\\mathsf{LOCAL}$ model based on a generalization of a technique of Marks \\cite{DetMarks}. Then, we use ideas arising in the finitary setting---connected to the adaptation of Marks' technique---to obtain new results back in the Borel context.\nFor an introduction to Marks' technique and a high-level discussion about the challenges in adapting the technique to the standard distributed setting as well as our solution via the new notion of an ID graph, we refer the reader to \\cref{sec:intromarks}.\n\n\\begin{comment}\nIn \\cite{DetMarks}, Marks showed that there exist $\\Delta$-regular Borel forests that do not admit a Borel proper $\\Delta$-coloring. As it was indicated above, this is a striking difference between $\\mathsf{MEASURE}$ or $\\mathsf{BAIRE}$ and $\\mathsf{BOREL}$, as such colorings exist in the former classes, see \\cite{BrooksMeas}. Since prior to Marks' breakthrough (essentially) any lower bound on Borel chromatic numbers utilized measure theoretic or Baire category arguments, his approach was (and had to be) based on a new technique: \\emph{determinacy of infinite games}. \n\nFix a payoff set $A \\subseteq \\mathbb{R}$. Consider the infinite game in which two players alternatingly determine digits of a real number. The first player wins iff the resulting real is in $A$. A rather counter-intuitive fact of set theory is that there exists a set $A$ for which none of the players has a winning strategy, i.e., the game is not determined. However, a deep theorem of Martin \\cite{martin} says that if $A$ is Borel, then one of the players does have such a strategy. \nMarks has constructed $\\Delta$-regular Borel forests and games with Borel payoff sets, for which the existence of a winning strategy for either of the players contradicts the existence of a Borel proper $\\Delta$-coloring. \n\nIn what follows, we generalize and adapt his games to the finitary setting, thus obtaining new lower bounds for LOCAL algorithms; then we use ideas inspired by the LOCAL technique of ID graphs to prove a new theorem in the Borel context. \n\nNote that Marks considered Borel versions of certain locally checkable problems (e.g., coloring, edge grabbing, perfect matching). As discussed above, for these problems, lower bounds were independently obtained via the round elimination technique in the LOCAL context, see e.g., \\cite{brandt19automatic}\\todo{cite more stuff??-Z}. Remarkably, the core ideas behind Brandt's and Marks' argument exhibit a surprising degree of similarity. \n\n\\todo{add the story about going from borel to local, and then back from local to borel}\n\nHowever, our extension of Marks' technique yields lower bounds for fairly natural problems that were resistant to round elimination.\nMoreover, as with our approach the lower bound is obtained by checking whether the considered LCL satisfies a certain combinatorial property called \\emph{playability condition} (see Definition~\\ref{def:Playability}), we can use it to obtain lower bounds for problems that have complicated descriptions---something that the round elimination technique in its current form is ill-suited for. \\todo{Maybe this sentence should rather be somewhere in the intro?}\n\n\n\n\\end{comment}\n\n\n\n\n\n\n\nThe setting we consider in this section is $\\Delta$-regular trees that come with a proper $\\Delta$-edge coloring with colors from $[\\Delta]$.\nAll lower bounds discussed already hold under these restrictions (and therefore also in any more general setting).\n\nRecall that an LCL $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ is given by specifying a list of allowed vertex configurations and a list of allowed edge configurations (see Definition \\ref{def:LCL_trees}).\nTo make our lower bounds as widely applicable as possible, we replace the latter list by a separate list for each of the $\\Delta$ edge colors; in other words, we consider LCLs where the correctness of the output is allowed to depend on the input that is provided by the edge colors.\nHence, formally, in this section, an LCL is more generally defined: it is a tuple $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ where $\\Sigma$ and $\\mathcal{V}$ are as before, while $\\mathcal{E} = (\\mathcal{E}_{\\alpha})_{\\alpha \\in [\\Delta]}$ is now a $\\Delta$-tuple of sets $\\mathcal{E}_{\\alpha}$ consisting of cardinality-$2$ multisets.\nSimilarly as before, a half-edge labeling (see Definition \\ref{def:descriptive} for the Borel context) with labels from $\\Sigma$ is a correct solution for $\\Pi$ if, for each vertex $v$, the multiset of labels assigned to the half-edges incident to $v$ is contained in $\\mathcal{V}$, and, for each edge $e$, the multiset of labels assigned to the half-edges belonging to $e$ is contained in $\\mathcal{E}_{\\alpha}$, where $\\alpha$ is the color of the edge $e$.\n\nThe idea of our approach is to identify a condition that an LCL $\\Pi$ necessarily has to satisfy if it is solvable in $O(\\log^* n)$ rounds in the $\\mathsf{LOCAL}$ model.\nShowing that $\\Pi$ does not satisfy this condition then immediately implies an $\\Omega(\\log n)$ deterministic and $\\Omega(\\log \\log n)$ randomized lower bound, by Theorem~\\ref{thm:basicLOCAL}.\n\nIn order to define our condition we need to introduce the notion of a configuration graph: a \\emph{configuration graph} is a $\\Delta$-tuple $\\mathbb{P}=(\\mathbb{P}_\\alpha)_{\\alpha\\in \\Delta}$ of graphs, where the vertex set of each of the graphs $\\mathbb{P}_\\alpha$ is the set of subsets of $\\Sigma$, and there is an edge in $\\mathbb{P}_\\alpha$ connecting two vertices $S,T$ if and only if there are $\\mathtt{a}\\in S$ and $\\mathtt{b}\\in T$ such that $\\{\\mathtt{a},\\mathtt{b}\\}\\in {\\mathcal{E}}_\\alpha$, note that loops are allowed.\n(Naturally, we will consider any two vertices of different $\\mathbb{P}_\\alpha$ to be distinct, even if they correspond to the same subset of $\\Sigma$.)\n\nNow we are set to define the aforementioned condition. The intuition behind the playability condition is the following: assume that there exists a local algorithm $\\mathcal{A}$ that solves $\\Pi$ using the $t$ neighbourhood of a given vertex. We are going to define a family of two player games. The game will depend on some $S \\subseteq \\Sigma$. $\\operatorname{Alice}$ and $\\operatorname{Bob}$ will assign labels (or IDs) to the vertices in the $t$-neighbourhood (in some way specified later on, depending on $\\alpha \\in [\\Delta]$). When the assignment is complete, we evaluate $\\mathcal{A}$ on the root of the obtained labelled graph, this way obtaining an element $s$ of $\\Sigma$, and decide who is the winner based on $s \n\\in S$ or not. Naturally, it depends on $S$ and $\\alpha$, which player has a winning strategy. This gives rise to a two coloring of vertices of $\\mathbb{P}_\\alpha$ by colors $\\operatorname{Alice}$ and $\\operatorname{Bob}$. The failure of the playability condition will guarantee that using a strategy stealing argument one can derive a contradiction.\n\n\n\n\n\\begin{definition}[Playability condition]\\label{def:Playability}\nWe say that an LCL $\\Pi$ is \\emph{playable} if for every $\\alpha\\in [\\Delta]$ there is a coloring $\\Lambda_{\\alpha}$ of the vertices of $\\mathbb{P}_{\\alpha}$ with two colors $\\{\\operatorname{Alice},\\operatorname{Bob}\\}$ such that the following conditions are satisfied:\n\\begin{enumerate}\n \\item [(A)] For any tuple $(S_{\\alpha})_{\\alpha \\in [\\Delta]} \\in V(\\mathbb{P}_1) \\times \\dots \\times V(\\mathbb{P}_{\\Delta})$ satisfying $\\Lambda_\\alpha(S_\\alpha)=\\operatorname{Alice}$ for each $\\alpha \\in [\\Delta]$, there exist $\\mathtt{a}_\\alpha\\not\\in S_\\alpha$ such that $\\{\\mathtt{a}_\\alpha\\}_{\\alpha\\in \\Delta}\\in {\\mathcal{V}}$, and\n \\item [(B)] for any $\\alpha \\in [\\Delta]$, and any tuple $(S,T) \\in V(\\mathbb{P}_{\\alpha}) \\times V(\\mathbb{P}_{\\alpha})$ satisfying $\\Lambda_\\alpha(S)=\\Lambda_\\alpha(T)=\\operatorname{Bob}$, we have that $(S,T)$ is an edge of $\\mathbb{P}_{\\alpha}$.\n\\end{enumerate}\n\n\\end{definition}\n\nOur aim in this section is to show the following general results.\n\n\\begin{theorem}\\label{th:MainLOCAL}\nLet $\\Pi$ be an LCL that is not playable.\nThen $\\Pi$ is not in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\n\\end{theorem}\n\nUsing ideas from the proof of this result, we can formulate an analogous theorem in the Borel context.\nLet us mention that while \\cref{th:MainLOCAL} is a consequence of \\cref{thm:MainBorel} by \\cite{Bernshteyn2021LLL}, we prefer to state the theorems separately.\nThis is because the proof of the Borel version of the theorem uses heavily complex results such as the Borel determinacy theorem, theory of local-global limits and some set-theoretical considerations about measurable sets.\nThis is in a stark contrast with the proof of \\cref{th:MainLOCAL} that uses `merely' the existence of large girth graphs and determinacy of finite games.\nHowever, the ideas surrounding these concepts are the same in both cases.\n\n\\begin{theorem}\\label{thm:MainBorel}\nLet $\\Pi$ be an LCL that is not playable\nThen $\\Pi$ is not in the class $\\mathsf{BOREL}$.\n\\end{theorem}\n\nThe main application of these abstract results is to find lower bounds for \\emph{(graph) homomorphism LCLs} aka \\emph{edge constraint LCLs}.\nWe already defined this class in \\cref{subsec:local_problems} but we recall the definition in the formalism of this section to avoid any confusion.\nLet $G$ be a finite graph.\nThen $\\Pi_G=(\\Sigma,\\mathcal{V},\\mathcal{E})$ is defined by letting\n\\begin{enumerate}\n\\item \\label{p:vertex} ${\\mathcal{V}}=\\{({\\mathtt{a}},\\dots,{\\mathtt{a}}):\\mathtt{a}\\in \\Sigma\\}$,\n\\item $\\Sigma=V(G)$,\n\\item $\\forall \\alpha \\in [\\Delta] \\ (\\mathcal{E}_\\alpha=E(G)).$\n\\end{enumerate}\nAn LCL for which \\eqref{p:vertex} holds is called a \\emph{vertex-LCL}. There is a one-to-one correspondence between vertex-LCLs for which $\\forall \n\\alpha,\\beta \\ (\\mathcal{E}_\\alpha=\\mathcal{E}_\\beta)$ and LCLs of the form $\\Pi_G$. (Indeed, to vertex-LCLs with this property one can associate a graph $G$ whose vertices are the labels in $\\Sigma$ and where two vertices $\\ell, \\ell'$ are connected by an edge if and only if $\\{ \\ell, \\ell' \\} \\in \\mathcal{E}$.)\nNote that if $\\Pi$ is a vertex-LCL, then condition (A) in Definition~\\ref{def:Playability} is equivalent to the statement that no tuple $\\{S_\\alpha\\}_{\\alpha\\in \\Delta}$ that satisfies the assumption of (A) can cover $\\Sigma$, i.e., that $\\Sigma \\nsubseteq S_1 \\cup \\dots \\cup S_{\\Delta}$ for such a tuple.\nMoreover, if $\\Pi_G$ is a homomorphism LCL, then $\\mathbb{P}_\\alpha=\\mathbb{P}_{\\beta}$ for every $\\alpha,\\beta\\in \\Delta$.\n\nThe simplest example of a graph homomorphism LCL is given by setting $G$ to be the clique on $k$ vertices; the associated LCL $\\Pi_G$ is simply the problem of finding a proper $k$-coloring of the vertices of the input graph.\nNow we can easily derive Marks' original result from our general theorem.\n\n\\begin{corollary}[Marks \\cite{DetMarks}]\\label{cor:detmarks}\nLet $G$ be a finite graph that has chromatic number at most $\\Delta$. Then $\\Pi_G$ is not playable. In particular, there is a $\\Delta$-regular Borel forest that have Borel chromatic number $\\Delta+1$. \n\\end{corollary}\n\\begin{proof}\nLet $A_1,\\dots, A_\\Delta$ be some independent sets that cover $G$, and $\\Lambda_1, \\dots, \\Lambda_{\\Delta}$ arbitrary colorings of the vertices of $\\mathbb{P}_1, \\dots, \\mathbb{P}_{\\Delta}$, respectively, with colors from $\\{\\operatorname{Alice},\\operatorname{Bob}\\}$.\nIt follows that $\\Lambda_\\alpha(A_\\alpha)=\\operatorname{Alice}$ for every $\\alpha\\in \\Delta$, since otherwise condition (B) in Definition~\\ref{def:Playability} is violated with $S=T=A_\\alpha$.\nBut then condition (A) does not hold.\n\\end{proof}\n\nAs our main application we describe graphs with chromatic number larger than $\\Delta$ such that $\\Pi_G$ is not playable.\nThis rules out the hope that the complexity of $\\Pi_G$ is connected with chromatic number being larger than $\\Delta$.\nIn Section \\ref{subsec:homlcls} we show the following.\nNote that the case $k=\\Delta$ is \\cref{cor:detmarks}.\n\n\\begin{theorem}\n\\label{th:homomorphism}\nLet $\\Delta > 2$ and $\\Delta < k \\leq 2\\Delta-2$. There exists a graph $G_k$ with $\\chi(G_k)= k$, such that $\\Pi_{G_k}$ is not playable and $\\Pi_{G_k} \\in \\mathsf{RLOCAL}(O(\\log\\log n ))$. In particular, $\\Pi_{G_k}\\not\\in \\mathsf{BOREL}$ and $\\Pi_{G_k}\\not\\in\\mathsf{LOCAL}(O(\\log^*(n)))$.\n\\end{theorem}\n\nInterestingly, recent results connected to counterexamples to Hedetniemi's conjecture yield the same statement asymptotically, as $\\Delta \\to \\infty$ (see Remark \\ref{r:hedet}).\n\n\\begin{remark}\nIt can be shown that for $\\Delta=3$ both the Chv\\' atal and Gr\\\" otsch graphs are suitable examples for $k=4$.\n\\end{remark}\n\n\\begin{remark}\nAs another application of his technique Marks showed in \\cite{DetMarks} that there is a Borel $\\Delta$-regular forest that does not admit Borel perfect matching.\nThis holds even when we assume that the forest is Borel bipartite, i.e., it has Borel chromatic number $2$.\nIn order to show this result Marks works with free joins of colored hyperedges, that is, Cayley graphs of free products of cyclic groups.\nOne should think of two types of triangles ($3$-hyperedges) that are joined in such a way that every vertex is contained in both types and there are no cycles.\nWe remark that the playability condition can be formulated in this setting.\nSimilarly, one can derive a version of \\cref{thm:MainBorel}.\nHowever, we do not have any application of this generalization.\n\\end{remark}\n\n\n\n\n\n\n\n\n\\subsection{Applications of playability to homomorphism LCLs}\\label{subsec:homlcls}\n\nIn this section we find the graph examples from \\cref{th:homomorphism}.\nFirst we introduce a condition $\\Delta$-(*) that is a weaker condition than having chromatic number at most $\\Delta$, but still implies that the homomorphism LCL is not playable. Then, we will show that--similarly to the way the complete graph on $\\Delta$-many vertices, $K_\\Delta$, is maximal among graphs of chromatic number $\\leq \\Delta$--there exists a maximal graph (under homorphisms) with property $\\Delta$-(*). Recall that we assume $\\Delta >2$.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=.6\\textwidth]{fig\/h3.pdf}\n \\caption{The maximal graph with the property $\\Delta$-(*) for $\\Delta=3$.\n \\label{fig:h3}\n\\end{figure}\n\n\n\n\n\\begin{definition}[Property $\\Delta$-(*)]\nLet $\\Delta>2$ and $G=(V,E)$ be a finite graph.\nWe say that $G$ satisfies property $\\Delta$-(*) if there are sets $S_0,S_1\\subseteq V$ such that $G$ restricted to $V \\setminus S_i$ has chromatic number at most $(\\Delta-1)$ for $i\\in \\{0,1\\}$, and there is no edge between $S_0$ and $S_1$.\n\\end{definition}\n\nNote that $\\chi(G) \\leq \\Delta$ implies $\\Delta$-(*): indeed, if $A_1,\\dots,A_\\Delta$ are independent sets that cover $V(G)$, we can set $S_0=S_1=A_1$.\n\nOn the other hand, we claim that if $G$ satisfies $\\Delta$-(*) then $\\chi(G)\\le 2\\Delta-2$. In order to see this, take $S_0,S_1 \\subseteq V(G)$ witnessing $\\Delta$-(*). Then, as there is no edge between $S_0$ and $S_1$, so in particular, between $S_0 \\setminus S_1$ and $S_0 \\cap S_1$, it follows that the chromatic number of $G$'s restriction to $S_0$ is $\\leq \\Delta-1$. But then we can construct proper $\\Delta-1$-colorings of $S_0$ and $V(G) \\setminus S_0$, which shows our claim.\n\n\n\\begin{proposition}\\label{pr:CONdition*}\nLet $G$ be a graph satisfying $\\Delta$-(*).\nThen $\\Pi_G$ is not playable.\n\\end{proposition}\n\\begin{proof}\nFix $S_0,S_1$ as in the definition of $\\Delta$-(*) and assume for a contradiction that colorings $\\Lambda_1, \\dots, \\Lambda_\\Delta$ as described in Definition~\\ref{def:Playability} exist.\nBy Property $\\Delta$-(*), there exist independent sets $A_1, \\dots, A_{\\Delta-1}$ such that $S_0$ together with the $A_i$ covers $G$, i.e., such that $S_0 \\cup A_1 \\cup \\dots \\cup A_{\\Delta-1} = V(G)$.\nFor each $\\alpha \\in [\\Delta-1]$, we must have $\\Lambda_\\alpha(A_{\\alpha})=\\operatorname{Alice}$, since otherwise condition (B) in Definition~\\ref{def:Playability} is violated.\nConsequently $\\Lambda_{\\Delta}(S_0)=\\operatorname{Bob}$, otherwise condition (A) is violated.\nSimilarly $\\Lambda_\\Delta(S_1)=\\operatorname{Bob}$.\nThis shows that $\\Lambda_\\Delta$ does not satisfy condition (B) with $S = S_0$ and $T = S_1$.\n\\end{proof}\n\nNext we describe maximal examples of graphs that satisfy the condition $\\Delta$-(*). That is, we define a graph $H_\\Delta$ that satisfies $\\Delta$-(*), its chromatic number is $2\\Delta-2$ and every other graph that satisfies $\\Delta$-(*) admits a homomorphism in $H_\\Delta$.\nThe graph $H_{3}$ is depicted in \\cref{fig:h3}.\n\nRecall that the (categorical) product $G \\times H$ of graphs $G,H$ is the graph on $V(G) \\times V(H)$, such that $((g,h),(g',h')) \\in E(G \\times H)$ iff $(g,g') \\in E(G)$ and $(h,h') \\in E(H)$.\n \n Write $P$ for the product $K_{\\Delta-1}\\times K_{\\Delta-1}$. Let $V_0$ and $V_1$ be vertex disjoint copies of $K_{\\Delta-1}$.\nWe think of vertices in $V_i$ and $P$ as having labels from $[\\Delta-1]$ and $[\\Delta-1] \\times [\\Delta-1]$, respectively.\nThe graph $H_\\Delta$ is the disjoint union of $V_0$, $V_1$, $P$ and an extra vertex $\\dagger$ that is connected by an edge to every vertex in $P$, and additionally, if $v$ is a vertex in $V_0$ with label $i\\in [\\Delta-1]$, then we connect it by an edge with $(i',j)\\in P$ for every $i'\\not=i$ and $j\\in [\\Delta-1]$, and if $v$ is a vertex in $V_1$ with label $j\\in \\Delta-1$, then we connect it by an edge with $(i,j')\\in P$ for every $j'\\not=j$ and $i \\in [\\Delta-1]$.\n\n\n\\begin{proposition}\\label{pr:MinimalExample}\n\\begin{enumerate}\n \\item \\label{prop:hdelta}$H_\\Delta$ satisfies $\\Delta$-(*).\n \\item \\label{prop:hdeltachrom} $\\chi(H_\\Delta)=2\\Delta-2$.\n \\item \\label{prop:hdeltamax} A graph $G$ satisfies $\\Delta$-(*) if and only if it admits a homomorphism to $H_\\Delta$.\n\\end{enumerate}\n\n\\end{proposition}\n\\begin{proof}\n\\eqref{prop:hdelta} Set $S_0=V(V_0)\\cup \\{\\dagger\\}$ and $S_1=V(V_1)\\cup \\{\\dagger\\}$.\nBy the definition there are no edges between $S_0$ and $S_1$.\nConsider now, e.g., $V(H_\\Delta)\\setminus S_0$.\nLet $A_j$ consist of all elements in $P$ that have second coordinate equal to $j$ together with the vertex in $V_1$ that has the label $j$.\nBy the definition, the set $A_i$ is independent and $\\bigcup_{i\\in [\\Delta-1]}A_i$ covers $H_\\Delta\\setminus S_0$, and similarly for $S_1$.\n\n\\eqref{prop:hdeltachrom} By \\eqref{prop:hdelta} and the claim after the definition of $\\Delta$-(*), it is enough to show that $\\chi(H_\\Delta)\\geq 2\\Delta-2$. Towards a contradiction, assume that $c$ is a proper coloring of $H_\\Delta$ with $<2\\Delta-2$-many colors. Note the vertex $\\dagger$ guarantees that $|c(V(P))|\\leq 2\\Delta-4$, and also $\\Delta-1 \\leq |c(V(P))|$.\n\nFirst we claim that there are no indices $i,j\\in [\\Delta-1]$ (even with $i=j$) such that $c(i,r)\\not=c(i,s)$ and $c(r,j)\\not=c(s,j)$ for every $s\\not=r$: indeed, otherwise, by the definition of $P$ we had $c(i,r)\\not=c(s,j)$ for every $r,s$ unless $(i,r)=(s,j)$, which would the upper bound on the size of $c(V(P))$.\n\nTherefore, without loss of generality, we may assume that for every $i\\in [\\Delta-1]$ there is a color $\\alpha_i$ and two indices $j_i \\neq j'_i$ such that $c(i,j_i)=c(i,j'_i)=\\alpha_i$. It follows form the definition of $P$ and $j_i \\neq j'_i$ that\n $\\alpha_i\\not=\\alpha_{i'}$ whenever $i\\not= i'$. \n \n Moreover, note that any vertex in $V_1$ is connected to at least one of the vertices $(i,j_i)$ and $(i,j'_i)$, hence none of the colors $\\{\\alpha_i\\}_{i \\in [\\Delta-1]}$ can appear on $V_1$. Consequently, since $V_1$ is isomorphic to $K_{\\Delta-1}$ we need to use at least $\\Delta-1$ additional colors, a contradiction.\n\n\\eqref{prop:hdeltamax} First note that if $G$ admits a homomorphism into $H_\\Delta$, then the pullbacks of the sets witnessing $\\Delta$-(*) will witness that $G$ has $\\Delta$-(*).\n\nConversely, let $G$ be a graph that satisfies $\\Delta$-(*).\nFix the corresponding sets $S_0,S_1$ together with $(\\Delta-1)$-colorings $c_0,c_1$ of their complements.\nWe construct a homomorphism $\\Theta$ from $G$ to $H_\\Delta$. Let\n\\[\\Theta(v)=\n\\begin{cases} \\dagger &\\mbox{if } v \\in S_0 \\cap S_1, \\\\\nc_0(v) & \\mbox{if } v \\in S_1 \\setminus S_0, \\\\\nc_1(v) & \\mbox{if } v \\in S_0 \\setminus S_1,\n\\\\\n(c_0(v),c_1(v)) & \\mbox{if } v \\not \\in S_0 \\cup S_1.\n\\end{cases} \\]\n\n\nObserve that $S=S_0\\cap S_1$ is an independent set such that there is no edge between $S$ and $S_0 \\cup S_1$.\nUsing this observation, one easily checks case-by-case that $\\Theta$ is indeed a homomorphism.\n\n \n \n \n \n \n \n\\end{proof}\n\nNow, combining what we have so far, we can easily prove Theorem \\ref{th:homomorphism}.\n\n\n\\begin{proof}[Proof of Theorem \\ref{th:homomorphism}]\nIt follows that $\\Pi_{H_\\Delta}$ is not playable from \\cref{pr:MinimalExample}, \\cref{pr:CONdition*}. It is easy to see that if for a graph $G$ the LCL $\\Pi_G$ is not playable then $\\Pi_{G'}$ is not playable for every subgraph $G'$ of $G$. Since erasing a vertex decreases the chromatic number with at most one, for each $k\\leq 2\\Delta-2$ there is a subgraph $G_k$ of $H_\\Delta$ with $\\chi(G_k)=k$, such that $\\Pi_{G_k}$ is not playable.\n\nIt follows from Theorems \\ref{th:MainLOCAL} and \\ref{thm:MainBorel} that there is a $\\Delta$-regular Borel forest that admits no Borel homomorphism to any graph of $G_k$ and that $\\Pi_{G_k} \\not \\in \\mathsf{LOCAL}(O(\\log^*(n)))$.\n\nFinally, note that if $k \\geq \\Delta$ then $G_k$ can be chosen so that it contains $K_\\Delta$, yielding $\\Pi_{G_k} \\in \\mathsf{RLOCAL}(O(\\log\\log(n))$.\n\\end{proof}\n\n\\begin{remark}\n\\label{r:hedet}Recall that Hedetniemi's conjecture is the statement that if $G,H$ are finite graphs then $\\chi(G \\times H)=\\min\\{\\chi(G),\\chi(H)\\}$. This conjecture has been recently disproven by Shitov \\cite{shitov}, and strong counterexamples have been constructed later (see, \\cite{tardif2019note,zhu2021note}). We claim that these imply for $\\varepsilon>0$ the existence of finite graphs $H$ with $\\chi(H) \\geq (2-\\varepsilon)\\Delta$ to which $\\Delta$-regular Borel forests cannot have a homomorphism in BOREL, for every large enough $\\Delta$. Indeed, if a $\\Delta$-regular Borel forest admitted a Borel homomorphism to each finite graph of chromatic number at least $(2-\\varepsilon)\\Delta$, it would have such a homomorphism to their product as well. Thus, we would obtain that the chromatic number of the product of any graphs of chromatic number $(2-\\varepsilon)\\Delta$ is at least $\\Delta+1$. This contradicts Zhu's result \\cite{zhu2021note}, which states that the chromatic number of the product of graphs with chromatic number $n$ can drop to $\\approx \\frac{n}{2}$.\n\n\\end{remark}\n\n\\begin{remark}\nA natural attempt to construct graphs with large girth and not playable homomorphisms problem would be to consider random $d$-regular graphs of size $n$ for a large enough $n$. However, it is not hard to see that setting $\\Lambda(A)=\\operatorname{Alice}$ if and only if $|A|<\\frac{n}{d}$ shows that this approach cannot work.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{th:MainLOCAL}}\\label{sec:marksproof}\n\nIn this section we prove Theorem \\ref{th:MainLOCAL}, by applying Marks' game technique in the LOCAL setting. In order to define our games, we will need certain auxiliary graphs, the so-called \\emph{ID graphs}. \nThe purpose of these graphs is to define a ``playground'' for the games that we consider.\nNamely, vertices in the game are labeled by vertices from the ID graph in such a way that the at the end we obtain a homomorphism from our underlying graph to the ID graph. \n\n\n\n\n\\begin{definition}\nA pair $\\idgraphcolored{n}{t}{r}=(\\idgraph{n}{t}{r},c)$ is called an \\emph{ID graph}, if\n\\begin{enumerate}\n \\item \\label{p:idgirth} $\\idgraph{n}{t}{r}$ is graph with girth at least $2t+2$,\n \\item \\label{p:idsize} $|V(\\idgraph{n}{t}{r})|\\le n$,\n \\item \\label{p:idlabels} $c$ is a $\\Delta$-edge-coloring of $\\idgraph{n}{t}{r}$, such that every vertex is adjacent to at least one edge of each color,\n \\item \\label{p:idratio} for each $\\alpha \\in [\\Delta]$ the ratio of a maximal independent set of $\\idgraph{n}{t}{r}^{\\alpha}=(V(\\idgraph{n}{t}{r}), E(\\idgraph{n}{t}{r}) \\cap c^{-1}(\\alpha))$ is at most $r$ (i.e., $\\idgraph{n}{t}{r}^\\alpha$ is the graph formed by $\\alpha$-colored edges).\n\\end{enumerate}\n\\end{definition}\n\nBefore we define the game we show that ID graphs exist.\n\n\\begin{proposition}\\label{pr:IDgraph}\nLet $t_n\\in o(\\log(n))$, $r>0$, $\\Delta\\geq 2$.\nThen there is an ID graph $\\idgraphcolored{n}{t_n}{r}$ for every $n\\in \\mathbb{N}$ sufficiently large.\n\\end{proposition}\n\\begin{proof}\nWe use the \\emph{configuration model} for regular random graphs, see \\cite{wormald1999models}.\nThis model is defined as follows.\nLet $n$ be even.\nThen a $d$-regular random sample on $n$-vertices is just a union of $d$-many independent uniform random perfect matchings.\nNote that in this model we allow parallel edges.\n\nIt was proved by Bollob\\'as \\cite{bollobas} that the independence ratio of a random $d$-regular graph is at most $\\frac{2\\log(d)}{d}$ a.a.s.\nMoreover, this quantity is concentrated by an easy application of McDiarmid's result \\cite{mcdiarmid2002concentration}, i.e., \n\\begin{equation}\n \\P\\left(\\left|X-\\textrm{\\textbf{E}}(X)\\right|\\ge \\sqrt{n}\\right)<2\\exp\\left(-\\frac{n}{d}\\right),\n\\end{equation}\nwhere $X$ is the random variable that counts the size of a maximal independent set.\nTherefore for fixed $n$ large enough we have that the independence ratio of a random sample is at most $\\frac{3\\log(d)}{d}$ with probability at least $\\left(1-2\\exp\\left(-\\frac{n}{d}\\right)\\right)$.\n\n\nPick a $d$ large enough such that $\\frac{3\\log(d)}{d} & 1-\\left(1-2\\exp\\left(-\\frac{n}{d}\\right)\\right)^\\Delta\n\\end{split}\n\\end{equation}\nas $n\\to \\infty$.\nThis shows that there exists such a graph $\\idgraphcolored{t_n}{n}{r}$ with non-zero probability.\n\\end{proof}\n\n\n\n\nNext, we define the games. As mentioned before, the games are going to depend on the following parameters: an algorithm $\\mathcal{A}_n$ of local complexity $t\\in o(\\log(n))$, an ID graph $\\idgraphcolored{n}{t}{r}$, $\\alpha\\in \\Delta$, $\\sigma\\in V(\\idgraph{n}{t}{r})$ and $S \\subseteq \\Sigma$.\n(We will view $\\mathcal{A}_n$, $\\idgraphcolored{n}{t}{r}$ as fixed, and the rest of the parameters as variables).\n\nThe game\n$$\\mathbb{G}(\\mathcal{A}_n,n,t,\\idgraphcolored{n}{t}{r})[\\alpha,\\sigma,S]$$\nis defined as follows: two players, $\\operatorname{Alice}$ and $\\operatorname{Bob}$ assign labels to the vertices of a rooted $\\Delta$-regular tree of diameter $t$. The labels are vertices of $\\idgraph{n}{t}{r}$ and the root is labeled $\\sigma$. In the $k$-th round, where $00$. $\\idgraphingcolored{r}=(\\idgraphing{r},c)$ is an \\emph{ID graphing}, if\n\n\\begin{enumerate}\n \\item $\\idgraphing{r}$ is an acyclic locally finite Borel graphing on a standard probability measure space $(X,\\mu)$,\n \\item \\label{p:idgraphinglabels} $c$ is a Borel $\\Delta$-edge-coloring of $\\idgraphing{r}$, such that every vertex is adjacent to at least one edge of each color,\n \\item \\label{p:idgraphingratio} for each $\\alpha \\in [\\Delta]$ the $\\mu$-measure of a maximal independent set of $\\idgraphing{r}^{\\alpha}=(V(\\idgraphing{r}), E(\\idgraphing{r}) \\cap c^{-1}(\\alpha))$ is at most $r$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{proposition}\n\\label{p:idgraphing}\nFor each $r>0$ there exists an ID graphing $\\idgraphingcolored{r}$.\n\\end{proposition}\n\\begin{proof}\nLet $\\idgraphing{r}$ be a \\emph{local-global limit} of the random graphs constructed in \\cref{pr:IDgraph} (see, e.g., \\cite{hatamilovaszszegedy} for the basic results about local-global convergence).\nIt is not hard to check that this limit satisfies the required properties.\n\\end{proof}\n\n\nNow we are ready to prove the theorem. The proof will closely follow the argument given in the proof of the LOCAL version, i.e., Theorem \\ref{th:MainLOCAL}, but can be understood without reading the latter.\n\n\\begin{proof}[Proof of \\cref{thm:MainBorel}]\nLet $\\Pi$ be an LCL that is not playable, and $N\\in \\mathbb{N}$ be the number of all possible colorings of vertices of $(\\mathbb{P}_{\\alpha})_{\\alpha\\in \\Delta}$ with two colors, that is, $N=2^{\\sum_\\alpha |V(\\mathbb{P}_\\alpha)|}$.\nSet $r:=\\frac{1}{N+1}$.\n\nWe define a Borel acyclic $\\Delta$-regular graph $\\mathcal{G}$, with edges properly colored by $\\Delta$, that does not admit a Borel solution of $\\Pi$.\nVertices of $\\mathcal{G}$ are pairs $(x,A)$, where $x\\in X$ is a vertex of $\\idgraphing{r}$ and $A$ is a countable subgraph of $\\idgraphing{r}$ that is a $\\Delta$-regular tree that contains $x$ and the edge coloring of $\\idgraphingcolored{r}$ induces a proper edge coloring of $A$.\nWe say that $(x,A)$ and $(y,B)$ are connected by an $\\alpha$-edge in $\\mathcal{G}$ if $A=B$, $x,y$ are adjacent in $A$ and the edge that connects them has color $\\alpha\\in \\Delta$.\n\nSuppose for a contradiction that $\\mathcal{A}$ is a Borel function that solves $\\Pi$ on $\\mathcal{G}$.\n\nNext, we define a family of games parametrized by $\\alpha \\in [\\Delta]$, $x \\in V(\\idgraphing{r})$ and $S \\subseteq \\Sigma$. For the reader familiar with Marks' construction, let us point out that for a fixed $x$, the games are analogues to the ones he defines, with the following differences: allowed moves are vertices of the ID graphing $\\idgraphing{r}$ and restricted by its edge relation, and the winning condition is defined by a set of labels, not just merely one label.\n\nSo, the game\n$$\\mathbb{G}(\\mathcal{A},\\idgraphingcolored{r})[\\alpha,x,S]$$ \nis defined as follows: $\\operatorname{Alice}$ and $\\operatorname{Bob}$ alternatingly label vertices of a $\\Delta$-regular rooted tree. The root is labelled by $x$, and the labels come from $V(\\idgraphing{r})$. In the $k$-th round, first $\\operatorname{Alice}$ labels vertices of distance $k$ from the root on the side of the $\\alpha$ edge.\nAfter that, $\\operatorname{Bob}$ labels all remaining vertices of distance $k$, etc (see \\cref{fig:game}). We also require the assignment of labels to give rise to an edge-color preserving homomorphism to $\\idgraphingcolored{r}$. \n\nIt follows from the acyclicity of $\\idgraphing{r}$ that a play of a game determines a $\\Delta$-regular rooted subtree of $\\idgraphing{r}$ to which the restriction of the edge-coloring is proper. That is, it determines a vertex $(x,A)$ of $\\mathcal{G}$. Let $\\operatorname{Alice}$ win iff the output of $\\mathcal{A}$ on the half-edge determined by $(x,A)$ and the color $\\alpha$ is \\textbf{not} in $S$.\n\nDefine the function $\\Lambda^x_{\\alpha}:\\mathbb{P}_\\alpha\\to \\{\\operatorname{Alice},\\operatorname{Bob}\\}$ assigning the player to some $S \\in V(\\mathbb{P}_\\alpha)$ who has a winning strategy in $\\mathbb{G}(\\mathcal{A},\\idgraphingcolored{r})[\\alpha,x,S]$.\nNote that, since $\\idgraphing{r}$ is locally finite, each player has only finitely many choices at each position. Thus, it follows from Borel Determinacy Theorem that $\\Lambda^x_{\\alpha}$ is well defined.\n\nNow we show the analogue of Proposition \\ref{pr:(A)satisfied}.\n\n\\begin{proposition}\\label{pr:Borel(A)satisfied}\n$(\\Lambda^{x}_{\\alpha})_{\\alpha\\in \\Delta}$ satisfies (A) in \\cref{def:Playability}.\n\\end{proposition}\n\\begin{proof}\nAssume that $(S_\\alpha)_{\\alpha\\in \\Delta}$ is such that $\\Lambda^x_\\alpha(S_\\alpha)=\\operatorname{Alice}$.\nThis means that $\\operatorname{Alice}$ has winning strategy in all the games corresponding to $S_\\alpha$. Letting these strategies play against each other in the obvious way, produces a vertex $(x,A)$ of $\\mathcal{G}$.\nSince $\\mathcal{A}$ solves $\\Pi$ it has to output labeling of half edges $(\\mathtt{a}_{\\alpha})_{\\alpha\\in \\Delta}\\in \\mathcal{N}$, where $\\mathtt{a}_\\alpha$ is a label on the half edge that start at $(x,A)$ and has color $\\alpha$.\nNote that we must have $\\mathtt{a}_\\alpha\\not\\in S_\\alpha$ by the definition of winning strategy for $\\operatorname{Alice}$.\nThis shows that (A) of Definition \\ref{def:Playability} holds.\n\\end{proof}\n\n Let $f$ be \n defined by $$x\\mapsto (\\Lambda^x_{\\alpha})_{\\alpha\\in\\Delta}.$$ Note that $f$ has a finite range. Using the fact that the allowed moves for each player can be determined in a Borel way, uniformly in $x$, it is not hard to see that for each element $s$ in the range, $f^{-1}(s)$ is in the algebra generated by sets that can be obtained by applying game quantifiers to Borel sets (for the definition see \\cite[Section 20.D]{kechrisclassical}). It has been shown by Solovay \\cite{Solovay} and independently by Fenstad-Norman \\cite{fenstadnorman} that such sets are provably $\\Delta^1_2$, and consequently, measurable. Therefore, $f$ is a measurable map.\n \n As the number of sequences of functions $(\\Lambda^x_{\\alpha})_{\\alpha\\in\\Delta}$ is $\\leq N$, by the choice of $r$, there exists a Borel set $Y$ with $\\mu(Y) > r$, such that $f$ is constant on $Y$.\n \nSince $\\Pi$ is not playable Proposition \\ref{pr:Borel(A)satisfied} gives that there is an $\\alpha\\in \\Delta$ and $S,T\\in \\mathbb{P}_\\alpha$ that violate condition (B).\nRecall that the measure of an independent Borel set of $\\idgraphing{r}^\\alpha$ is at most $r$.\nThat means that there are $x,y\\in Y$ such that $(x,y)$ is an $\\alpha$-edge in $\\idgraphing{r}$.\nWe let the winning strategies of $\\operatorname{Bob}$ in the games\n$$\\mathbb{G}(\\mathcal{A},\\idgraphingcolored{r})[\\alpha,x,S], \\ \\mathbb{G}(\\mathcal{A},\\idgraphingcolored{r})[\\alpha,y,T],$$\nplay against each other, where we start with an $\\alpha$ edge with endpoints labeled by $x,y$.\nThis produces a labeling of a $\\Delta$-regular tree $A$ with labels from $\\idgraphing{r}$ such that $(x,A)$ and $(y,A)$ span an $\\alpha$-edge in $\\idgraphing{r}$.\nApplying $\\mathcal{A}$ on the half-edge determined by $(x,A)$ and the color $\\alpha$, we obtain $\\mathtt{a}_0\\in S$.\nSimilarly, we produce $\\mathtt{a}_1\\in T$ for $(y,A)$.\nHowever, $(\\mathtt{a}_0,\\mathtt{a}_1)\\not\\in {\\bf E}_\\alpha$ by the definition of an edge in $\\mathbb{P}_{\\alpha}$.\nThis shows that $\\mathcal{A}$ does not produce a Borel solution to $\\Pi$.\n\\end{proof}\n\n\\begin{remark}\nOne can give an alternative proof of the existence of a Borel $\\Delta$-regular forest that does not admit a Borel homomorphism to $G_k$ that avoids using the graphing $\\idgraphing{r}$ and uses the original example described by Marks \\cite{DetMarks} instead.\nThe reason is that the example graphs $G_k$ contain $K_\\Delta$, as discussed in the proof of \\cref{th:homomorphism}.\nThis takes care of the ``non-free'' as in the argument of Marks.\nThen it is enough to use the pigeonhole principle on $\\mathbb{N}$ instead of the independence ratio reasoning.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Separation Of Various Complexity Classes}\n\\label{sec:separating_examples}\n\nIn this section we provide examples of several problems that separate some classes from \\cref{fig:big_picture_trees}.\nThe examples show two things. First, we have $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon) \\not= \\mathsf{OLOCAL}(O(\\log\\log 1\/\\varepsilon))$. \nThis shows that there are problems such that their worst case complexity is at least $\\Theta(\\log n)$ on finite $\\Delta$-regular trees, but their average local complexity is constant. \nSecond, we show that there are problems in the class $\\mathsf{BOREL}$ that are not in the class $\\mathsf{RLOCAL}(O(\\log\\log n)) = \\mathsf{RLOCAL}(o(\\log n))$, on $\\Delta$-regular trees. This shows that one cannot in general hope that results from (Borel) measurable combinatorics can be turned into very efficient (sublogarithmic) distributed algorithms. \n\n\\subsection{Preliminaries}\n\\label{subsec:preliminaries_separating}\n\nWe first show that there is a strong connection between randomized local complexities and uniform local complexities. Afterwards, we introduce a generic construction that turns any LCL into a new LCL. We later use this generic transformation to construct an LCL that is contained in the set $\\mathsf{BOREL} \\setminus \\mathsf{RLOCAL}(O(\\log \\log n))$.\n\n\\paragraph{Uniform vs Local Randomized Complexity} \n\nWe will now discuss the connections between uniform and randomized complexities. \nNote that the easy part of the connection between the two concepts is turning uniform local algorithms into randomized ones, as formalized in the following proposition. \n\n\\begin{proposition}\n\\label{prop:bee_gees}\nWe have $\\mathsf{OLOCAL}(t(\\varepsilon)) \\subseteq \\mathsf{RLOCAL}(t(1\/n^{O(1)}))$. \n\\end{proposition}\n\\begin{proof}\nWe claim that an uniform local algorithm $\\mathcal{A}$ with an uniform local complexity of $t(\\varepsilon)$ can be turned into a local randomized algorithm $\\mathcal{A}'$ with a local complexity of $t(1\/n^{O(1)})$. \n\nThe algorithm $\\mathcal{A}'$ simulates $\\mathcal{A}$ on an infinite $\\Delta$-regular tree -- each vertex $u$ of degree less than $\\Delta$ in the original tree pretends that the tree continues past its virtual half-edges and the random bits of $u$ are used to simulate the random bits in this virtual subtree. \nChoosing $\\varepsilon = 1\/n^{O(1)}$, one gets that the probability of $\\mathcal{A}'$ needing to look further than $t(\\varepsilon)$ for any vertex is bounded by $1\/n^{O(1)}$, as needed in the definition of the randomized local complexity. \n\\end{proof}\n\n\nOn the other hand, we will use the following proposition from \\cite{grebik_rozhon2021toasts_and_tails}. It informally states that the existence of \\emph{any} uniform local algorithm together with the existence of a sufficiently fast randomized local algorithm for a given LCL $\\Pi$ directly implies an upper bound on the uniform complexity of $\\Pi$.\n\\begin{proposition}[\\cite{grebik_rozhon2021toasts_and_tails}]\n\\label{prop:local->uniform}\nLet $\\mathcal{A}$ be an uniform local algorithm solving an LCL problem $\\Pi$ such that its \\emph{randomized} local complexity on finite $\\Delta$-regular trees is $t(n)$ for $t(n) = o(\\log n)$. \nThen, the uniform local complexity of $\\mathcal{A}$ on infinite $\\Delta$-regular trees is $O(t(1\/\\varepsilon))$.\n\\end{proposition}\n\n\\cref{prop:local->uniform} makes our life simpler, since we can take a known randomized distributed local algorithm with local complexity $g(n) = o(\\log n)$ and check if it works without the knowledge of $n$. If yes, this automatically implies that the uniform local complexity of the algorithm is $h(\\varepsilon) = O(g(1\/\\varepsilon))$. \nIn particular, combining \\cref{prop:local->uniform} with the work of \\cite{chang_kopelowitz_pettie2019exp_separation,chang_pettie2019time_hierarchy_trees_rand_speedup} and verifying that the algorithms of \nFischer and Ghaffari~\\cite[Section 3.1.1]{fischer-ghaffari-lll} and Chang~et~al.~\\cite[Section 5.1]{ChangHLPU20} work without the knowledge of $n$, we obtain the following result. \n\n\n\\begin{theorem}\n\\label{thm:classification_local_uniform_trees}\nWe have:\n\\begin{itemize}\n \\item $\\mathsf{LOCAL}(O(1)) = \\mathsf{OLOCAL}(O(1))$\n \\item $\\mathsf{RLOCAL}(O(\\log^* n)) = \\mathsf{OLOCAL}(O(\\log^* 1\/\\varepsilon))$\n \\item $\\mathsf{RLOCAL}(O(\\log\\log n)) = \\mathsf{OLOCAL}(O(\\log\\log 1\/\\varepsilon))$\n\\end{itemize}\nMoreover, there are no other possible uniform local complexities for $t(\\varepsilon) = o(\\log 1\/\\varepsilon)$. \n\\end{theorem}\nWe note that the first two items are proven in \\cite{grebik_rozhon2021toasts_and_tails}.\nFor the third item it suffices by known reductions to find an uniform algorithm\nsolving a version of the distributed Lov\\'asz Local Lemma (LLL) on so-called tree-structured dependency graphs considered in~\\cite{ChangHLPU20}.\n\n\\begin{proof}[Proof sketch]\nThe proof follows from \\cref{prop:bee_gees} and the following ideas. \nThe first item follows from the fact that any local algorithm with local complexity $O(1)$ can simply be made uniform. More specifically, there exists a constant $n_0$ --- depending only on the problem $\\Pi$ --- such that the algorithm, being told the size of the graph is $n_0$, is correct on \\emph{any} graph of size $n \\ge n_0$. \n\nFor the second item, by the work of \\cite{chang_kopelowitz_pettie2019exp_separation,chang_pettie2019time_hierarchy_trees_rand_speedup}, it suffices to check that there is an uniform distributed $(\\Delta + 1)$-coloring algorithm with uniform local complexity $O(\\log^* 1\/\\varepsilon)$. Such an algorithm was given in \\cite{HolroydSchrammWilson2017FinitaryColoring}, or follows from the work in \\cite{Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring} and \\cref{prop:local->uniform}. \n\nSimilarly, for the third item, by the work of \\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} it suffices to check that there is an uniform distributed algorithm for a specific LLL problem on trees\nwith uniform local complexity $O(\\log\\log 1\/\\varepsilon)$.\nSuch an algorithm can be obtained by combining the randomized \\emph{pre-shattering} algorithm of Fischer and Ghaffari~\\cite[Section 3.1.1]{fischer-ghaffari-lll} and the deterministic \\emph{post-shattering} algorithm of Chang~et~al.~\\cite[Section 5.1]{ChangHLPU20} in a \\emph{graph shattering} framework~\\cite{barenboim2016locality}, which solves the LLL problem \nwith local complexity $O(\\log\\log n)$.\nBy \\cref{prop:local->uniform}, it suffices to check that this algorithm can be made to work even if it does not know the size of the graph $n$. We defer the details to \\cref{sec:LLL}.\nThis finishes the proof of \\cref{thm:classification_local_uniform_trees}. \n\\end{proof}\n\n\n\n\n\n\n\n\\paragraph{Adding Paths}\nBefore we proceed to show some separation results, we define a certain construction that turns any LCL problem $\\Pi$ into a new LCL problem $\\overline{\\Pi}$ with the following property. If the original problem $\\Pi$ cannot be solved by a fast local algorithm, then the same holds for $\\overline{\\Pi}$. However, $\\overline{\\Pi}$ might be strictly easier to solve than $\\Pi$ for $\\mathsf{BOREL}$ constructions. \n\n\\begin{definition}\nLet $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ be an LCL. \nWe define an LCL $\\overline{\\Pi}=(\\Sigma',\\mathcal{V}',\\mathcal{E}')$ as follows.\nLet $\\Sigma'$ be $\\Sigma$ together with one new label.\nLet $\\mathcal{V}'$ be the union of $\\mathcal{V}$ together with any cardinality-$\\Delta$ multiset that contains the new label exactly two times.\nLet $\\mathcal{E}'$ be the union of $\\mathcal{E}$ together with the cardinality-$2$ multiset that contains the new label twice.\n\\end{definition}\n\nIn other words, the new label determines doubly infinite lines in infinite $\\Delta$-regular trees, or lines that start and end in virtual half-edges in finite $\\Delta$-regular trees.\nMoreover, a vertex that is on such a line does not have to satisfy any other vertex constraint.\nWe call these vertices \\emph{line-vertices} and each edge on a line a \\emph{line-edge}.\n\n\n\\begin{proposition}\n\\label{pr:AddingLineLLL}\nLet $\\Pi$ be an LCL problem such that $\\overline{\\Pi}\\in \\mathsf{RLOCAL}(t(n))$ for $t(n) = o(\\log(n))$. Then also $\\Pi \\in \\mathsf{RLOCAL}(O(t(n)))$.\n\\end{proposition}\n\\begin{proof}\nLet $\\overline{\\mathcal{A}}$ be a randomized $\\mathsf{LOCAL}$ algorithm that solves $\\overline{\\Pi}$ in $t(n) = o(\\log n)$ rounds with probability at least $1 - 1\/n^C$ for some sufficiently large constant $C$. We will construct an algorithm $\\mathcal{A}$ for $\\Pi$ with complexity $t(n)$ that is correct with probability $1 - \\frac{4}{n^{C\/3 - 2}}$. The success probability of $\\mathcal{A}$ can then be boosted by ``lying to it'' that the number of vertices is $n^{O(1)}$ instead of $n$; this increases the running time by at most a constant factor and boosts the success probability back to $1 - 1\/n^C$. \n\nConsider a $\\Delta$-regular rooted finite tree $T$ of depth $10t(n)$ and let $u$ be its root vertex. \nNote that $|T| \\le \\Delta^{10t(n)+1} < n$, for $n$ large enough. \n\n\nWe start by proving that when running $\\overline{\\mathcal{A}}$ on the tree $T$, then $u$ is most likely not a line-vertex. This observation then allows us to turn $\\overline{\\mathcal{A}}$ into an algorithm $\\mathcal{A}$ that solves $\\Pi$ on any $\\Delta$-regular input tree.\nLet $X$ be the indicator of $\\mathcal{A}$ marking $u$ as a line-vertex. Moreover, for $i \\in [\\Delta]$, let $Y_i$ be the indicator variable for the following event. The number of line edges in the $i$-th subtree of $u$ with one endpoint at depth $5t(n)$ and the other endpoint at depth $5t(n) + 1$ is odd.\n\nBy a simple parity argument we can relate the value of $X$ with the values of $Y_1,Y_2,\\ldots,Y_\\Delta$ as follows. If $X = 0$, that is, $u$ is not a line-vertex, then all of the $Y_i$'s have to be $0$, as each path in the tree $T$ is completely contained in one of the $\\Delta$ subtrees of $u$. On the other hand, if $u$ is a line-vertex, then there exists exactly one path, the one containing $u$, that is not completely contained in one of the $\\Delta$ subtrees of $u$. This in turn implies that exactly two of the $\\Delta$ variables $Y_1,Y_2,\\ldots,Y_\\Delta$ are equal to $1$.\n\nThe random variables $Y_1,Y_2,\\ldots,Y_\\Delta$ are identically distributed and mutually independent. Hence, if $\\P(Y_i = 1) > \\frac{1}{n^{C\/3}}$, then the probability that there are at least $3$ $Y_i$'s equal to $1$ is strictly greater than $\\left(\\frac{1}{n^{C\/3}}\\right)^3 = \\frac{1}{n^C}$. This is a contradiction, as in that case $\\overline{\\mathcal{A}}$ does not produce a valid output, which according to our assumption happens with probability at most $\\frac{1}{n^C}$. \n\nThus, we can conclude that $\\P(Y_i = 1) \\leq \\frac{1}{n^{C\/3}}$. By a union bound, this implies that all of the $Y_i$'s are zero with probability at least $\\frac{1}{n^{C\/3 - 1}}$, and in that case $u$ is not a line-vertex. \n\nFinally, the algorithm $\\mathcal{A}$ simulates $\\overline{\\mathcal{A}}$ as if the neighborhood of each vertex was a $\\Delta$-regular branching tree up to depth at least $t(n)$. \n\nIt remains to analyze the probability that $\\mathcal{A}$ produces a valid solution for the LCL problem $\\Pi$. To that end, let $v$ denote an arbitrary vertex of the input tree. The probability that the output of $v$'s half edges satisfy the vertex constraint of the LCL problem $\\overline{\\Pi}$ is at least $1 - 1\/n^C$. Moreover, in the case that $v$ is not a line-vertex, which happens with probability at least $1 - \\frac{1}{n^{C\/3 - 1}}$, the output of $v$'s half edges even satisfy the vertex constraint of the LCL problem $\\Pi$. Hence, by a union bound the vertex constraint of the LCL problem $\\Pi$ around $v$ is satisfied with probability at least $1 - \\frac{2}{n^{C\/3 - 1}}$. With exactly the same reasoning, one can argue that the probability that the output of $\\mathcal{A}$ satisfies the edge constraint of the LCL problem $\\Pi$ at a given edge is also at least $1 - \\frac{2}{n^{C\/3 - 1}}$. Finally, doing a union bound over the $n$ vertex constraints and $n-1$ edge constraints it follows that $\\mathcal{A}$ produces a valid solution for $\\Pi$ with probability at least $1 - \\frac{4}{n^{C\/3 - 2}}$.\n\\end{proof}\n\n\\begin{remark}\nThe ultimate way how to solve LCLs of the form $\\overline{\\Pi}$ is to find a spanning forest of lines.\nThis is because once we have a spanning forest of lines, then we might declare every vertex to be a line-vertex and label every half-edge that is contained on a line with the new symbol in $\\overline{\\Pi}$.\nThe remaining half-edges can be labeled arbitrarily in such a way that the edge constraints are satisfied.\n\nPut otherwise, once we are able to construct a spanning forest of lines and $\\mathcal{E}\\not=\\emptyset$, where $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$, then $\\overline{\\Pi}$ can be solved.\nMoreover, in that case the complexity of $\\overline{\\Pi}$ is upper bounded by the complexity of finding the spanning forest of lines.\n\nLyons and Nazarov \\cite{lyons2011perfect} showed that $\\Pi_{\\text{pm}}\\in \\mathsf{fiid}$ and it is discussed in \\cite{LyonsTrees} that the construction can be iterated $(\\Delta-2)$-many times. After removing each edge that is contained in one of the $\\Delta-2$ perfect matchings each vertex has a degree of two.\nHence, it is possible to construct a spanning forest of lines as fiid.\nConsequently, $\\overline{\\Pi}\\in \\mathsf{fiid}$ for every $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ with $\\mathcal{E}\\not=\\emptyset$.\n\\end{remark}\n\n\n\\subsection{$\\mathsf{LOCAL}(O(\\log^* n)) \\not= \\mathsf{BOREL}$}\n\nWe now formalize the proof that $\\mathsf{LOCAL}(O(\\log^* n)) \\not= \\mathsf{BOREL}$ from \\cref{sec:introseparation}. \nRecall that $\\Pi_{\\chi,\\Delta}$ is the proper vertex $\\Delta$-coloring problem.\nThe following claim directly follows by \\cref{pr:AddingLineLLL} together with the fact that $\\Pi_{\\chi,\\Delta}\\not\\in \\mathsf{LOCAL}(O(\\log^* n))$.\n\n\\begin{claim}\nWe have $\\overline{\\Pi_{\\chi,\\Delta}}\\not \\in \\mathsf{LOCAL}(O(\\log^* n))$.\n\\end{claim}\n\nOn the other hand we show that adding lines helps to find a Borel solution.\nThis already provides a simple example of a problem in the set $\\mathsf{BOREL} \\setminus \\mathsf{LOCAL}(O(\\log^* n))$.\n\n\\begin{proposition}\nWe have $\\overline{\\Pi_{\\chi,\\Delta}} \\in \\mathsf{BOREL}$. \n\\end{proposition}\n\\begin{proof}\nBy \\cite{KST}, every Borel graph of finite maximum degree admits a Borel maximal independent set.\nLet $\\mathcal{G}$ be a Borel $\\Delta$-regular acyclic graph on a standard Borel space $X$.\nBy an iterative application of the fact above we find Borel sets $A_1,\\dots,A_{\\Delta-2}$ such that $A_1$ is a maximal independent set in $\\mathcal{G}$ and $A_i$ is a maximal independent set in $\\mathcal{G}\\setminus \\bigcup_{j1$.\nWe think of $A_1,\\dots,A_{\\Delta-2}$ as color classes for the first $\\Delta-2$ colors.\nLet $B=X\\setminus \\bigcup_{i\\in [\\Delta-2]}A_i$ and let $\\mathcal{H}$ be the subgraph of $\\mathcal{G}$ determined by $B$.\nIt is easy to see that the maximum degree of $\\mathcal{H}$ is $2$.\nIn particular, $\\mathcal{H}$ consists of finite paths, one-ended infinite paths or doubly infinite paths.\nWe use the extra label in $\\overline{\\Pi_{\\chi,\\Delta}}$ to mark the doubly infinite paths.\nIt remains to use the remaining $2$ colors in $[\\Delta]$ to define a proper vertex $2$-coloring of the finite and one-ended paths.\nIt is a standard argument that this can be done in a Borel way and it is easy to verify that, altogether, we found a Borel $\\overline{\\Pi_{\\chi,\\Delta}}$-coloring of $\\mathcal{G}$.\n\\end{proof}\n\n\n\n\\subsection{Examples and Lower Bound}\n\nIn this subsection we define two LCLs and show that they are not in the class $\\mathsf{RLOCAL}(o(\\log n))$.\nIn the next subsection we show that one of them is in the class $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon)$ and the other in the class $\\mathsf{BOREL}$.\nBoth examples that we present are based on the following relaxation of the perfect matching problem.\n\n\\begin{definition}[Perfect matching in power-$2$ graph]\nLet $\\Pi^2_{\\text{pm}}$ be the perfect matching problem in the power-$2$ graph, i.e., in the graph that we obtain from the input graph by adding an edge between any two vertices that have distance at most $2$ in the input graph\n\\end{definition}\n\nWe show that $\\Pi^2_{\\text{pm}}$ is not contained in $\\mathsf{RLOCAL}(o(\\log n))$. \\cref{pr:AddingLineLLL} then directly implies that $\\overline{\\Pi^2_{\\text{pm}}}$ is not contained in $\\mathsf{RLOCAL}(o(\\log n))$ as well.\nOn the other hand, we later use a one-ended spanning forest decomposition to show that $\\Pi^2_{\\text{pm}}$ is in $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon)$ and a one or two-ended spanning forest decomposition to show that $\\overline{\\Pi^2_{\\text{pm}}}$ is in $\\mathsf{BOREL}$.\n\n\nWe first show the lower bound result.\nThe proof is based on a simple parity argument as in the proof of \\cref{pr:AddingLineLLL}.\n\n\\begin{theorem}\nThe problems $\\Pi^2_{\\text{pm}}$ and $\\overline{\\Pi^2_{\\text{pm}}}$ are not in the class $\\mathsf{RLOCAL}(o(\\log n))$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose that the theorem statement does not hold for $\\Pi^2_{\\text{pm}}$.\nThen, by \\cref{thm:basicLOCAL}, there is a distributed randomized algorithm solving $\\Pi^2_{\\text{pm}}$ with probability at least $1 - 1\/n$. By telling the algorithm that the size of the input graph is $n^{2\\Delta}$ instead of $n$, we can further boost the success probability to $1 - 1\/n^{2\\Delta}$. The resulting round complexity is at most a constant factor larger, as $\\Delta = O(1)$.\nWe can furthermore assume that the resulting algorithm $\\mathcal{A}$ will always output for each vertex $v$ a vertex $M(v) \\neq v$ in $v$'s $2$-hop neighborhood, even if $\\mathcal{A}$ fails to produce a valid solution. The vertex $M(v)$ is the vertex $v$ decides to be matched to, and if $\\mathcal{A}$ produces a valid solution it holds that $M(M(v)) = v$.\n\nConsider a fully branching $\\Delta$-regular tree $T$ of depth $10t(n)$. Note that $|T| < n$ for $n$ large enough. Let $u$ be the root of $T$ and $T_i$ denote the $i$-th subtree of $u$ (so that $u \\not\\in T_i$).\nLet $S_i$ denote the set of vertices $v$ in $T_i$ such that both $v$ and $M(v)$ have distance at most $2t(n)$ from $u$.\nNote that $M(v)$ is not necessarily a vertex of $T_i$.\nLet $Y_i$ be the indicator of whether $S_i$ is even.\n\nObserve that, if $\\mathcal{A}$ does not fail on the given input graph, then the definition of the $S_i$ implies that every vertex in $\\{ u \\} \\cup \\bigcup_{i \\in [\\Delta]} S_i$ is matched to a vertex in $\\{ u \\} \\cup \\bigcup_{i \\in [\\Delta]} S_i$.\nHence, the number of vertices in $\\{ u \\} \\cup \\bigcup_{i \\in [\\Delta]} S_i$ is even, which implies that we cannot have $Y_1 = Y_2 = \\ldots = Y_\\Delta = 1$ unless $\\mathcal{A}$ fails.\nNote that $Y_i$ depends only on the output of $\\mathcal{A}$ at the vertices in $T_i$ that have a distance of precisely $2t(n) - 1$ or $2t(n)$ to $u$ (as all other vertices in $T_i$ are guaranteed to be in $S_i$). \nHence, the events $Y_i = 1$, $i \\in [\\Delta]$, are independent, and, since $\\P(Y_1 = 1) = \\ldots = \\P(Y_{\\Delta} = 1)$ (as all vertices in all $T_i$ see the same topology in their $t(n)$-hop view), we obtain $(\\P(Y_1 = 1))^{\\Delta} \\le 1\/n^{2\\Delta}$, which implies $\\P(Y_i = 1) \\le 1\/n^2$, for any $i \\in [\\Delta]$.\n\n\nHence, with probability at least $1 - \\Delta \/n^2$ we have $Y_1=Y_2=\\ldots = Y_\\Delta = 0$, by a union bound.\nLet $v_1, v_2, \\dots, v_\\Delta$ denote the neighbors of $u$ with $v_i$ being the root of subtree $T_i$.\nNote that if $\\mathcal{A}$ does not fail and $Y_1=Y_2=\\ldots = Y_\\Delta = 0$, then it has to be the case that $u$ is matched to a vertex in some subtree $T_i$ (not necessarily $v_i$), while any vertex from $N_{-i}(u) := \\{v_1, v_2, \\dots, v_{i-1}, v_{i+1}, \\dots, v_\\Delta\\}$ is matched with another vertex from $N_{-i}(u)$ (hence, we already get that $\\Delta$ needs to be odd).\nThis is because each subtree needs to have at least one vertex matched to a different subtree (since $|S_i|$ is odd for each $i \\in [\\Delta]$).\nIf $M(u)$ is a vertex in $T_i$ and, for any $v \\in N_{-i}(u)$, we have $M(v) \\in N_{-i}(u)$, we say that $u$ orients the edge going to $v_i$ outwards.\n\nConsider a path $(u_0 = u, u_1, \\dots, u_k)$, where $k = 2t(n) + 3$.\nBy the argument provided above, the probability that $u$ orients an edge outwards is at least $1 - \\Delta\/n^2 - 1\/n^{2\\Delta}$, and, if $u$ orients an edge outwards, the probability that this edge is not $uu_1$ is $(\\Delta - 1)\/\\Delta$, by symmetry.\nHence, we obtain (for sufficiently large $n$) that $\\P(\\mathcal{E}_1) \\geq 1\/2$, where $\\mathcal{E}_1$ denotes the event that vertex $u$ orients an edge different from $uu_1$ outwards.\nWith an analogous argument, we obtain that $\\P(\\mathcal{E}_2) \\geq 1\/2$, where $\\mathcal{E}_2$ denotes the event that vertex $u_k$ orients an edge different from $u_ku_{k-1}$ outwards.\nSince $\\mathcal{E}_1$ and $\\mathcal{E}_2$ are independent (due to the fact that the distance between any neighbor of $u$ and any neighbor of $u_k$ is at least $2t(n) + 1$), we obtain $\\P(\\mathcal{E}_1 \\cap \\mathcal{E}_2) \\geq 1\/4$.\nMoreover, the probability that all vertices in $\\{ u_1, \\dots, u_{k-1} \\}$ orient an edge outwards is at least $1 - (2t(n)+2)(\\Delta\/n^2 + 1\/n^{2\\Delta})$, which is at least $4\/5$ for sufficiently large $n$.\nThus, by a union bound, there is a probability of at least $1 - 3\/4 - 1\/5 - 1\/n^{2\\Delta}$ that $\\mathcal{A}$ does not fail, all vertices in $(u, u_1, \\dots, u_k)$ orient an edge outwards, and the outward oriented edges chosen by $u$ and $u_k$ are not $uu_1$ and $u_ku_{k-1}$, respectively.\nFor sufficiently large $n$, the indicated probability is strictly larger than $0$.\nWe will obtain a contradiction and conclude the proof for $\\Pi^2_{\\text{pm}}$ by showing that there is no correct solution with the described properties.\n\nAssume in the following that the solution provided by $\\mathcal{A}$ is correct and $u, u_1, \\dots, u_k$ satisfy the mentioned properties.\nSince $u$ orients an edge different from $uu_1$ outwards, $u_1$ must be matched to a neighbor of $u$, which implies that $u_1$ orients edge $u_1u$ outwards.\nUsing an analogous argumentation, we obtain inductively that $u_j$ orients edge $u_ju_{j-1}$ outwards, for all $2 \\leq j \\leq k$.\nHowever, this yields a contradiction to the fact that the edge that $u_k$ orients outwards is different from $u_ku_{k-1}$, concluding the proof (for $\\Pi^2_{\\text{pm}}$).\n\nThe theorem statement for $\\overline{\\Pi^2_{\\text{pm}}}$ follows by applying \\cref{pr:AddingLineLLL}.\n\n\n\\end{proof}\n\n\n\n\n\n\\subsection{Upper Bounds Using Forest Decompositions}\n\nWe prove an upper bound for the problems $\\Pi^2_{\\text{pm}}$ and $\\overline{\\Pi^2_{\\text{pm}}}$ defined in the previous subsection.\nWe use a technique of decomposing the input graph in a spanning forest with some additional properties, i.e., one or two ended. This technique was used in \\cite{BrooksMeas} to prove Brooks' theorem in a measurable context.\nNamely, they proved that if $\\mathcal{G}$ is a Borel $\\Delta$-regular acyclic graph and $\\mu$ is a Borel probability measure, then it is possible to erase some edges of $\\mathcal{G}$ in such a way that the remaining graph is a one-ended spanning forest on a $\\mu$-conull set.\nIt is not hard to see that one can solve $\\Pi^2_{\\text{pm}}$ on one-ended trees in an inductive manner, starting with the leaf vertices. Consequently we have $\\Pi^2_{\\text{pm}}\\in \\mathsf{MEASURE}$.\nWe provide a quantitative version of this result and thereby show that the problem of constructing a one-ended forest is contained in $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon)$.\nA variation of the construction shows that it is possible to find a one or two-ended spanning forest in a Borel way.\nThis allows us to show that $\\overline{\\Pi^2_{\\text{pm}}}\\in \\mathsf{BOREL}$.\nAs an application of the decomposition technique we show that Vizing's theorem, i.e., proper $\\Delta+1$ edge coloring $\\Pi_{\\chi',\\Delta+1}$, for $\\Delta=3$ is in the class $\\mathsf{OLOCAL}(\\operatorname{poly} \\log 1\/\\varepsilon)$.\n\n\n\\subsubsection{Uniform Complexity of the One-Forest Decomposition }\nWe start with the definition of a one-ended spanning forest.\nIt can be viewed as a variation of the edge grabbing problem $\\Pi_{\\text{edgegrab}}$.\nNamely, if a vertex $v$ grabs an incident edge $e$, then we think of $e$ as being oriented away from $v$ and $v$ selecting the other endpoint of $e$ as its parent.\nSuppose that $\\mathcal{T}$ is a solution of $\\Pi_{\\text{edgegrab}}$ on $T_\\Delta$.\nWe denote with $\\mathcal{T}^{\\leftarrow}(v)$ the subtree with root $v$.\n\n\n\\begin{definition}[One-ended spanning forest]\nWe say that a solution $\\mathcal{T}$ of $\\Pi_{\\text{edgegrab}}$ is a \\emph{one-ended spanning forest} if $\\mathcal{T}^{\\leftarrow}(v)$ is finite for every $v\\in T_\\Delta$ (see \\cref{fig:one_ended}). \n\n\\end{definition}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=.4\\textwidth]{fig\/one_ended.pdf}\n \\caption{An example of a one-ended forest decomposition of an infinite 3-regular tree. }\n \\label{fig:one_ended}\n\\end{figure}\n\n\nNote that on an infinite $\\Delta$-regular tree every connected component of a one-ended spanning forest must be infinite.\nFurthermore, recall that we discussed above that it is possible to construct a one-ended spanning forest in $\\mathsf{MEASURE}$ by \\cite{BrooksMeas}.\nNext we formulate our quantitative version.\n\n\n\n\n\n\\begin{restatable}{theorem}{oneendedtreetail}\n\\label{thm:one_ended_forest}\nThe one-ended spanning forest can be constructed by an uniform local algorithm with an uniform local complexity of $O(\\operatorname{poly}\\log1\/\\varepsilon)$. \nMore precisely, there is an uniform distributed algorithm $\\mathcal{A}$ that computes a one-ended spanning forest and if we define $R(v)$ to be the smallest coding radius that allows $v$ to compute $\\mathcal{T}^{\\leftarrow}(v)$, then\n\\[\nP(R(v) > O(\\operatorname{poly} \\log 1\/\\varepsilon)) \\le \\varepsilon. \n\\]\n\\end{restatable}\n\nThe formal proof of \\cref{thm:one_ended_forest} can be found in \\cref{sec:formalproof}.\nWe first show the main applications of the result.\nNamely, we show that $\\Pi^2_{\\text{pm}}$ can be solved inductively starting from the leaves of the one-ended trees. This proves that $\\Pi^2_{\\text{pm}}$ is in $\\mathsf{OLOCAL}(\\operatorname{poly} \\log 1\/\\varepsilon)$ and hence $\\mathsf{OLOCAL}(\\operatorname{poly} \\log 1\/\\varepsilon) \\setminus \\mathsf{RLOCAL}(o(\\log n))$ is non-empty.\n\n\\begin{theorem}\\label{thm:twomatchFFIID}\nThe problem $\\Pi^2_{\\text{pm}}$ is in the class $\\mathsf{MEASURE}$ and $\\mathsf{OLOCAL}(O(\\operatorname{poly} \\log 1\/\\varepsilon))$.\n\\end{theorem}\n\\begin{proof}\nFirst we show that every infinite one-ended directed tree $T$ admits an inductive solution from the leaves to the direction of infinity. More precisely, we define a power-$2$ perfect matching on the infinite one-ended directed tree $T$ such that the following holds. Let $v$ be an arbitrary vertex and $T^{\\leftarrow}(v)$ the finite subtree rooted at $v$. Then, all vertices in $T^{\\leftarrow}(v)$, with the possible exception of $v$, are matched to a vertex in $T^{\\leftarrow}(v)$. Moreover, for each vertex in $T^{\\leftarrow}(v) \\setminus \\{v\\}$, it is possible to determine to which vertex it is matched by only considering the subtree $T^{\\leftarrow}(v)$. \nWe show that it is possible to define a perfect matching in that way by induction on the height of the subtree $T^{\\leftarrow}(v)$. \n\nWe first start with the leaves and declare them as unmatched in the first step of the induction.\n\nNow, consider an arbitrary vertex $v$ with $R \\geq 1$ children that we denote by $v_1, v_2, \\ldots, v_R$.\nBy the induction hypothesis, all vertices in $T^{\\leftarrow}(v_i) \\setminus \\{v_i\\}$ are matched with a vertex in $T^{\\leftarrow}(v_i)$ for $i \\in [R]$ and it is possible to compute that matching given $T^{\\leftarrow}(v)$. However, some of the children of $v$ are possibly still unmatched. If the number of unmatched children of $v$ is even, then we simply pair them up in an arbitrary but fixed way and we leave $v$ unmatched. Otherwise, if the number of unmatched children of $v$ is odd, we pair up $v$ with one of its unmatched children and we pair up all the remaining unmatched children of $v$ in an arbitrary but fixed way. This construction clearly satisfies the criteria stated above. In particular, the resulting matching is a perfect matching, as every vertex gets matched eventually.\n\n\nBy \\cref{thm:one_ended_forest} there is an uniform distributed algorithm of complexity $O(\\operatorname{poly} \\log 1\/\\varepsilon)$ that computes a one-ended spanning forest $\\mathcal{T}$ on $T_\\Delta$.\nMoreover, every vertex $v\\in T_\\Delta$ can compute where it is matched with an uniform local complexity of $O(\\operatorname{poly} \\log 1\/\\varepsilon)$.\nThis follows from the discussion above, as a vertex $v$ can determine where it is matched once it has computed $\\mathcal{T}^{\\leftarrow}(w)$ for each $w$ in its neighborhood.\nHence, $\\Pi^2_{\\text{pm}}$ is in the class $\\mathsf{OLOCAL}(O(\\operatorname{poly} \\log 1\/\\varepsilon))$.\n\nSimilarly, the one-ended spanning forest $\\mathcal{T}\\subseteq \\mathcal{G}$ can be constructed in the class $\\mathsf{MEASURE}$ by \\cite{BrooksMeas}.\nBy the discussion above, this immediately implies that $\\Pi^2_{\\text{pm}}$ is in the class $\\mathsf{MEASURE}$. \n\\end{proof}\n\n\n\n\n\\subsubsection{Decomposition in $\\mathsf{BOREL}$}\nNext, we show that a similar, but weaker, decomposition can be done in a Borel way.\nNamely, we say that a subset of edges of an acyclic infinite graph $G$, denoted by $\\mathcal{T}$, is a \\emph{one or two-ended spanning forest} if every vertex $v\\in G$ is contained in an infinite connected component of $\\mathcal{T}$ and each infinite connected component of $\\mathcal{T}$ has exactly one or two directions to infinity.\nLet $S$ be such a connected component.\nNote that if $S$ has one end, then we can solve $\\Pi^2_{\\text{pm}}$ on $S$ as above.\nIf $S$ has two ends, then $S$ contains a doubly infinite path $P$ with the property that erasing $P$ splits $S\\setminus P$ into finite connected components.\nIn order to solve $\\overline{\\Pi^2_{\\text{pm}}}$ we simply solve $\\Pi^2_{\\text{pm}}$ on the finite connected components of $S\\setminus P$ (with some of the vertices in $S\\setminus P$ being potentially matched with vertices on the path $P$) and declare vertices on $P$ to be line vertices.\n\nThe high-level idea to find a one or two-ended spanning forest is to do the same construction as in the measure case and understand what happens with edges that do not disappear after countably many steps.\nA slight modification in the construction guarantees that what remains are doubly infinite paths.\n\n\\begin{restatable}{theorem}{oneendedforestborel}\n\\label{thm:OneTwoEndBorel}\nLet $\\mathcal{G}$ be a Borel $\\Delta$-regular forest. \nThen there is a Borel one or two-ended spanning forest $\\mathcal{T}\\subseteq \\mathcal{G}$.\n\\end{restatable}\n\nThe proof of \\cref{thm:OneTwoEndBorel} can be found in \\cref{sec:oneortwo}.\nWe remark that the notation used in the proof is close to the notation in the proof that gives a measurable construction of a one-ended spanning forest in \\cite{BrooksMeas}.\nNext, we show more formally how \\cref{thm:OneTwoEndBorel} implies that $\\overline{\\Pi^2_{\\text{pm}}}\\in \\mathsf{BOREL}$.\n\n\\begin{theorem}\nThe problem $\\overline{\\Pi^2_{\\text{pm}}}$ is in the class $\\mathsf{BOREL}$.\n\\end{theorem}\n\\begin{proof}\n\nLet $\\mathcal{T}$ be the one or two-ended spanning forest given by \\cref{thm:OneTwoEndBorel} and $S$ be a connected component of $\\mathcal{T}$.\nIf $S$ is one-ended, then we use the first part of the proof of \\cref{thm:twomatchFFIID} to find a solution of $\\Pi^2_{\\text{pm}}$ on $S$.\nSince deciding that $S$ is one-ended as well as computing an orientation towards infinity can be done in a Borel way, this yields a Borel labeling.\n\nSuppose that $S$ is two-ended.\nThen, there exists a doubly infinite path $P$ in $S$ with the property that the connected components of $S\\setminus P$ are finite.\nMoreover, it is possible to detect $P$ in a Borel way.\nThat is, declaring the vertices on $P$ to be line vertices and using the special symbol in $\\overline{\\Pi^2_{\\text{pm}}}$ for the half-edges on $P$ yields a Borel measurable labeling.\nLet $C\\not=\\emptyset$ be one of the finite components in $S\\setminus P$ and $v\\in C$ be the vertex of distance $1$ from $P$.\nOrient the edges in $C$ towards $v$, this can be done in a Borel way since $v$ is uniquely determined for $C$.\nThen the first part of the proof of \\cref{thm:twomatchFFIID} shows that one can inductively find a solution to $\\Pi^2_{\\text{pm}}$ on $C$ in such a way that all vertices, possibly up to $v$, are matched.\nIf $v$ is matched, then we are done.\nIf $v$ is not matched, then we add the edge that connects $v$ with $P$ to the matching. Note that it is possible that multiple vertices are matched with the same path vertex, but this is not a problem according to the definition of $\\overline{\\Pi^2_{\\text{pm}}}$. \nIt follows that this defines a Borel function on $H(\\mathcal{G})$ that solves $\\overline{\\Pi^2_{\\text{pm}}}$.\n\\end{proof}\n\n\n\n\n\n\\subsubsection{Vizing's Theorem for $\\Delta=3$}\nFinding a measurable or local version of Vizing's Theorem, i.e., proper edge $(\\Delta+1)$-coloring $\\Pi_{\\chi',\\Delta+1}$, was studied recently in \\cite{grebik2020measurable,weilacher2021borel,BernshteynVizing}.\nIt is however not known, even on trees, whether $\\Pi_{\\chi',\\Delta+1}$ is in $\\mathsf{RLOCAL}(O(\\log\\log n))$.\nHere we use the one-ended forest construction to show that $\\Pi_{\\chi',\\Delta+1}$ is in the class $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon)$ for $\\Delta=3$.\n\n\n\n\\iffalse\n\\paragraph{Composition of Uniform Local Algorithms}\nWhen designing algorithms, one often composes several simple algorithms to obtain a more sophisticated algorithm. For example, an algorithm $\\mathcal{A}$ might first invoke algorithm $\\mathcal{A}_1$ and after $\\mathcal{A}_1$ finished, $\\mathcal{A}$ invokes algorithm $\\mathcal{A}_2$ with the input of $\\mathcal{A}_2$ being equal to the output of $\\mathcal{A}_1$. Then, the final output of $\\mathcal{A}$ is simply the output of $\\mathcal{A}_2$. We call such a composition a sequential composition. In the context of the classic deterministic local model, the round complexity of $\\mathcal{A}$ is simply the sum of the round complexities of the two algorithms $\\mathcal{A}_1$ and $\\mathcal{A}_2$ that are used as a subroutine. Here, we prove two such composition results in the context of uniform local algorithms. These general results will help us in \\cref{app:one_ended} to analyze the uniform complexity of an uniform algorithm that computes a so-called one-ended forest in a principled manner.\n\n\n\n\\begin{lemma}[Parallel Composition]\n\\label{lem:parallel_composition}\nLet $A_1$ and $A_2$ be two distributed uniform algorithms with an uniform local complexity of $t_{A_1}(\\varepsilon)$ and $t_{A_2}(\\varepsilon)$, respectively. \nLet $A$ be the parallel composition of $A_1$ and $A_2$. That is, the coding radius of algorithm $\\mathcal{A}$ at a vertex $u$ is the smallest radius such that $A$ knows the local answer of both $A_1$ and $A_2$ at $u$. \nThen, $t_A(\\varepsilon) \\leq \\max(t_{A_1}(\\varepsilon\/2), t_{A_2}(\\varepsilon\/2))$.\n\\end{lemma}\n\\begin{proof}\nThe probability that the coding radius of algorithm $\\mathcal{A}_1$ at vertex $u$ is larger than $\\max(t_{A_1}(\\varepsilon\/2), t_{A_2}(\\varepsilon\/2))$ is at most $\\varepsilon\/2$ and the same holds for the coding radius of algorithm $\\mathcal{A}_2$ at vertex $u$. Hence, by a simple union bound, the probability that the coding radius of algorithm $\\mathcal{A}$ at a vertex $u$ is larger than $\\max(t_{A_1}(\\varepsilon\/2), t_{A_2}(\\varepsilon\/2))$ is at most $\\varepsilon\/2 + \\varepsilon\/2 = \\varepsilon$. \\\\ Hence, $t_A(\\varepsilon) \\leq \\max(t_{A_1}(\\varepsilon\/2), t_{A_2}(\\varepsilon\/2))$, as desired.\n\\end{proof}\n\n\\fi\n\n\n\n\\begin{proposition}\nLet $\\Delta=3$. We have $\\Pi_{\\chi',\\Delta+1} \\in\\mathsf{OLOCAL}(\\operatorname{poly} \\log 1\/\\varepsilon)$.\n\\end{proposition}\n\\begin{proof}[Proof sketch.]\nBy \\cref{thm:one_ended_forest}, we can compute a one-ended forest decomposition $\\mathcal{T}$ with uniform local complexity $O(\\operatorname{poly} \\log 1\/\\varepsilon)$. \nNote that every vertex has at least one edge in $\\mathcal{T}$, hence the edges in $G \\setminus \\mathcal{T}$ form paths. \nThese paths can be $3$-edge colored by using the uniform version of Linial's coloring algorithm. This algorithm has an uniform local complexity of $O(\\log^* 1\/\\varepsilon)$. By \\cref{lem:sequential_composition}, the overall complexity is $O(\\operatorname{poly} \\log (\\Delta^{\\log^* 1\/\\varepsilon} \/ \\varepsilon) + \\log^* 1\/\\varepsilon)) = O(\\operatorname{poly} \\log 1\/\\varepsilon)$. \n\nFinally, we color the edges of $T$. We start from the leaves and color the edges inductively. In particular, whenever we consider a vertex $v$ we color the at most two edges in $T^{\\leftarrow}(v)$ below $v$. \nNote that there always is at least one uncolored edge going from $v$ in the direction of $T$. Hence, we can color the at most two edges below $v$ in $T^{\\leftarrow}(v)$ greedily -- each one neighbors with at most $3$ colored edges at any time. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{thm:one_ended_forest}} \n\\label{sec:formalproof}\n\n\nIn this section we formally prove \\cref{thm:one_ended_forest}. \n\n\n\\oneendedtreetail*\n\nWe follow the construction of the one-ended spanning forest in \\cite{BrooksMeas}.\nThe construction proceeds in rounds. Before each round, some subset of the vertices have already decided which incident edge to grab, and we refer to these vertices as settled vertices. All the remaining vertices are called unsettled. The goal in each round is to make a large fraction of the unsettled vertices settled. To achieve this goal, our construction relies on certain properties that the graph induced by all the unsettled vertices satisfies. \n\nOne important such property is that the graph induced by all the unsettled vertices is expanding. That is, the number of vertices contained in the neighborhood around a given vertex grows exponentially with the radius. The intuitive reason why this is a desirable property is the following. Our algorithm will select a subset of the unsettled vertices and clusters each unsettled vertex to the closest selected vertex. As the graph is expanding and any two vertices in the selected subset are sufficiently far away, there will be a lot of edges leaving a given cluster. For most of these inter-cluster edges, both of the endpoints will become settled. This in turn allows one to give a lower bound on the fraction of vertices that become settled in each cluster.\n\nTo ensure that the graph induced by all the unsettled vertices is expanding, a first condition we impose on the graph induced by all the unsettled vertices is that it has a minimum degree of at least $2$. While this condition is a first step in the right direction, it does not completely suffice to ensure the desired expansion, as an infinite path has a minimum degree of $2$ but does not expand sufficiently. Hence, our algorithm will keep track of a special subset of the unsettled vertices, the so-called hub vertices. Each hub vertex has a degree of $\\Delta$ in the graph induced by all the unsettled vertices. That is, all of the neighbors of a hub vertex are unsettled as well. Moreover, each unsettled vertex has a bounded distance to the closest hub vertex, where the specific upper bound on the distance to the closest hub vertex increases with each round. As we assume $\\Delta > 2$, the conditions stated above suffice to show that the graph induced by all the unsettled vertices expands.\n\nNext, we explain in more detail how a vertex decides which edge to grab. Concretely, if a vertex becomes settled, it grabs an incident edge such that the other endpoint of that edge is strictly closer to the closest unsettled vertex as the vertex itself. For example, if a vertex becomes settled and there is exactly one neighbor that is still unsettled, then the vertex will grab the edge that the vertex shares with its unsettled neighbor. Grabbing edges in that way, one can show two things. First, for a settled vertex $v$, the set $T^{\\leftarrow}(v)$ of vertices behind $v$ does not change once $v$ becomes settled. The intuitive reason for this is that the directed edges point towards the unsettled vertices and therefore the unsettled vertices lie before $v$ and not after $v$. Moreover, one can also show that $T^{\\leftarrow}(v)$ only contains finitely many vertices. The reason for this is that at the moment a vertex $v$ becomes settled, there exists an unsettled vertex that is sufficiently close to $v$, where the exact upper bound on that distance again depends on the specific round in which $v$ becomes settled. \n\nWhat remains to be discussed is at which moment a vertex decides to become settled. As written above, in each round the algorithm considers a subset of the unsettled vertices and each unsettled vertex is clustered to the closest vertex in that subset. This subset only contains hub vertices and it corresponds to an MIS on the graph with the vertex set being equal to the set of hub vertices and where two hub vertices are connected by an edge if they are sufficiently close in the graph induced by all the unsettled vertices. Now, each cluster center decides to connect to exactly $\\Delta$ different neighboring clusters, one in each of its $\\Delta$ subtrees. Remember that as each cluster center is a hub vertex, all of its neighbors are unsettled as well. Now, each unsettled vertex becomes settled except for those that lie on the unique path between two cluster centers such that one of the cluster centers decided to connect to the other one.\n\n\n\n\n\n\n\\paragraph{Construction}\nThe algorithm proceeds in rounds. After each round, a vertex is either settled or unsettled and a settled vertex remains settled in subsequent rounds. Moreover, some unsettled vertices are so-called hub vertices. We denote the set of unsettled, settled and hub vertices after the $i$-th round with $U_i$, $S_i$ and $H_i$, respectively. We set $U_0 = H_0 = V$ prior to the first round and we always set $S_i = V \\setminus U_i$. \\cref{fig:constr1} illustrates the situation after round $i$. Moreover, we denote with $O_i$ a partial orientation of the edges of the infinite $\\Delta$-regular input tree $T$. In the beginning, $O_0$ corresponds to the partial orientation with no oriented edges. Each vertex is incident to at most $1$ outwards oriented edge in $O_i$. If a vertex is incident to an outwards oriented edge in $O_i$, then the other endpoint of the edge will be its parent in the one-ended forest decomposition. For each vertex $v$ in $S_i$, we denote with $T_{v,i}$ the smallest set that satisfies the following conditions. First, $v \\in T_{v,i}$. Moreover, if $u \\in T_{v,i}$ and $\\{w,u\\}$ is an edge that according to $O_i$ is oriented from $w$ to $u$, then $w \\in T_{v,i}$. We later show that $T_{v,i}$ contains exactly those vertices that are contained in the subtree $\\mathcal{T}^\\leftarrow(v)$.\n\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics{fig\/one_ended_construction.pdf}\n \\caption{The picture illustrates the situation after some round $i$ of the construction. The vertices on the paths are all contained in $U_i$, while the vertices corresponding to the big black dots are additionally contained in $H_i$. Note that the length of each path is at least $d_i$. However, the distance between $u$ and $v$ in the original graph can be much smaller.}\n \\label{fig:constr1}\n\\end{figure}\n\n\nAfter the $i$-th round, the construction satisfies the following invariants.\n\n\n\n\\begin{enumerate}\n \n \\item For each $v \\in S_{i-1}$, we have $T_{v,i} = T_{v,i-1}$.\n \n \\item Let $v$ be an arbitrary vertex in $S_i$. Then, $T_{v,i}$ contains finitely many vertices and furthermore $T_{v,i} \\subseteq S_i$.\n \n \\item The minimum degree of the graph $T[U_i]$ is at least $2$ and each vertex in $H_i$ has a degree of $\\Delta$ in $T[U_i]$.\n \n \\item Each vertex in $U_i$ has a distance of at most $\\sum_{j=0}^i d_j$ to the closest vertex in $H_i$ in the graph $T[U_i]$.\n\\end{enumerate}\n\nWe now describe how to compute $U_i, S_i$, $H_i$ and $O_i$. The construction during the $i$-th round is illustrated in $\\cref{fig:constr2}$. Note that we can assume that the invariants stated above are satisfied after the $(i-1)$-th round. In the $i$-th round we have a parameter $d_i$ that we later set to $2^{2^i}$.\n\n\\begin{enumerate}\n \\item $H_i$ is a subset of $H_{i-1}$ that satisfies the following property. No two vertices in $H_i$ have a distance of at most $d_i$ in $T[U_{i-1}]$. Moreover, each vertex in $H_{i-1} \\setminus H_i$ has a distance of at most $d_i$ to the closest vertex in $H_i$ in the graph $T[U_{i-1}]$. Note that we can compute $H_i$ by computing an MIS in the graph with vertex set $H_{i-1}$ and where two vertices are connected iff they have a distance of at most $d_i$ in the graph $T[U_{i-1}]$. Hence, we can use Ghaffari's MIS algorithm \\cite{ghaffari2016MIS} to compute the set $H_i$.\n \\item Next, we describe how to compute $U_i$. \n We assign each vertex $u \\in U_{i-1}$ to the closest vertex in the graph $T[U_{i-1}]$ that is contained in $H_i$, with ties being broken arbitrarily. We note that there exists a node in $H_i$ with a distance of at most $(\\sum_{j=0}^{i-1} d_j) + d_i$ to $u$. To see why, note that Invariant $(4)$ from round $i - 1$ implies that there exists a vertex $w$ in $H_{i-1}$ with a distance of at most $\\sum_{j=0}^{i-1} d_j$ to $u$. We are done if $w$ is also contained in $H_i$. If not, then it follows from the way we compute $H_i$ that there exists a vertex in $H_i$ with a distance of at most $d_i$ to $w$, but then the triangle inequality implies that the distance from $u$ to that vertex is at most $(\\sum_{j=0}^{i-1} d_j) + d_i$, as desired. For each vertex $v \\in H_i$, we denote with $C(v)$ the set of all vertices in $U_{i-1}$ that got assigned to $v$. Now, let $E_v$ denote the set of edges that have exactly one endpoint in $C(v)$. We can partition $E_v$ into $E_{v,1} \\sqcup E_{v,2} \\sqcup \\ldots \\sqcup E_{v,\\Delta}$, where for $\\ell \\in [\\Delta]$, $E_{v,\\ell}$ contains all the edges in $E_v$ that are contained in the $\\ell$-th subtree of $v$. \n Invariants $(3)$ and $(4)$ after the $(i-1)$-th round imply that $E_{v,\\ell} \\neq \\emptyset$. Moreover, $E_{v,\\ell}$ contains only finitely many edges, as we have shown above that the cluster radius is upper bounded by $\\sum_{j=0}^{i} d_j$.\n \n Now, for $\\ell \\in [\\Delta]$, we choose an edge $e_\\ell$ uniformly at random from the set $E_{v,\\ell}$. Let $u_\\ell \\neq v$ be the unique vertex in $H_i$ such that one endpoint of $e_\\ell$ is contained in $C(u_\\ell)$. We denote with $P_{v,\\ell}$ the set of vertices that are contained in the unique path between $v$ and $u_\\ell$. Finally, we set $U_i = \\bigcup_{v \\in H_i, \\ell \\in [\\Delta]} P_{v,\\ell}$.\n \\item It remains to describe how to compute the partial orientation $O_i$. All edges that are oriented in $O_{i-1}$ will be oriented in the same direction in $O_i$. Additionally, we orient for each vertex $u$ that got settled in the $i$-th round, i.e., $u \\in S_i \\cap U_{i-1}$, exactly one incident edge away from $u$. Let $w$ be the closest vertex to $u$ in $T[U_{i-1}]$ that is contained in $U_i$, with ties being broken arbitrarily. We note that it follows from the discussions above that the distance between $u$ and $w$ in the graph $T[U_{i-1}]$ is at most $\\sum_{j=0}^{i} d_j$. We now orient the edge incident to $u$ that is on the unique path between $u$ and $w$ outwards. We note that this orientation is well-defined in the sense that we only orient edges that were not oriented before and that we don't have an edge such that both endpoints of that edge want to orient the edge outwards.\n\\end{enumerate}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics{fig\/one_ended_construction2.pdf}\n \\caption{The picture illustrates the situation during some round $i$ of the construction. The vertices on all paths are contained in $U_{i-1}$ and the vertices corresponding to the big black dots are contained in $H_{i-1}$. The vertices $v$, $u_1$, $u_2$ and $u_3$ are the only vertices in the picture that are contained in $H_i$. All the vertices on the red paths remain unsettled during round $i$ while all the vertices on the black paths become settled. Each vertex that becomes settled during the $i$-th round orients one incident edge outwards, illustrated by the green edges. Each green edge is oriented towards the closest vertex that is still unsettled after the $i$-th round. We note that some of the edges in the picture are oriented away from $v$.}\n \\label{fig:constr2}\n\\end{figure}\n\n\nWe now prove by induction that our procedure satisfies all the invariants.\nPrior to the first round, all invariants are trivially satisfied. Hence, it remains to show that the invariants hold after the $i$-th round, given that the invariants hold after the $(i-1)$-th round. \n\n\\begin{enumerate}\n\n \\item Let $v \\in S_{i-1}$ be arbitrary. As $O_i$ is an extension of $O_{i-1}$, it follows from the definition of $T_{v,i-1}$ and $T_{v,i}$ that $T_{v,i-1} \\subseteq T_{v,i}$. Now assume that $T_{v,i} \\not \\subseteq T_{v,i-1}$. This would imply the existence of an edge $\\{u,w\\}$ such that $u \\in T_{v,i-1}$ and the edge was oriented during the $i$-th round from $w$ to $u$. As $u \\in T_{v,i-1}$, Invariant $(2)$ from the $(i-1)$-th round implies $u \\in S_{i-1}$. This is a contradiction as during the $i$-th round only edges with both endpoints in $U_{i-1}$ can be oriented. Therefore it holds that $T_{v,i} = T_{v,i-1}$, as desired.\n\n \\item Let $v \\in S_i$ be arbitrary. If it also holds that $v \\in S_{i-1}$, then the invariant from round $i-1$ implies that $T_{v,i-1} = T_{v,i}$ contains finitely many vertices and $T_{v,i-1} \\subseteq S_{i-1} \\subseteq S_i$. Thus, it suffices to consider the case that $v \\in S_i \\cap U_{i-1}$. Note that $T_{v,i} \\cap U_i = \\emptyset$. Otherwise there would exist an edge that is oriented away from a vertex in $U_i$ according to $O_i$, but this cannot happen according to the algorithm description. Hence, $T_{v,i} \\subseteq S_i$. Now, for the sake of contradiction, assume that $T_{v,i}$ contains infinitely many vertices. From the definition of $T_{v,i}$, this implies that there exists a sequence of vertices $(v_k)_{k \\geq 1}$ with $v = v_1$ such that for each $k \\geq 1$, $\\{v_{k+1},v_k\\}$ is an edge in $T$ that is oriented from $v_{k+1}$ to $v_k$ according to $O_i$. For each $k \\geq 1$ we have $v_k \\in S_i$. We furthermore know that $v_k \\in U_{i-1}$, as otherwise $T_{v_k,i} = T_{v_k,i-1}$ would contain infinitely many vertices, a contradiction. From the way we orient the edges, a simple induction proof implies that the closest vertex of $v_k$ in the graph $T[U_{i-1}]$ that is contained in $U_i$ has a distance of at least $k$. However, we previously discussed that the distance between $v_k$ and the closest vertex in $U_i$ in the graph $T[U_{i-1}]$ is at most $\\sum_{j=0}^{i} d_j$. This is a contradiction. Hence, $T_{v,i}$ contains finitely many vertices.\n \n \\item It follows directly from the description of how to compute $U_i$ and $H_i$ that the minimum degree of the graph $T[U_i]$ is at least $2$ and that each vertex in $H_i$ has a degree of $\\Delta$ in $T[U_i]$.\n \n \\item Let $u \\in U_i$ be arbitrary. We need to show that $u$ has a distance of at most $\\sum_{j=0}^i d_j$ to the closest vertex in $H_i$ in the graph $T[U_i]$. Let $v$ be the vertex with $u \\in C(v)$. We know that there is no vertex in $H_i$ that is closer to $u$ in $T[U_{i-1}]$ than $v$. Hence, the distance between $u$ and $v$ is at most $\\sum_{j=0}^i d_j$ in the graph $T[U_{i-1}]$. Moreover, from the description of how we compute $U_i$, it follows that all the vertices on the unique path between $u$ and $v$ are contained in $U_i$. Hence, each vertex in $U_i$ has a distance of at most $\\sum_{j=0}^i d_j$ to the closest vertex in $H_i$ in the graph $T[U_i]$, as desired.\n \\end{enumerate}\n\nWe derive an upper bound on the coding radius of the algorithm in three steps. First, for each $i \\in \\mathbb{N}$ and $\\varepsilon > 0$, we derive an upper bound on the coding radius for computing with probability at least $1 - \\varepsilon$ for a given vertex in which of the sets $S_i, U_i$ and $H_i$ it is contained in and for each incident edge whether and how it is oriented in $O_i$, given that each vertex receives as additional input in which of the sets $S_{i-1}, U_{i-1}$ and $H_{i-1}$ it is contained in and for each incident edge whether and how it is oriented in $O_{i-1}$. Given this upper bound on the coding radius, we can use the sequential composition lemma (\\cref{lem:sequential_composition}) and a simple induction proof to give an upper bound on the coding radius for computing with probability at least $1 - \\varepsilon$ for a given vertex in which of the sets $S_i, U_i$ and $H_i$ it is contained in and for each incident edge whether and how it is oriented in $O_i$, this time without providing any additional input. \n\nSecond, we analyze after how many rounds a given vertex is settled with probability at least $1- \\varepsilon$ for a given $\\varepsilon > 0$. \n\nFinally, we combine these two upper bounds to prove \\cref{thm:one_ended_forest}.\n \n \n\\begin{lemma}\n\\label{lem:tail1}\nFor each $i \\in \\mathbb{N}$ and $\\varepsilon \\in (0,0.01]$, let $g_i(\\varepsilon)$ denote the smallest coding radius such that with probability at least $1 - \\varepsilon$ one knows for a given vertex in which of the sets $S_i, U_i$ and $H_i$ it is contained in and for each incident edge whether and how it is oriented in $O_i$, given that each vertex receives as additional input in which of the sets $S_{i-1}, U_{i-1}$ and $H_{i-1}$ it is contained in and for each incident edge whether and how it is oriented in $O_{i-1}$. It holds that $g_i(\\varepsilon) = O(d^2_i \\log (\\Delta\/\\varepsilon))$.\n\\end{lemma}\n\n\\begin{proof}\n When running Ghaffari's uniform MIS algorithm on some graph $G'$ with maximum degree $\\Delta'$, each vertex knows with probability at least $1-\\varepsilon$ whether it is contained in the MIS or not after $O(\\log (\\Delta') + \\log (1\/\\varepsilon))$ communication rounds in $G'$ (\\cite{ghaffari2016MIS}, Theorem 1.1). For the MIS computed during the $i$-th round, $\\Delta' =\\Delta^{O(d_i)}$ and each communication round in $G'$ can be simulated with $O(d_i)$ communication rounds in the tree $T$. Hence, to know for a given vertex with probability at least $1-\\varepsilon$ whether it is contained in the MIS or not, it suffices consider the $O(d_i^2\\log(\\Delta) + d_i\\log(1\/\\varepsilon))$-hop neighborhood around that vertex. Now, let $u$ be an arbitrary vertex. In order to compute in which of the sets $S_i, U_i$ and $H_i$ $u$ is contained in and for each incident edge of $u$ whether and how it is oriented in $O_i$, we not only need to know whether $u$ is in the MIS or not. However, it suffices if we know for all the vertices in the $O(d_i)$-hop neighborhood of $u$ whether they are contained in the MIS or not (on top of knowing for each vertex in the neighborhood in which of the sets $S_{i-1}, U_{i-1}$ and $H_{i-1}$ it is contained in and for each incident edge whether and how it is oriented in $O_{i-1}$). Hence, by a simple union bound over the at most $\\Delta^{O(d_i)}$ vertices in the $O(d_i)$-hop neighborhood around $u$, we obtain $g_i(\\varepsilon) = O(d_i) + O(d_i^2\\log(\\Delta) + d_i\\log(\\Delta^{O(d_i)}\/\\varepsilon)) = O(d_i^2 \\log(\\Delta\/\\varepsilon))$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:tail2}\nFor each $i \\in \\mathbb{N}$ and $\\varepsilon \\in (0,0.01]$, let $h_i(\\varepsilon)$ denote the smallest coding radius such that with probability at least $1 - \\varepsilon$ one knows for a given vertex in which of the sets $S_i, U_i$ and $H_i$ it is contained in and for each incident edge whether and how it is oriented in $O_i$. Then, there exists a constant $c$ independent of $\\Delta$ such that $h_i(\\varepsilon) \\leq 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta\/\\varepsilon)$.\n\\end{lemma}\n\n\\begin{proof}\nBy prove the statement by induction on $i$. For a large enough constant $c$, it holds that $h_1(\\varepsilon) \\leq 2^{2^{3}} \\cdot(c\\log(\\Delta)) \\log(\\Delta\/\\varepsilon)$. Now, consider some arbitrary $i$ and assume that $h_i(\\varepsilon) \\leq 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta\/\\varepsilon)$ for some large enough constant $c$. We show that this implies $h_{i+1}(\\varepsilon) \\leq 2^{2^{(i+1)+2}} \\cdot(c\\log(\\Delta))^{i+1} \\log(\\Delta\/\\varepsilon)$.\nBy the sequential composition lemma (\\cref{lem:sequential_composition}) and assuming that $c$ is large enough, we have\n\n\\begin{align*}\n h_{i+1}(\\varepsilon) &\\leq h_{i}((\\varepsilon\/2)\/\\Delta^{g_{i+1}(\\varepsilon\/2) + 1}) + g_{i+1}(\\varepsilon\/2) \\\\\n &\\leq 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta \\cdot \\Delta^{g_{i+1}(\\varepsilon\/2) + 1}\/(\\varepsilon\/2)) + g_{i+1}(\\varepsilon\/2) \\\\\n &\\leq 2 \\cdot 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta \\cdot \\Delta^{g_{i+1}(\\varepsilon\/2) + 1}\/(\\varepsilon\/2)) \\\\\n &\\leq 2 \\cdot 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta^{(c\/10) \\cdot 2^{2^{i+2}}\\log(\\Delta\/\\varepsilon)}\/(\\varepsilon\/2)) \\\\\n &\\leq 4 \\cdot 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta^{(c\/10) \\cdot 2^{2^{i+2}}\\log(\\Delta\/\\varepsilon)}) \\\\\n &\\leq (4\/10) \\cdot 2^{2^{i+2}} \\cdot 2^{2^{i+2}} (c \\log(\\Delta))^{i+1} \\log(\\Delta\/\\varepsilon) \\\\\n &\\leq 2^{2^{(i+1)+2}} \\cdot(c\\log(\\Delta))^{i+1} \\log(\\Delta\/\\varepsilon),\n\\end{align*}\n\nas desired.\n\n\\end{proof}\n\n\n \n \n\n\\begin{lemma}\\label{lm:tail0}\nLet $u$ be an arbitrary vertex. For each $\\varepsilon \\in (0,0.01]$, let $f(\\varepsilon)$ denote the smallest $i \\in \\mathbb{N}$ such that $u$ is settled after the $i$-th round with probability at least $1 - \\varepsilon$. There exists a fixed $c \\in \\mathbb{R}$ independent of $\\Delta$ such that $f(\\varepsilon) \\leq \\lceil 1 + \\log \\log \\frac{1}{c} \\log_\\Delta(1\/\\varepsilon)\\rceil$.\n\\end{lemma}\n\n\\begin{proof}\n\nLet $i \\in \\mathbb{N}$ be arbitrary. We show that a given vertex is settled after the $i$-th round with probability at least $1 - O(1\/\\Delta^{\\Omega(d_i\/d_{i-1})})$. For the sake of analysis, we run the algorithm on a finite $\\Delta$-regular high-girth graph instead of an infinite $\\Delta$-regular tree. For now, we additionally assume that no vertex realizes that we don't run the algorithm on an infinite $\\Delta$-regular tree. That is, we assume that for each vertex the coding radius to compute all its local information after the $i$-th round is much smaller than the girth of the graph. \n\nWith this assumption, we give a deterministic upper bound on the fraction of vertices that are not settled after the $i$-th round. On the one hand, the number of vertices that are not settled after the $i$-th round can be upper bounded by $|H_i| \\cdot \\Delta \\cdot O(d_i)$. On the other hand, we will show that the fraction of vertices that are contained in $H_i$ is upper bounded by $1\/\\Delta^{\\Omega(d_i\/d_{i-1})}$. We do this by showing that $C(v)$ contains $\\Delta^{\\Omega(d_i\/d_{i-1})}$ many vertices for a given $v \\in H_i$. Combining these two bounds directly implies that the fraction of unsettled vertices is smaller than \n\n\\[\\frac{\\Delta \\cdot O(d_i)}{\\Delta^{\\Omega(d_i\/d_{i-1})}} = 1\/\\Delta^{\\Omega(d_i\/d_{i-1})}.\\]\n\n\nLet $v \\in H_i$ be arbitrary. We show that $|C(v)| = \\Delta^{\\Omega(d_i\/d_{i-1})}$. Using Invariants (3) and (4) together with a simple induction argument, one can show that there are at least $(\\Delta-1)^{\\lfloor D \/ ((2\\sum_{j=0}^{i-1}d_{j}) + 1)\\rfloor}$ vertices contained in $H_{i-1}$ and whose distance to $v$ is at most $D$ in the graph induced by the vertices in $U_{i-1}$. Furthermore, from the way we defined the clustering $C(v)$ and the fact that two vertices in $H_i$ have a distance of at least $d_i$ in the graph $T[U_{i-1}]$, it follows that all vertices in $U_{i-1}$ having a distance of at most $d_i\/2 -1$ to $v$ are contained in the cluster $C(v)$. Hence, the total number of vertices in $C(v)$ is at least $(\\Delta-1)^{\\lfloor (d_i\/2 - 1) \/ ((2\\sum_{j=0}^{i-1}d_{j}) + 1)\\rfloor} = \\Delta^{\\Omega(d_i\/d_{i-1})}$, as promised.\n\nNow we remove the assumption that no vertex realizes that we don't run the algorithm on an infinite $\\Delta$-regular tree. If a vertex does realize that we don't run the algorithm on an infinite $\\Delta$-regular tree, then we consider the vertex as being unsettled after the $i$-th round. By considering graphs with increasing girth, we can make the expected fraction of vertices that realize that we are not on an infinite $\\Delta$-regular tree arbitrarily small. Combining this observation with the previous discussion, this implies that the expected fraction of vertices that are not settled after the $i$-th round is at most $1\/\\Delta^{\\Omega(d_i\/d_{i-1})}$. By symmetry, each vertex has the same probability of being settled after the $i$-th round. Hence, the probability that a given vertex is settled after the $i$-th round when run on a graph with sufficiently large girth is $1 - 1\/\\Delta^{\\Omega(d_i\/d_{i-1})}$, and the same holds for each vertex when we run the algorithm on an infinite $\\Delta$-regular tree. Thus, there exists a constant $c$ such that the probability that a given vertex is unsettled after the $i$-th round is at most $1\/\\Delta^{c \\cdot d_i\/d_{i-1}} = 1\/\\Delta^{c \\cdot 2^{2^{i-1}}}$. Setting $i = \\lceil 1 + \\log \\log \\frac{1}{c} \\log_\\Delta(1\/\\varepsilon)\\rceil$ finishes the proof.\n\\end{proof}\n\n\n\n\n\nWe are now finally ready to finish the proof of \\cref{thm:one_ended_forest}. Let $u$ be an arbitrary vertex and $\\varepsilon \\in (0,0.01]$. We need to compute an upper bound on the coding radius that is necessary for $u$ to know all the vertices in $\\mathcal{T}^{\\leftarrow}(u)$ with probability at least $1-\\varepsilon$. To compute such an upper bound for the required coding radius it suffices to find an $i^*$ and an $R^*$ such that the following holds. First, $u$ is settled after $i^*$ rounds with probability at least $1 - \\varepsilon\/2$. Second, $u$ knows for each edge in its $O(d_{i^*})$-hop neighborhood whether it is oriented in the partial orientation $O_{i^*}$, and if yes, in which direction, by only considering its $R^*$-hop neighborhood with probability at least $1 - \\varepsilon\/2$. By a union bound, both of these events occur with probability at least $1-\\varepsilon$. Moreover, if both events occur $u$ knows all the vertices in the set $\\mathcal{T}^{\\leftarrow}(u)$. The reason is as follows. If $u$ is settled after round $i^*$, then it follows from the previous analysis that only vertices in the $O(d_{i^*})$-hop neighborhood of $u$ can be contained in $\\mathcal{T}^{\\leftarrow}(u)$. Moreover, for each vertex in its $O(d_{i^*})$-hop neighborhood, $u$ can determine if the vertex is contained in $\\mathcal{T}^{\\leftarrow}(u)$ if it knows all the edge orientations on the unique path between itself and that vertex after the $i^*$-th round. Hence, it remains to find concrete values for $i^*$ and $R^*$. According to \\cref{lm:tail0}, we can choose $i^* = \\lceil 1 + \\log \\log \\frac{1}{c} \\log_\\Delta (2\/\\varepsilon)\\rceil$ for some large enough constant $c$. Moreover, it follows from a union bound that all vertices in the $O(d_{i^*})$-hop neighborhood around $u$ know with probability at least $1-\\varepsilon\/2$ the orientation of all its incident edges according to $O_{i^*}$ by only considering their $h_{i^*}((\\varepsilon\/2)\/\\Delta^{O(d_{i^*})})$-hop neighborhood. Hence, we can set \n\n\\begin{align*}\nR^* &= O(d_{i^*}) + h_{i^*}((\\varepsilon\/2)\/\\Delta^{O(d_{i^*})}) \\\\\n &\\leq O(d_{i^*}) + 2^{2^{i^* + 2}} \\cdot (c \\log(\\Delta))^{i^*}\\log(\\Delta^{O(d_{i^*})}\/\\varepsilon) \\\\\n &= O(2^{2^{i^*}}\\cdot 2^{2^{i^* + 2}} (c \\log(\\Delta))^{i^*}\\log(\\Delta\/\\varepsilon)) \\\\\n &= O(2^{2^{i^* + 3}} (c \\log(\\Delta))^{i^*}\\log(\\Delta\/\\varepsilon)) \\\\\n &= O(2^{2^{\\log \\log \\frac{1}{c} \\log (2\/\\varepsilon) + 5}} (c \\log(\\Delta))^{\\log \\log \\frac{1}{c} \\log (2\/\\varepsilon) + 2}\\log(\\Delta\/\\varepsilon)) \\\\\n &= \\log(1\/\\varepsilon)^{32} \\cdot \\log(\\Delta\/\\varepsilon) \\cdot \\log\\log(1\/\\varepsilon)^{O(\\log\\log(\\Delta))}.\n\\end{align*}\n\nAs $\\Delta = O(1)$, it therefore holds that\n\n\\[P(R(v)) = \\operatorname{poly}(\\log(1\/\\varepsilon)) \\geq 1 - \\varepsilon, \\]\n\nas desired.\n\n\n\n\n\n\n\\section{$\\mathsf{BAIRE} = \\mathsf{LOCAL}(O(\\log n))$}\n\\label{sec:baire}\n\nIn this section, we show that on $\\Delta$-regular trees the classes $\\mathsf{BAIRE}$ and $\\mathsf{LOCAL}(O(\\log(n)))$ are the same.\nAt first glance, this result looks rather counter-intuitive.\nThis is because in finite $\\Delta$-regular trees every vertex can see a leaf of distance $O(\\log(n))$, while there are no leaves at all in an infinite $\\Delta$-regular tree.\nHowever, there is an intuitive reasons why these classes are the same: in both setups there is a technique to decompose an input graph into a hierarchy of subsets. Furthermore, the existence of a solution that is defined inductively with respect to these decompositions can be characterized by the same combinatorial condition of Bernshteyn \\cite{Bernshteyn_work_in_progress}.\nWe start with a high-level overview of the decomposition techniques used in both contexts.\n\n\n\\paragraph{{\\sf Rake}\\ and {\\sf Compress}}\nThe hierarchical decomposition in the context of distributed computing is based on a variant of a decomposition algorithm of\nMiller and Reif~\\cite{MillerR89}.\nTheir original decomposition algorithm works as follows.\nStart with a tree $T$, and repeatedly apply the following two operations alternately: {\\sf Rake}\\ (remove all degree-1 vertices) and {\\sf Compress}\\ (remove all degree-2 vertices).\nThen $O(\\log n)$ iterations suffice to remove all vertices in $T$~\\cite{MillerR89}. To view it another way, this produces a decomposition of the vertex set $V$ into $2L - 1$ layers \\[V = \\VR{1} \\cup \\VC{1} \\cup \\VR{2} \\cup \\VC{2} \\cup \\VR{3} \\cup \\VC{3} \\cup \\cdots \\cup \\VR{L},\\] \nwith $L= O(\\log n)$, where $\\VR{i}$ is the set of vertices removed during the $i$-th {\\sf Rake}\\ operation and $\\VC{i}$ is the set of vertices removed during the $i$-th {\\sf Compress}\\ operation. We will use a variant~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} of this decomposition in the proof of \\cref{thm:ellfull_to_logn}.\n\n\n\n\nVariants of this decomposition turned out to be useful in designing $\\mathsf{LOCAL}$ algorithms~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup,ChangHLPU20,chang2020n1k_speedups}. \nIn our context, we assume that the given LCL satisfies a certain combinatorial condition and then find a solution inductively, in the reversed order of the construction of the decomposition.\nNamely, in the {\\sf Rake}\\ step we want to be able to existentially extend the inductive partial solution to all relative degree $1$-vertices (each $v \\in \\VR{i}$ has degree at most 1 in the subgraph induced by $\\VR{i} \\cup \\cdots \\cup \\VR{L}$) and in the {\\sf Compress}\\ step we want to extend the inductive partial solution to paths with endpoints labeled from the induction (the vertices in $\\VC{i}$ form degree-2 paths in the subgraph induced by $\\VC{i} \\cup \\cdots \\cup \\VR{L}$).\n\n\n\\paragraph{$\\TOAST$}\nFinding a hierarchical decomposition in the context of descriptive combinatorics is tightly connected with the notion of \\emph{Borel hyperfiniteness}.\nUnderstanding what Borel graphs are Borel hyperfinite is a major theme in descriptive set theory \\cite{doughertyjacksonkechris,gaojackson,conley2020borel}.\nIt is known that grids, and generally polynomial growth graphs are hyperfinite, while, e.g., acyclic graphs are not in general hyperfinite \\cite{jackson2002countable}.\nA strengthening of hyperfiniteness that is of interest to us is called a \\emph{toast} \\cite{gao2015forcing,conleymillerbound}.\nA $q$-toast, where $q\\in \\mathbb{N}$, of a graph $G$ is a collection $\\mathcal{D}$ of fintie subsets of $G$ with the property that (i) every pair of vertices is covered by an element of $\\mathcal{D}$ and (ii) the boundaries of every $D\\not=E\\in \\mathcal{D}$ are at least $q$ apart.\nThe idea to use a toast structure to solve LCLs appears in \\cite{conleymillerbound} and has many applications since then \\cite{gao2015forcing,marksunger}.\nThis approach has been formalized in \\cite{grebik_rozhon2021toasts_and_tails}, where the authors introduce $\\TOAST$ algorithms.\nRoughly speaking, an LCL $\\Pi$ admits a $\\TOAST$ algorithm if there is $q\\in \\mathbb{N}$ and a partial extending function (the function is given a finite subset of a tree that is partially colored and outputs an extension of this coloring on the whole finite subset) that has the property that whenever it is applied inductively to a $q$-toast, then it produces a $\\Pi$-coloring.\nAn advantage of this approach is that once we know that a given Borel graph admits, e.g., a Borel toast structure and a given LCL $\\Pi$ admits a $\\TOAST$ algorithm, then we may conclude that $\\Pi$ is in the class $\\mathsf{BOREL}$.\nSimilarly for $\\mathsf{MEASURE}$, $\\mathsf{BAIRE}$ or $\\mathsf{OLOCAL}$, we refer the reader to \\cite{grebik_rozhon2021toasts_and_tails} for more details and results concerning grids.\n\nIn the case of trees there is no way of constructing a Borel toast in general, however, it is a result of Hjorth and Kechris \\cite{hjorth1996borel} that every Borel graph is hyperfinite on a comeager set for every compatible Polish topology.\nA direct consequence of \\cite[Lemma~3.1]{marks2016baire} together with a standard construction of toast via Voronoi cells gives the following strengthening to toast.\nWe include a sketch of the proof for completeness.\n\n\\begin{restatable}{proposition}{BaireToast}\n\\label{pr:BaireToast}\nLet $\\mathcal{G}$ be a Borel graph on a Polish space $(X,\\tau)$ with degree bounded by $\\Delta\\in \\mathbb{N}$.\nThen for every $q>0$ there is a Borel $\\mathcal{G}$-invariant $\\tau$-comeager set $C$ on which $\\mathcal{G}$ admits a Borel $q$-toast.\n\\end{restatable}\n\\begin{proof}[Proof sketch]\nLet $\\{A_n\\}_{n\\in \\mathbb{N}}$ be a sequence of MIS with parameter $f(n)$ as in \\cite[Lemma~3.1]{marks2016baire} for a sufficiently fast growing function $f(n)$, e.g., $f(n)=(2q)^{n^2}$.\nThen $C=\\bigcup_{n\\in \\mathbb{N}}A_n$ is a Borel $\\tau$-comeager set that is $\\mathcal{G}$-invariant.\nWe produce a toast structure in a standard way, e.g., see~\\cite[Appendix~A]{grebik_rozhon2021toasts_and_tails}.\n\nLet $B_n(x)$ denote the ball of radius $f(n)\/3$ and $R_n(x)$ the ball of radius $f(n)\/4$ around $x\\in A_n$.\nIteratively, define cells $\\mathcal{D}_n=\\{C_{n}(x)\\}_{x\\in A_n}$ as follows.\nSet $C_1(x)=R_1(x)$.\nSuppose that $\\mathcal{D}_n$ has been defined and set\n\\begin{itemize}\n\t\\item $H^{n+1}(x,n+1):=R_{n+1}(x)$ for every $x\\in A_{n+1}$,\n\t\\item if $1\\le i\\le n$ and $\\left\\{H^{n+1}(x,i+1)\\right\\}_{x\\in A_{k+1}}$ has been defined, then we put\n\t$$H^{n+1}(x,i)=\\bigcup \\left\\{B_i(y):H^{n+1}(x,i+1)\\cap B_i(y)\\not=\\emptyset\\right\\}$$\n\tfor every $x\\in A_{n+1}$,\n\t\\item set $C_{n+1}(x):=H^{n+1}(x,1)$ for every $x\\in A_{n+1}$, this defines $\\mathcal{D}_{n+1}$.\n\\end{itemize}\nThe fact that $\\mathcal{D}=\\bigcup_{n\\in\\mathbb{N}} \\mathcal{D}_n$ is a $q$-toast on $C$ follows from the fact that $C=\\bigcup_{n\\in \\mathbb{N}}A_n$ together with the fact that the boundaries are $q$ separated which can be shown as in \\cite[Appendix~A]{grebik_rozhon2021toasts_and_tails}.\n\\end{proof}\n\n\n\nTherefore to understand LCLs in the class $\\mathsf{BAIRE}$ we need understand what LCLs on trees admit $\\TOAST$ algorithm.\nIt turns out that these notions are equivalent, again by using the combinatorial characterization of Bernshteyn \\cite{Bernshteyn_work_in_progress} that we now discuss.\n\n\n\\paragraph{Combinatorial Condition -- $\\ell$-full set}\nIn both decompositions, described above, we need to extend a partial coloring along paths that have their endpoints colored from the inductive step.\nThe precise formulation of the combinatorial condition that captures this demand was extracted by Bernshteyn \\cite{Bernshteyn_work_in_progress}.\nHe proved that it characterizes the class $\\mathsf{BAIRE}$ for Cayley graphs of virtually free groups.\nNote that this class contains, e.g., $\\Delta$-regular trees with a proper edge $\\Delta$-coloring.\n\n\\begin{definition}[Combinatorial condition -- an $\\ell$-full set]\n\\label{def:ellfull}\nLet $\\Pi = (\\Sigma, \\mathcal{V}, \\mathcal{E})$ be an LCL and $\\ell\\geq 2$.\nA set $\\mathcal{V}' \\subseteq \\mathcal{V}$ is \\emph{$\\ell$-full} whenever the following is satisfied.\nTake a path with at least $\\ell$ vertices, and add half-edges to it so that each vertex has degree $\\Delta$.\nTake any $c_1, c_2 \\in \\mathcal{V}'$ and label arbitrarily the half-edges around the endpoints with $c_1$ and $c_2$, respectively.\nThen there is a way to label the half-edges around the remaining $\\ell-2$ vertices with configurations from $\\mathcal{V}'$ such that all the $\\ell-1$ edges on the path have valid edge configuration on them. \n\\end{definition}\n\nNow we are ready to formulate the result that combines Bernshteyn's result \\cite{Bernshteyn_work_in_progress} (equivalence between (1.) and (2.), and the moreover part) with the main results of this section.\nThis also shows the remaining implications in \\cref{fig:big_picture_trees}.\n\n\\begin{theorem}\\label{thm:MainBaireLog}\nLet $\\Pi$ be an LCL on regular trees. \nThen the following are equivalent:\n\\begin{enumerate}\n \\item $\\Pi\\in \\mathsf{BAIRE}$,\n \\item $\\Pi$ admits a $\\TOAST$ algorithm,\n \\item $\\Pi$ admits an $\\ell$-full set,\n \\item $\\Pi\\in \\mathsf{LOCAL}(O(\\log(n)))$.\n\\end{enumerate}\nMoreover, any of the equivalent conditions is necessary for $\\Pi\\in \\mathsf{fiid}$.\n\\end{theorem}\n\nNext we discuss the proof of \\cref{thm:MainBaireLog}.\nWe refer the reader to Bernshteyn's paper \\cite{Bernshteyn_work_in_progress} for full proofs in the case of $\\mathsf{BAIRE}$ and $\\mathsf{fiid}$, here we only sketch the argument for completeness.\nWe also note that instead of using the toast construction, he used a path decomposition of acyclic graphs of Conley, Marks and Unger \\cite{conley2020measurable}.\n\n\n\\subsection{Sufficiency}\n\nWe start by showing that the combinatorial condition is sufficient for $\\mathsf{BAIRE}$ and $\\mathsf{LOCAL}(O(\\log(n)))$.\nNamely, it follows from the next results together with \\cref{pr:BaireToast} that (2.) implies all the other conditions in \\cref{thm:MainBaireLog}.\nAs discussed above the main idea is to color inductively along the decompositions.\n\n\\begin{proposition}\\label{pr:toastable}\nLet $\\Pi=(\\Sigma, \\mathcal{V}, \\mathcal{E})$ be an LCL that admits $\\ell$-full set $\\mathcal{V}'\\subseteq \\mathcal{V}$ for some $\\ell>0$.\nThen $\\Pi$ admits a $\\TOAST$ algorithm that produces a $\\Pi$-coloring for every $(2\\ell+2)$-toast $\\mathcal{D}$.\n\\end{proposition}\n\\begin{proof}[Proof sketch]\nOur aim is to build a partial extending function.\nSet $q:=2\\ell+2$.\nLet $E$ be a piece in a $q$-toast $\\mathcal{D}$ and suppose that $D_1,\\dots,D_k\\in \\mathcal{D}$ are subsets of $E$ such that the boundaries are separated.\nSuppose, moreover, that we have defined inductively a coloring of half-edges of vertices in $D=\\bigcup D_i$ using only vertex configurations from $\\mathcal{V}'$ such that every edge configuration $\\mathcal{E}$ is satisfied for every edge in $D$.\n\nWe handle each connected component of $E\\setminus D$ separately.\nLet $A$ be one of them.\nLet $u\\in A$ be a boundary vertex of $E$.\nSuch an vertex exists since every vertex in $E$ has degree $\\Delta$.\nThe distance of $u$ and any $D_i$ is at least $2\\ell+2$ for every $i\\in [k]$.\nWe orient all the edges from $A$ towards $u$.\nMoreover if $v_i\\in A$ is a boundary vertex of some $D_i$ we assign to $v_i$ a path $V_i$ of length $\\ell$ towards $u$.\nNote that $V_i$ and $V_j$ have distance at least $1$, in particular, are disjoint for $i\\not=j\\in [k]$ .\nNow, until you encounter some path $V_i$, color any in manner half-edges of vertices in $A$ inductively starting at $u$ in such a way that edge configurations $\\mathcal{E}$ are satisfied on every edge and only vertex configurations from $\\mathcal{V}'$ are used.\nUse the definition of $\\ell$-full set to find a coloring of any such $V_i$ and continue in a similar manner until the whole $A$ is colored.\n\\end{proof}\n\n\n\n\n\\begin{proposition}[$\\ell$-full $\\Rightarrow$ $\\mathsf{LOCAL}(O(\\log n))$]\\label{thm:ellfull_to_logn}\nLet $\\Pi = (\\Sigma, \\mathcal{V}, \\mathcal{E})$ be an LCL with an $\\ell$-full set $\\mathcal{V}' \\subseteq \\mathcal{V}$. Then $\\Pi$ can be solved in $O(\\log n)$ rounds in $\\mathsf{LOCAL}$.\n\\end{proposition}\n\\begin{proof}\nThe proof uses a variant of the rake-and-compress decomposition considered in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}.\n\n\\paragraph{The Decomposition}\nThe decomposition is parameterized an integer $\\ell' \\geq 1$, and it decomposes the vertices of $T$ into $2L - 1$ layers \\[V = \\VR{1} \\cup \\VC{1} \\cup \\VR{2} \\cup \\VC{2} \\cup \\VR{3} \\cup \\VC{3} \\cup \\cdots \\cup \\VR{L},\\] \nwith $L= O(\\log n)$. We write $\\GC{i}$ to denote the subtree induced by the vertices $\\left(\\bigcup_{j=i+1}^{L} \\VR{j}\\right) \\cup \\left( \\bigcup_{j=i}^{L-1} \\VC{j}\\right)$. Similarly, $\\GR{i}$ is the subtree induced by the vertices \n$\\left(\\bigcup_{j=i}^L \\VR{j}\\right) \\cup \\left( \\bigcup_{j=i}^{L-1} \\VC{j}\\right)$.\nThe sets $\\VR{i}$ and $\\VC{i}$ are required to satisfy the following requirements. \n\\begin{itemize}\n \\item Each $v \\in \\VR{i}$ has degree at most one in the graph $\\GR{i}$.\n \\item Each $v \\in \\VC{i}$ has degree exactly two in the graph $\\GC{i}$. Moreover, the $\\VC{i}$-vertices in $\\GC{i}$ form paths with $s$ vertices, with $\\ell' \\leq s \\leq 2 \\ell'$.\n\\end{itemize}\nFor any given constant $\\ell' \\geq 1$, it was shown in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} that such a decomposition of a tree $T$ can be computed in $O(\\log n)$ rounds. See \\cref{fig:rake_and_compress} for an example of such a decomposition with $\\ell' = 4$.\n\n\\paragraph{The Algorithm}\nGiven such a decomposition with $\\ell' = \\max\\{1, \\ell - 2\\}$, $\\Pi$ can be solved in $O(\\log n)$ rounds by labeling the vertices in this order: $\\VR{L}$, $\\VC{L-1}$, $\\VR{L-1}$, $\\ldots$, $\\VR{1}$, as follows. The algorithm only uses the vertex configurations in the $\\ell$-full set $\\mathcal{V}'$.\n\n\\paragraph{Labeling $\\VR{i}$}\nSuppose all vertices in $\\VR{L}, \\VC{L-1}, \\VR{L-1}, \\ldots, \\VC{i}$ have been labeled using $\\mathcal{V}'$. Recall that each $v \\in \\VR{i}$ has degree at most one in the graph $\\GR{i}$.\nIf $v \\in \\VR{i}$ has no neighbor in $\\VR{L} \\cup \\VC{L-1} \\cup \\VR{L-1} \\cup \\cdots \\cup \\VC{i}$, then we can label the half edges surrounding $v$ by any $c \\in \\mathcal{V}'$.\nOtherwise, $v \\in \\VR{i}$ has exactly one neighbor $u$ in $\\VR{L} \\cup \\VC{L-1} \\cup \\VR{L-1} \\cup \\cdots \\cup \\VC{i}$. Suppose the vertex configuration of $u$ is $c$, where the half-edge label on $\\{u,v\\}$ is $\\mathtt{a} \\in c$. A simple observation from the definition of $\\ell$-full sets is that for any $c \\in \\mathcal{V}'$ and any $\\mathtt{a} \\in c$, there exist $c' \\in \\mathcal{V}'$ and $\\mathtt{a}' \\in c'$ in such a way that $\\{\\mathtt{a}, \\mathtt{a}'\\} \\in \\mathcal{E}$.\nHence we can label the half edges surrounding $v$ by $c' \\in \\mathcal{V}'$ where the half-edge label on $\\{u,v\\}$ is $\\mathtt{a}' \\in c'$. \n\n\n\\paragraph{Labeling $\\VC{i}$} Suppose all vertices in $\\VR{L}, \\VC{L-1}, \\VR{L-1}, \\ldots, \\VR{i+1}$ have been labeled using $\\mathcal{V}'$. Recall that the $\\VC{i}$-vertices in $\\GC{i}$ form degree-2 paths $P=(v_1, v_2, \\ldots, v_s)$, with $\\ell' \\leq s \\leq 2 \\ell'$.\nLet $P'= (x, v_1, v_2, \\ldots, v_s, y)$ be the path resulting from appending to $P$ the neighbors of the two end-points of $P$ in $\\GC{i}$. The two vertices $x$ and $y$ are in $\\VR{L} \\cup \\VC{L-1} \\cup \\VR{L-1} \\cup \\cdots \\cup \\VR{i+1}$, so they have been assigned half-edge labels using $\\mathcal{V}'$. Since $P'$ contains at least $\\ell'+2 \\geq \\ell$ vertices, the definition of $\\ell$-full sets ensures that we can label $v_1, v_2, \\ldots, v_s$ using vertex configurations in $\\mathcal{V}'$ in such a way that the half-edge labels on $\\{x, v_1\\}, \\{v_1, v_2\\}, \\ldots, \\{v_s, y\\}$ are all in $\\mathcal{E}$.\n\\end{proof}\n\n\n\n\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width= .8\\textwidth]{fig\/y1.pdf}\n \\caption{The variant of the rake-and-compress decomposition used in the proof of \\cref{thm:ellfull_to_logn}. }\n \\label{fig:rake_and_compress}\n\\end{figure}\n\n\n\n\\subsection{Necessity}\n\nWe start by sketching that (2.) in \\cref{thm:MainBaireLog} is necessary for $\\mathsf{BAIRE}$ and $\\mathsf{fiid}$.\n\n\n\\begin{theorem}[Bernshteyn \\cite{Bernshteyn_work_in_progress}]\nLet $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ be an LCL and suppose that $\\Pi\\in \\mathsf{BAIRE}$ or $\\Pi\\in \\mathsf{fiid}$.\nThen $\\Pi$ admits an $\\ell$-full set $\\mathcal{V}'\\subseteq\\mathcal{V}$ for some $\\ell>0$.\n\\end{theorem}\n\\begin{proof}[Proof Sketch]\n\n\nWe start with $\\mathsf{BAIRE}$.\nSuppose that every Borel acyclic $\\Delta$-regular graph admits a Borel solution on a $\\tau$-comeager set for every compatible Polish topology $\\tau$.\nIn particular, this holds for the Borel graph induced by the standard generators of the free product of $\\Delta$-copies of $\\mathbb{Z}_2$ on the free part of the shift action on the alphabet $\\{0,1\\}$ endowed with the product topology.\nLet $F$ be such a solution.\nWrite $\\mathcal{V}'\\subseteq \\mathcal{V}$ for the configurations of half-edge labels around vertices that $F$ outputs on a non-meager set.\nLet $C$ be a comeager set on which $F$ is continuous. Then, every element of $\\mathcal{V}'$ is encoded by some finite window in the shift on $C$, that is, for each element there are a $k \\in \\mathbb{N}$ and function $s:B(1,k) \\to \\{0,1\\}$ such that $F$ is constant on the set $N_s \\cap C$ (where $N_s$ is the basic open neighbourhood determined by $s$, and $B(1,k)$ is the $k$-neighbourhood of the identity in the Cayley graph of the group).\nSince $\\mathcal{V}'$ is finite, we can take $t>0$ to be the maximum of such $k$'s.\nIt follows by standard arguments that $\\mathcal{V}'$ is $\\ell$-full for $\\ell>2t+1$.\n\nA similar argument works for the $\\mathsf{fiid}$, however, for the sake of brevity, we sketch a shorter argument that uses the fact that there must be a correlation decay for factors of iid's. Let $\\Pi\\in \\mathsf{fiid}$.\nThat is, there is an $\\operatorname{Aut}(T)$-equivariant measurable function from iid's on $T$ (without colored edges this time) into the space of $\\Pi$-colorings.\nLet $\\mathcal{V}'$ be the set of half-edges configurations around vertices that have non-zero probability to appear.\nLet $u,v\\in T$ be vertices of distance $k_0\\in \\mathbb{N}$.\nBy \\cite{BGHV} the correlation between the configurations around $u$ and $v$ tends to $0$ as $k_0\\to \\infty$.\nThis means that if the distance is big enough, then all possible pairs of $\\mathcal{V}'$ configurations need to appear.\n\\end{proof}\n\nTo finish the proof of \\cref{thm:MainBaireLog} we need to demonstrate the following theorem. Note that $\\mathsf{LOCAL}(n^{o(1)}) = \\mathsf{LOCAL}(O(\\log n))$ according to the $\\omega(\\log n)$ -- $n^{o(1)}$ complexity gap~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}.\n\n\\begin{restatable}{theorem}{logcomb}\\label{thm:logn_to_ellfull}\nLet $\\Pi = (\\Sigma, \\mathcal{V}, \\mathcal{E})$ be an LCL solvable in $\\mathsf{LOCAL}(n^{o(1)})$ rounds.\nThen there exists an $\\ell$-full set $\\mathcal{V}' \\subseteq \\mathcal{V}$ for some $\\ell \\geq 2$.\n\\end{restatable}\n\n\nThe rest of the section is devoted to the proof of \\cref{thm:logn_to_ellfull}. We start with the high-level idea of the proof.\nA natural attempt for showing $\\mathsf{LOCAL}(n^{o(1)})$ $\\Rightarrow$ $\\ell$-full is to simply take any $\\mathsf{LOCAL}(n^{o(1)})$ algorithm $\\mathcal{A}$ solving $\\Pi$, and then take $\\mathcal{V}'$ to be all vertex configurations that can possibly occur in an output of $\\mathcal{A}$. It is not hard to see that this approach does not work in general, because the algorithm might use a special strategy to label vertices with degree smaller than $\\Delta$. Specifically, there might be some vertex configuration $c$ used by $\\mathcal{A}$ so that some $\\mathtt{a} \\in c$ will only be used to label \\emph{virtual} half edges. It will be problematic to include $c$ in $\\mathcal{V}'$.\n\nTo cope with this issue, we do not deal with general bounded-degree trees. Instead, we construct recursively a sequence $(W^\\ast_1, W^\\ast_2, \\ldots, W^\\ast_L)$ of sets of rooted, layered, and \\emph{partially labeled} tree in a special manner. A tree $T$ is included in $W^\\ast_i$ if it can be constructed by gluing a multiset of rooted trees in $W^\\ast_{i-1}$ and a new root vertex $r$ in a certain fixed manner. A vertex is said to be in layer $i$ if it is introduced during the $i$-th step of the construction, i.e., it is introduced as the root $r$ during the construction of $W^\\ast_i$ from $W^\\ast_{i-1}$. All remaining vertices are said to be in layer 0.\n\n\nWe show that each $T \\in W^\\ast_L$ admits a correct labeling that extends the given partial labeling, as these partial labelings are computed by a simulation of $\\mathcal{A}$.\nMoreover, in these correct labelings, the variety of possible configurations of half-edge labels around vertices in different non-zero layers is the same for each layer.\nThis includes vertices of non-zero layer whose half-edges are labeled by the given partial labeling.\nWe simply pick $\\mathcal{V}'$ to be the set of all configurations of half-edge labels around vertices that can appear in a non-zero layer in a correct labeling of a tree $T \\in W^\\ast_L$.\nOur construction ensures that each $c \\in \\mathcal{V}'$ appears as the labeling of some degree-$\\Delta$ vertex in some tree that we consider.\n\n\nThe proof that $\\mathcal{V}'$ is an $\\ell$-full set is based on finding paths using vertices of non-zero layers connecting two vertices with any two vertex configurations in $\\mathcal{V}'$ in different lengths. These paths exist because the way rooted trees in $W^\\ast_{i-1}$ are glued together in the construction of $W^\\ast_{i}$ is sufficiently flexible.\nThe reason that we need $\\mathcal{A}$ to have complexity $\\mathsf{LOCAL}(n^{o(1)})$ is that the construction of the trees can be parameterized by a number $w$ so that all the trees have size polynomial in $w$ and the vertices needed to be assigned labeling are at least distance $w$ apart from each other. Since the number of rounds of $\\mathcal{A}$ executed on trees of size $w^{O(1)}$ is much less than $w$, each labeling assignment can be calculated locally and independently. \nThe construction of the trees as well as the analysis are based on a machinery developed in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}.\n Specifically, we will consider the equivalence relation $\\overset{\\star}{\\sim}$ defined in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} and prove some of its properties, including a pumping lemma for bipolar trees. The exact definition of $\\overset{\\star}{\\sim}$ in this paper is different from the one in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} because the mathematical formalism describing LCL problems in this paper is different from the one in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}.\nAfter that, we will consider a procedure for gluing trees parameterized by a labeling function $f$ similar to the one used in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}. \nWe will apply this procedure iteratively to generate a set of trees. We will show that the desired $\\ell$-full set $\\mathcal{V}' \\subseteq \\mathcal{V}$ can be constructed by considering the set of all possible correct labeling of these trees. \n \n\n\\paragraph{The Equivalence Relation $\\overset{\\star}{\\sim}$} \nWe consider trees with a list of designated vertices $v_1, v_2, \\ldots, v_k$ called \\emph{poles}. A \\emph{rooted tree} is a tree with one pole $r$, and a \\emph{bipolar tree} is a tree with two poles $s$ and $t$. \nFor a tree $T$ with its poles $S=(v_1, v_2, \\ldots, v_k)$ with $\\deg(v_i) = d_i < \\Delta$, we denote by $h_{(T,S)}$ the function that maps each choice of the \\emph{virtual} half-edge labeling surrounding the poles of $T$ to $\\textsf{YES}$ or $\\textsf{NO}$, indicating whether such a partial labeling can be completed into a correct complete labeling of $T$. \nMore specifically, consider \n\\[X=(\\mathcal{I}_1, \\mathcal{I}_2, \\ldots, \\mathcal{I}_k),\\]\nwhere $\\mathcal{I}_i$ is a size-$(\\Delta - d_i)$ multiset of labels in $\\Sigma$, for each $1 \\leq i \\leq k$. Then $h_{(T,S)}(X) = \\textsf{YES}$ if there is a correct labeling of $T$ such that the $\\Delta - d_i$ virtual half-edge labels surrounding $v_i$ are labeled by $\\mathcal{I}_i$, for each $1 \\leq i \\leq k$. This definition can be generalized to the case $T$ is already partially labeled in the sense that some of the half-edge labels have been fixed. In this case, $h_{(T,S)}(X) = \\textsf{NO}$ whenever $X$ is incompatible with the given partial labeling.\n\nLet $T$ be a tree with poles $S = (v_1, v_2, \\ldots, v_k)$ and let $T'$ be another tree with poles $S' = (v_1', v_2', \\ldots, v_k')$ such that $\\deg(v_i) = \\deg(v_i') = d_i$ for each $1 \\leq i \\leq k$.\nThen we write $T_1 \\overset{\\star}{\\sim} T_2$ if $h_{(T,S)} = h_{(T',S')}$.\n\nGiven an LCL problem $\\Pi$, it is clear that the number of equivalence classes of rooted trees and bipolar trees w.r.t.~$\\overset{\\star}{\\sim}$ is finite.\nFor a rooted tree $T$, denote by $\\textsf{Class}_1(T)$ the equivalence class of $T$. For a bipolar tree $H$, denote by $\\texttt{type}(H)$ the equivalence class of $H$.\n\n\n\n\n\\paragraph{Subtree Replacement}\nThe following lemma provides a sufficient condition that the equivalence class of a tree $T$ is invariant of the equivalence class of its subtree $T'$.\nWe note that a real half edge in $T$ might become virtual in its subtree $T'$.\nConsider a vertex $v$ in $T'$ and its neighbor $u$ that is in $T$ but not in $T'$. Then the half edge $(v, \\{u,v\\})$ is real in $T$ and virtual in $T'$.\n\n\n\\begin{lemma}[Replacing subtrees]\\label{lem:replace}\nLet $T$ be a tree with poles $S$.\nLet $T'$ be a connected subtree of $T$ induced by $U \\subseteq V$, where $V$ is the set of vertices in $T$.\nWe identify a list of designated vertices $S' = (v_1', v_2', \\ldots, v_k')$ in $U$ satisfying the following two conditions to be the poles of $T'$.\n\\begin{itemize}\n \\item $S \\cap U \\subseteq S'$.\n \\item Each edge $e=\\{u,v\\}$ connecting $u \\in U$ and $ v \\in V \\setminus U$ must satisfy $u \\in S'$.\n\\end{itemize}\nLet $T''$ be another tree with poles $S'' = (v_1'', v_2'', \\ldots, v_k'')$ that is in the same equivalence class as $T'$.\nLet $T^\\ast$ be the result of replacing $T'$ by $T''$ in $T$ by identifying $v_i' = v_i''$ for each $1 \\leq i \\leq k$.\nThen $T^\\ast$ is in the same equivalence class as $T$. \n\\end{lemma}\n\\begin{proof}\nTo prove the lemma, by symmetry, it suffices to show that starting from any correct labeling $\\mathcal{L}$ of $T$, it is possible to find a correct labeling $\\mathcal{L}^\\ast$ of $T^\\ast$ in such a way that the multiset of the\nvirtual half-edge labels surrounding each pole in $S$ remain the same.\n\nSuch a correct labeling $\\mathcal{L}^\\ast$ of $T^\\ast$ is constructed as follows. If $v \\in V \\setminus U$, then we simply adopt the given labeling $\\mathcal{L}$ of $v$ in $T$.\nNext, consider the vertices in $U$.\nSet $X=(\\mathcal{I}_1, \\mathcal{I}_2, \\ldots, \\mathcal{I}_k)$ to be the one compatible with the labeling $\\mathcal{L}$ of $T$ restricted to the subtree $T'$ in the sense that $\\mathcal{I}_i$ is the multiset of the virtual half-edge labels surrounding the pole $v_i'$ of $T'$, for each $1 \\leq i \\leq k$. We must have $h_{T',S'}(X) = \\textsf{YES}$.\nSince $T'$ and $T''$ are in the same equivalence class, we have $h_{T'',S''}(X) = \\textsf{YES}$, and so we can find a correct labeling $\\mathcal{L}''$ of $T''$ that is also compatible with $X$. Combining this labeling $\\mathcal{L}''$ of the vertices $U$ in $T''$ with the labeling $\\mathcal{L}$ of the vertices $V \\setminus U$ in $T$, we obtain a desired labeling $\\mathcal{L}^\\ast$ of $T^\\ast$.\n\nWe verify that $\\mathcal{L}^\\ast$ gives a correct labeling of $T^\\ast$. Clearly, the size-$\\Delta$ multiset that labels each vertex $v \\in V$ is in $\\mathcal{V}$, by the correctness of $\\mathcal{L}$ and $\\mathcal{L}''$. Consider any edge $e=\\{u,v\\}$ in $T^\\ast$. Similarly, if $\\{u,v\\} \\subseteq V \\setminus U$ or $\\{u,v\\} \\cap (V \\setminus U) = \\emptyset$, then the size-$2$ multiset that labels $e$ is in $\\mathcal{E}$, by the correctness of $\\mathcal{L}$ and $\\mathcal{L}''$. For the case that $e=\\{u,v\\}$ connects a vertex $u \\notin V$ and a vertex $v \\in V \\setminus U$, we must have $u \\in S'$ by the lemma statement. Therefore, the label of the half edge $(u,e)$ is the same in both $\\mathcal{L}$ and $\\mathcal{L}^\\ast$ by our choice of $\\mathcal{L}''$. Thus, the size-$2$ multiset that labels $e$ is in $\\mathcal{E}$, by the correctness of $\\mathcal{L}$.\n\nWe verify that the \nvirtual half-edge labels surrounding each pole in $S$ are the same in both $\\mathcal{L}$ and $\\mathcal{L}^\\ast$. Consider a pole $v \\in S$. If $v \\in V \\setminus U$, then the labeling of $v$ is clearly the same in both $\\mathcal{L}$ and $\\mathcal{L}^\\ast$.\nIf $v \\notin V\\setminus U$, then the condition $S \\cap U \\subseteq S'$ in the statement implies that $v \\in S'$. In this case, the way we pick $\\mathcal{L}''$ ensures that the \nvirtual half-edge labels surrounding $v \\in S'$ are the same in both $\\mathcal{L}$ and $\\mathcal{L}^\\ast$.\n\\end{proof}\n\nIn view of the proof of \\cref{lem:replace}, as long as the conditions in \\cref{lem:replace} are met, we are able to abstract out a subtree by its equivalence class when reasoning about correct labelings of a tree. This observation will be applied repeatedly in the subsequent discussion.\n\n\n\\paragraph{A Pumping Lemma} We will prove a pumping lemma of bipolar trees using \\cref{lem:replace}. \nSuppose $T_i$ is a tree with a root $r_i$ for each $1 \\leq i \\leq k$, then $H=(T_1, T_2, \\ldots, T_k)$ denotes the bipolar tree resulting from concatenating the roots $r_1, r_2, \\ldots, r_k$ into a path $(r_1, r_2, \\ldots, r_k)$ and setting the two poles of $H$ by $s = r_1$ and $t = r_k$.\nA simple consequence of \\cref{lem:replace} is that $\\texttt{type}(H)$ is determined by $\\textsf{Class}_1(T_1), \\textsf{Class}_1(T_2), \\ldots, \\textsf{Class}_1(T_k)$.\nWe have the following pumping lemma. \n\n\\begin{lemma}[Pumping lemma]\\label{lem:pump}\nThere exists a finite number $ {\\ell_{\\operatorname{pump}}} > 0$ such that as long as $k \\geq {\\ell_{\\operatorname{pump}}} $, any bipolar tree $H=(T_1, T_2, \\ldots, T_k)$ can be decomposed into $H = X \\circ Y \\circ Z$ with $0 < |Y| < k$ so that $X \\circ Y^i \\circ Z$ is in the same equivalence class as $H$ for each $i \\geq 0$. \n\\end{lemma}\n\\begin{proof}\nSet $ {\\ell_{\\operatorname{pump}}} $ to be the number of equivalence classes for bipolar trees plus one. By the pigeon hole principle, there exist $1 \\leq a < b \\leq k$ such that $(T_1, T_2, \\ldots, T_a)$ and $(T_1, T_2, \\ldots, T_b)$ are in the same equivalence class. Set $X = (T_1, T_2, \\ldots, T_a)$, $Y = (T_{a+1}, T_{a+2}, \\ldots, T_b)$, and $Z = (T_{b+1}, T_{b+2}, \\ldots, T_k)$. As we already know that $\\texttt{type}(X) = \\texttt{type}(X\\circ Y)$, \\cref{lem:replace} implies that \n$\\texttt{type}(X\\circ Y) = \\texttt{type}(X^2\\circ Y)$ by replacing $X$ by $X\\circ Y$ in the bipolar tree $X\\circ Y$.\nSimilarly, $\\texttt{type}(X \\circ Y^i)$ is the same for for each $i \\geq 0$. Applying \\cref{lem:replace} again to replace $X \\circ Y$ by $X \\circ Y^i$ in $H = X \\circ Y \\circ Z$, we conclude that $X \\circ Y^i \\circ Z$ is in the same equivalence class as $H$ for each $i \\geq 0$. \n\\end{proof}\n\n\n\n\\paragraph{A Procedure for Gluing Trees} Suppose that we have a set of rooted trees $W$. We devise a procedure that generates a new set of rooted trees by gluing the rooted trees in $W$ together. \nThis procedure is parameterized by a labeling function $f$. Consider a bipolar tree \\[H = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)\\] \nwhere $T^m$ is formed by attaching the roots $r_1, r_2, \\ldots, r_{\\Delta - 2}$ of the rooted trees $T_1^m, T_2^m, \\ldots, T_{\\Delta -2}^m$ to the root $r^m$ of $T^m$.\n\n\nThe labeling function $f$ assigns the half-edge labels surrounding $r^m$ based on \n\\[\\texttt{type}(H^l), \\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m), \\texttt{type}(H^r)\\]\nwhere $H^l = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l)$ and $H^r = (T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)$.\n\nWe write $T^{m}_\\ast$ to denote the result of applying $f$ to label the root $r^m$ of $T^{m}$ in the bipolar tree $H$, and we write \\[H_\\ast = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T_\\ast^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)\\] to denote the result of applying $f$ to $H$. We make the following observation.\n\n\\begin{lemma}[Property of $f$]\\label{lem:labeling_function}\nThe two equivalence classes\n$\\textsf{Class}_1(T_\\ast^m)$ and $\\texttt{type}(H_\\ast)$ are determined by \\[\\texttt{type}(H^l), \\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m), \\texttt{type}(H^r),\\] and the labeling function $f$.\n\\end{lemma}\n\\begin{proof}\nBy \\cref{lem:replace}, once the half-edge labelings of the root $r^m$ of $T^{m}$ is fixed, $\\textsf{Class}_1(T_\\ast^m)$ is determined by $\\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m)$. \nTherefore, indeed $\\textsf{Class}_1(T_\\ast^m)$ is determined by\n\\[\\texttt{type}(H^l), \\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m), \\texttt{type}(H^r),\\] and the labeling function $f$.\nSimilarly, applying \\cref{lem:replace} to the decomposition of $H_\\ast$ into $H^l$, $T_\\ast^m$, and $H^r$, we infer that $\\texttt{type}(H_\\ast)$ depends only on $\\texttt{type}(H^l)$, $\\textsf{Class}_1(T_\\ast^m)$, and $\\texttt{type}(H^r)$.\n\\end{proof}\n\nThe three sets of trees $X_f(W)$, $Y_f(W)$, and $Z_f(W)$ are constructed as follows.\n\n\\begin{itemize}\n \\item $X_f(W)$ is the set of all rooted trees resulting from appending $\\Delta-2$ arbitrary rooted trees $T_1, T_2, \\ldots, T_{\\Delta-2}$ in $W$ to a new root vertex $r$.\n \\item $Y_f(W)$ is the set of bipolar trees constructed as follows. For each choice of $2 {\\ell_{\\operatorname{pump}}} +1$ rooted trees $T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r$ from $X_f(W)$, concatenate them into a bipolar tree \n \\[H = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r),\\] let $H_\\ast$ be the result of applying the labeling function $f$ to $H$, and then add $H_\\ast$ to $Y_f(W)$.\n \\item $Z_f(W)$ is the set of rooted trees constructed as follows. For each \\[H_\\ast = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T_\\ast^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r) \\in Y_f(W),\\] add $(T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T_\\ast^m)$ to $Z_f(W)$, where we set the root of $T_1^l$ as the root, and add $(T_\\ast^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)$ to $Z_f(W)$, where we set the root of ${T_{ {\\ell_{\\operatorname{pump}}} }^r}$ as the root. \n\\end{itemize}\n\nWe write $\\textsf{Class}_1(S) = \\bigcup_{T \\in S} \\{\\textsf{Class}_1(T)\\}$ and $\\texttt{type}(S) = \\bigcup_{H \\in S} \\{\\texttt{type}(H)\\}$, and \nwe make the following observation.\n\n\\begin{lemma}[Property of $X_f(W)$, $Y_f(W)$, and $Z_f(W)$]\\label{lem:sets_of_trees}\nThe three sets of equivalence classes $\\textsf{Class}_1(X_f(W))$, $\\texttt{type}(Y_f(W))$, and $\\textsf{Class}_1(Z_f(W))$ depend only on $\\textsf{Class}_1(W)$ and the labeling function $f$.\n\\end{lemma}\n\\begin{proof}\nThis is a simple consequence of \\cref{lem:replace,lem:labeling_function}.\n\\end{proof}\n\n\\paragraph{A Fixed Point $W^\\ast$}\nGiven a fixed labeling function $f$,\nwe want to find a set of rooted trees $W^\\ast$ that is a \\emph{fixed point} for the procedure $Z_f$ in the sense that \\[\\textsf{Class}_1(Z_f(W^\\ast)) = \\textsf{Class}_1(W^\\ast).\\] \nTo find such a set $W^\\ast$, we construct a two-dimensional array of rooted trees $\\{W_{i,j}\\}$, as follows.\n\n\\begin{itemize}\n \\item For the base case, $W_{1,1}$ consists of only the one-vertex rooted tree.\n \\item Given that $W_{i,j}$ as been constructed, we define $W_{i,j+1} = Z_f(W_{i,j})$.\n \\item Given that $W_{i,j}$ for all positive integers $j$ have been constructed, $W_{i+1,1}$ is defined as follows. Pick $b_i$ as the smallest index such that $\\textsf{Class}_1(W_{i, b_i}) = \\textsf{Class}_1(W_{i, a_i})$ for some $1 \\leq a_i < b_i$. By the pigeon hole principle, the index $b_i$ exists, and it is upper bounded by $2^C$, where $C$ is the number of equivalence classes for rooted trees. We set \n \\[W_{i+1,1} = W_{i, a_i} \\cup W_{i, a_i + 1} \\cup \\cdots \\cup W_{i, b_i - 1}.\\]\n\\end{itemize}\n\nWe show that the sequence $\\textsf{Class}_1(W_{i,a_i}), \\textsf{Class}_1(W_{i,a_i+1}), \\textsf{Class}_1(W_{i,a_i+2}), \\ldots$ is periodic with a period $c_i = b_i - a_i$.\n\n\\begin{lemma}\\label{lem:W-aux-1}\nFor any $i \\geq 1$ and for any $j \\geq a_i$, we have $\\textsf{Class}_1(W_{i,j}) = \\textsf{Class}_1(W_{i,j+c_i})$, where $c_i = b_i - a_i$.\n\\end{lemma}\n\\begin{proof}\nBy \\cref{lem:sets_of_trees}, $\\textsf{Class}_1(W_{i,j})$ depends only on $\\textsf{Class}_1(W_{i,j-1})$. Hence the lemma follows from the fact that $\\textsf{Class}_1(W_{i, b_i}) = \\textsf{Class}_1(W_{i, a_i})$.\n\\end{proof}\n\nNext, we show that $\\textsf{Class}_1(W_{2,1}) \\subseteq \\textsf{Class}_1(W_{3, 1}) \\subseteq \\textsf{Class}_1(W_{4, 1}) \\subseteq \\cdots$.\n\n\\begin{lemma}\\label{lem:W-aux-2}\nFor any $i \\geq 2$, we have $\\textsf{Class}_1(W_{i,1}) \\subseteq \\textsf{Class}_1(W_{i,j})$ for each $j > 1$, and so $\\textsf{Class}_1(W_{i,1}) \\subseteq \\textsf{Class}_1(W_{i+1, 1})$.\n\\end{lemma}\n\\begin{proof}\nSince $W_{i,1} = \\bigcup_{a_{i-1} \\leq l \\leq b_{i-1} - 1} W_{i-1, l}$, we have \\[\\bigcup_{a_{i-1}+j-1 \\leq l \\leq b_{i-1} +j} W_{i-1, l} \\subseteq W_{i,j}\\] according to the procedure of constructing $W_{i,j}$. By \\cref{lem:W-aux-1}, \\[ \\textsf{Class}_1\\left(\\bigcup_{a_{i-1} \\leq l \\leq b_{i-1} - 1} W_{i-1, l}\\right) = \\textsf{Class}_1\\left(\\bigcup_{a_{i-1}+j-1 \\leq l \\leq b_{i-1} +j} W_{i-1, l}\\right)\\] for all $j \\geq 1$, and so we have $\\textsf{Class}_1(W_{i,1}) \\subseteq \\textsf{Class}_1(W_{i,j})$ for each $j > 1$. The claim $\\textsf{Class}_1(W_{i,1}) \\subseteq \\textsf{Class}_1(W_{i+1, 1})$ follows from the fact that $W_{i+1,1} = \\bigcup_{a_{i} \\leq l \\leq b_{i} - 1} W_{i, l}$.\n\\end{proof}\n\nSet $i^\\ast$ to be the smallest index $i \\geq 2$ such that $\\textsf{Class}_1(W_{i,1}) = \\textsf{Class}_1(W_{i+1, 1})$. By the pigeon hole principle and \\cref{lem:W-aux-2}, the index $i^\\ast$ exists, and it is upper bounded by $C$, the number of equivalence classes for rooted trees. We set \\[W^\\ast = W_{i^\\ast, 1}.\\] \n\nThe following lemma shows that $\\textsf{Class}_1(Z_f(W^\\ast)) = \\textsf{Class}_1(W^\\ast)$, as needed.\n\n\\begin{lemma}\\label{lem:W-aux-3}\nFor any $i \\geq 2$, if $\\textsf{Class}_1(W_{i,1}) = \\textsf{Class}_1(W_{i+1, 1})$, then $\\textsf{Class}_1(W_{i,j})$ is the same for all $j \\geq 1$. \n\\end{lemma}\n\\begin{proof}\nBy \\cref{lem:W-aux-1} and the way we construct $W_{i+1, 1}$, we have \\[\\textsf{Class}_1(W_{i, j}) \\subseteq \\textsf{Class}_1(W_{i+1, 1}) \\ \\ \\text{for each} \\ \\ j \\geq a_i.\\]\nBy \\cref{lem:W-aux-2}, we have \\[\\textsf{Class}_1(W_{i, 1}) \\subseteq \\textsf{Class}_1(W_{i, j}) \\ \\ \\text{for each} \\ \\ j > 1.\\]\nTherefore, $\\textsf{Class}_1(W_{i,1}) = \\textsf{Class}_1(W_{i+1, 1})$ implies that $\\textsf{Class}_1(W_{i, j}) = \\textsf{Class}_1(W_{i,1})$ for each $j \\geq a_i$. Hence we must have $\\textsf{Class}_1(Z_f(W_{i,1})) = \\textsf{Class}_1(W_{i,1})$, and so $\\textsf{Class}_1(W_{i,j})$ is the same for all $j \\geq 1$. \n\\end{proof}\n\nWe remark that simply selecting $W^\\ast$ to be any set such that $\\textsf{Class}_1(Z_f(W^\\ast)) = \\textsf{Class}_1(W^\\ast)$ is not enough for our purpose. As we will later see, it is crucial that the set $W^\\ast$ is constructed by iteratively applying the function $Z_f$ and taking the union of previously constructed sets.\n\n\n\n\n\n\n\n\\paragraph{A Sequence of Sets of Trees}\nWe define $W_1^\\ast = W^\\ast$ and $W_i^\\ast = Z_f(W_{i-1}^\\ast)$ for each $1 < i \\leq L$, where $L$ is some sufficiently large number to be determined.\n\nThe way we choose $W^\\ast$ guarantees that $\\textsf{Class}_1(W_i^\\ast) = \\textsf{Class}_1(W^\\ast)$ for all $1 \\leq i \\leq L$. \nFor convenience, we write $X_i^\\ast = X_f(W_i^\\ast)$ and $Y_i^\\ast = Y_f(W_i^\\ast)$.\nSimilarly, $\\textsf{Class}_1(X_i^\\ast)$ is the same for all $1 \\leq i \\leq L$ and $\\texttt{type}(Y_i^\\ast)$ is the same for all $1 \\leq i \\leq L$.\n\n\nOur analysis will rely on the assumption that all rooted trees in $W^\\ast$ admit correct labelings. Whether this is true depends only on $\\textsf{Class}_1(W^\\ast)$, which depends only on the labeling function $f$. We say that $f$ is \\emph{feasible} if it leads to a set $W^\\ast$ where all the rooted trees therein admit correct labelings. The proof that a feasible labeling function $f$ exists is deferred.\n\nThe assumption that $f$ is feasible implies that all trees in $W_i^\\ast$, $X_i^\\ast$, and $Y_i^\\ast$, for all $1 \\leq i \\leq L$, admit correct labelings. All rooted trees in $X_i^\\ast$ admit correct labelings because they are subtrees of the rooted trees in $W_{i+1}^\\ast$. A correct labeling of any bipolar tree $H \\in Y_i^\\ast$ can be obtained by combining any correct labelings of the two rooted trees in $W_{i+1}^\\ast$ resulting from $H$.\n\n\n\n\n\\paragraph{Layers of Vertices}\nWe assign a layer number $\\lambda(v)$ to each vertex $v$ in a tree based on the step that $v$ is introduced in the construction \\[W_1^\\ast \\rightarrow W_2^\\ast \\rightarrow \\cdots \\rightarrow W_L^\\ast.\\]\n\nIf a vertex $v$ is introduced as the root vertex of a tree in $X_i^\\ast = X_f(W_i^\\ast)$, then we say that the layer number of $v$ is $\\lambda(v) = i \\in \\{1,2, \\ldots, L\\}$. A vertex $v$ has $\\lambda(v) = 0$ if it belongs to a tree in $W_1^\\ast$.\n\n\nFor any vertex $v$ with $\\lambda(v) = i$ in a tree $T \\in X_j^\\ast$ with $i \\leq j$, we write $T_v$ to denote the subtree of $T$ such that $T_v \\in X_i^\\ast$ where $v$ is the root of $T_v$. \n\n\n \nWe construct a sequence of sets $R_1, R_2, \\ldots, R_L$ as follows.\nWe go over all rooted trees $T \\in X_L^\\ast$, all possible correct labeling $\\mathcal{L}$ of $T$, and all vertices $v$ in $T$ with $\\lambda(v) = i \\in \\{1,2, \\ldots, L\\}$. Suppose that the $\\Delta-2$ rooted trees in the construction of $T_v \\in X_i^\\ast = X_f(W_i^\\ast)$ are $T_1$, $T_2$, $\\ldots$, $T_{\\Delta-2}$, and let $r_i$ be the root of $T_i$. Consider the following parameters.\n\\begin{itemize}\n \\item $c_i = \\textsf{Class}_1(T_i)$.\n \\item $\\mathtt{a}_i$ is the real half-edge label of $v$ in $T_v$ for the edge $\\{v, r_i\\}$, under the correct labeling $\\mathcal{L}$ of $T$ restricted to $T_v$.\n \\item $\\mathcal{I}$ is the size-$2$ multiset of virtual half-edge labels of $v$ in $T_v$, under the correct labeling $\\mathcal{L}$ of $T$ restricted to $T_v$.\n\\end{itemize}\nThen we add \n$(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I})$ to $R_i$.\n\n\n\n\n\\begin{lemma}\\label{lem:Rsets1}\nFor each $1 \\leq i < L$, $R_i$ is determined by $R_{i+1}$.\n\\end{lemma}\n\\begin{proof}\nWe consider the following alternative way of constructing $R_i$ from $R_{i+1}$.\nEach rooted tree $T'$ in $W_{i+1}^\\ast = Z_f(W_i^\\ast)$ can be described as follows.\n\\begin{itemize}\n \\item Start with a path $(r_1, r_2, \\ldots, r_{ {\\ell_{\\operatorname{pump}}} + 1})$, where $r_1$ is the root of $T'$.\n \\item For each $1 \\leq j \\leq {\\ell_{\\operatorname{pump}}} +1$, append $\\Delta-2$ rooted trees $T_{j,1}, T_{j,2}, \\ldots, T_{j, \\Delta-2} \\in W_i^\\ast$ to $r_j$.\n \\item Assign the labels to the half edges surrounding $r_{ {\\ell_{\\operatorname{pump}}} + 1}$ according to the labeling function $f$.\n\\end{itemize}\n\n Now, consider the function $\\phi$ that maps each equivalence class $c$ for rooted trees to a subset of $\\Sigma$ defined as follows:\n$\\mathtt{a} \\in \\phi(c)$ if there exist \\[(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I}) \\in R_{i+1}\\] and $1 \\leq j \\leq \\Delta -2$ such that $c = c_j$ and $\\mathtt{a} = \\mathtt{a}_j$.\n\nWe go over all possible $T' \\in W_{i+1}^\\ast = Z_f(W_i^\\ast)$.\nNote that the root $r$ of $T'$ has exactly one virtual half edge. For each $\\mathtt{b} \\in \\Sigma$ such that $\\{\\mathtt{a}, \\mathtt{b}\\} \\in \\mathcal{E}$ for some $\\mathtt{a} \\in \\phi(\\textsf{Class}_1(T'))$, we go over all possible correct labelings $\\mathcal{L}$ of $T'$ where the virtual half edge of $r$ is labeled $\\mathtt{b}$. For each $1 \\leq j \\leq {\\ell_{\\operatorname{pump}}} +1$, consider the following parameters.\n\\begin{itemize}\n \\item $c_l = \\textsf{Class}_1(T_{j,l})$.\n \\item $\\mathtt{a}_l$ is the half-edge label of $r_j$ for the edge $\\{r_j, r_{j,l}\\}$.\n \\item $\\mathcal{I}$ is the size-$2$ multiset of the remaining two half-edge labels of $v$.\n\\end{itemize}\nThen we add \n$(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I})$ to $R_i$. \n\nThis construction of $R_i$ is equivalent to the original construction of $R_i$ because a correct labeling of $T' \\in W_i^\\ast$ can be extended to a correct labeling of a tree $T \\in X_L^\\ast$ that contains $T'$ as a subtree if and only if the virtual half-edge label of the root $r$ of $T'$ is $\\mathtt{b} \\in \\Sigma$ such that $\\{\\mathtt{a}, \\mathtt{b}\\} \\in \\mathcal{E}$ for some $\\mathtt{a} \\in \\phi(\\textsf{Class}_1(T'))$.\n\n\nIt is clear that this construction of $R_i$ only depends on $\\textsf{Class}_1(W_i^\\ast)$, the labeling function $f$, and the function $\\phi$, which depends only on $R_{i+1}$. Since the labeling function $f$ is fixed and $\\textsf{Class}_1(W_i^\\ast)$ is the same for all $i$, we conclude that $R_{i}$ depends only on $R_{i+1}$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:Rsets2}\nWe have $R_1 \\subseteq R_2 \\subseteq \\cdots \\subseteq R_L$.\n\\end{lemma}\n\\begin{proof}\nFor the base case, we show that $R_{L-1} \\subseteq R_L$. In fact, our proof will show that $R_{i} \\subseteq R_L$ for each $1 \\leq i < L$. Consider any \n\\[(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I}) \\in R_i.\\] \nThen there is a rooted tree $T \\in X_i^\\ast$ that is formed by attaching $\\Delta-2$ rooted trees of equivalence classes $c_1, c_2, \\ldots, c_{\\Delta-2}$ to the root vertex so that if we label the half edges surrounding the root vertex according to $\\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}$, and $\\mathcal{I}$, then this partial labeling can be completed into a correct labeling of $T$.\n\nBecause $\\textsf{Class}_1(W_i^\\ast) = \\textsf{Class}_1(W_L^\\ast)$, there is also a rooted tree $T' \\in X_L^\\ast$ that is formed by attaching $\\Delta-2$ rooted trees of equivalence classes $c_1, c_2, \\ldots, c_{\\Delta-2}$ to the root vertex. Therefore, if we label the root vertex of $T'$ in the same way as we do for $T$, then this partial labeling can also be completed into a correct labeling of $T'$. Hence we must have \n\\[(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I}) \\in R_L.\\] \n\n\nNow, suppose that we already have $R_i \\subseteq R_{i+1}$ for some $1 < i < L$. We will show that $R_{i-1} \\subseteq R_{i}$. \nDenote by $\\phi_{i}$ and $\\phi_{i+1}$ the function $\\phi$ in \\cref{lem:Rsets1} constructed from $R_{i}$ and $R_{i+1}$. \nWe have $\\phi_i(c) \\subseteq \\phi_{i+1}(c)$ for each equivalence class $c$, because $R_i \\subseteq R_{i+1}$. Therefore, in view of the alternative construction described in the proof of \\cref{lem:Rsets1}, we have $R_{i-1} \\subseteq R_{i}$.\n\\end{proof}\n\n\n\n\\paragraph{The Set of Vertex Configurations $\\mathcal{V}'$}\nBy \\cref{lem:Rsets1,lem:Rsets2}, if we pick $L$ to be sufficient large, we can have $R_1 = R_2 = R_3$.\nMore specifically, if we pick \n\\[L \\geq C^{\\Delta-2} \\cdot \\binom{|\\Sigma|+1}{|\\Sigma|-1} + 3,\\]\nthen there exists an index $3 \\leq i \\leq L$ such that $R_i = R_{i-1}$, implying that $R_1 = R_2 = R_3$. Here $C$ is the number of equivalence classes for rooted trees and $\\binom{|\\Sigma|+1}{|\\Sigma|-1}$ is the number of size-$2$ multisets of elements from $\\Sigma$.\n\n\n\nThe set $\\mathcal{V}'$ is defined by including all size-$\\Delta$ multisets $\\{\\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}\\} \\cup \\mathcal{I}$ such that \\[(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I}) \\in R_{1}\\] for some $c_1, c_2, \\ldots, c_{\\Delta-2}$.\n\n\n\n\\paragraph{The Set $\\mathcal{V}'$ is $\\ell$-full}\nTo show that the set $\\mathcal{V}'$ is $\\ell$-full, we consider the subset $\\mathcal{V}^\\ast \\subseteq \\mathcal{V}'$ defined by the set of vertex configurations used by the labeling function $f$ in the construction of $Y_f(W_i^\\ast)$. The definition of $\\mathcal{V}^\\ast$ is invariant of $i$ as $\\textsf{Class}_1(W_i^\\ast)$ is the same for all $i$. Clearly, for each $x \\in \\mathcal{V}^\\ast$, there is a rooted tree $T \\in W_2^\\ast$ where the root of a subtree $T' \\in X_1^\\ast$ of $T$ has its size-$\\Delta$ multiset of half-edge labels fixed to be $x$ by $f$, and so $\\mathcal{V}^\\ast \\subseteq \\mathcal{V}'$.\n\nFor notational simplicity, we write $x \\overset{k}{\\leftrightarrow} x'$ if there is a correct labeling of a $k$-vertex path $(v_1, v_2, \\ldots, v_k)$ using only vertex configurations in $\\mathcal{V}'$ so that the vertex configuration of $v_1$ is $x$ and the vertex configuration of $v_k$ is $x'$. If it is further required that the half-edge label of $v_1$ for the edge $\\{v_1, v_2\\}$ is $\\mathtt{a} \\in x$, then we write $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} x'$. The notation $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{a}')$ is defined similarly.\n\n\n\\begin{lemma}\\label{lem:find_path_1}\nFor any $x \\in \\mathcal{V}' \\setminus \\mathcal{V}^\\ast$ and $\\mathtt{a} \\in x$, there exist a vertex configuration $x' \\in \\mathcal{V}^\\ast$ and a number $2 \\leq k \\leq 2 {\\ell_{\\operatorname{pump}}} + 1$ such that $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} x'$.\n\\end{lemma}\n\\begin{proof}\nLet $T \\in W_L^\\ast$ be chosen so that there is a correct labeling $\\mathcal{L}$ where $x$ is a vertex configuration of some vertex $v$ with $\\lambda(v) = 2$.\n\nTo prove the lemma, it suffices to show that for each of the $\\Delta$ neighbors $u$ of $v$, it is possible to find a path $P = (v,u, \\ldots, w)$ meeting the following conditions.\n\\begin{itemize}\n \\item $w$ is a vertex whose half-edge labels have been fixed by $f$. This ensures that the vertex configuration of $w$ is in $\\mathcal{V}^\\ast$.\n \\item All vertices in $P$ are within layers 1,2, and 3. This ensures that the vertex configuration of all vertices in $P$ are in $\\mathcal{V}'$.\n \\item The number of vertices $k$ in $P$ satisfies $2 \\leq k \\leq 2 {\\ell_{\\operatorname{pump}}} + 1$. \n\\end{itemize}\n\nWe divide the proof into three cases. Refer to \\cref{fig:ell_full_construction1} for an illustration, where squares are vertices whose half-edge labels have been fixed by $f$.\n\n\\paragraph{Case 1} Consider the subtree $T_v \\in X_2^\\ast$ of $T$ whose root is $v$. In view of the construction of the set $X_f(W_2^\\ast)$, $v$ has $\\Delta-2$ children $u_1, u_2, \\ldots, u_{\\Delta-2}$ in $T_v$, where the subtree $T_i$ rooted at $u_i$ is a rooted tree in $W_2^\\ast$. \n\nFor each $1 \\leq i \\leq \\Delta-2$, according to the structure of the trees in the set $W_2^\\ast = Z_f(W_1^\\ast)$, there is a path $(u_i = w_1, w_2, \\ldots, w_{ {\\ell_{\\operatorname{pump}}} +1})$ in $T_i$ containing only layer-1 vertices, where the half-edge labels of $w_{ {\\ell_{\\operatorname{pump}}} +1}$ have been fixed by $f$. Hence $P = (v, u_i = w_1, w_2, \\ldots, w_{ {\\ell_{\\operatorname{pump}}} +1})$ is a desired path with $k = {\\ell_{\\operatorname{pump}}} + 2$ vertices.\n \n \n \n \n \n\\paragraph{Case 2} Consider the subtree $T' \\in W_3^\\ast = Z_f(W_2^\\ast)$ that contains $v$ in $T$.\nSimilarly, according to the structure of the trees in the set $W_3^\\ast = Z_f(W_2^\\ast)$, there is a path $(r = w_1', w_2', \\ldots, w_{ {\\ell_{\\operatorname{pump}}} +1}')$ in $T'$ containing only layer-2 vertices so that $r$ is the root of $T'$, $v = w_i'$ for some $1 \\leq i' \\leq {\\ell_{\\operatorname{pump}}} +1$, and the half-edge labels of $w_{ {\\ell_{\\operatorname{pump}}} +1}'$ have been fixed by $f$.\nSince $x \\in \\mathcal{V}' \\setminus \\mathcal{V}^\\ast$, we have $v \\neq w_{ {\\ell_{\\operatorname{pump}}} +1}'$. Hence $P = (v = w_i', w_{i+1}', \\ldots, w_{ {\\ell_{\\operatorname{pump}}} +1}')$ is a desired path with $2 \\leq k \\leq {\\ell_{\\operatorname{pump}}} + 1$ vertices.\n\n \n \n \n\n\\paragraph{Case 3} There is only one remaining neighbor of $v$ to consider. In view of the construction of $X_f(W_3^\\ast)$, there is a layer-3 vertex $v'$ adjacent to the vertex $r$, the root of the tree $T' \\in W_3^\\ast$ considered in the previous case. If the half-edge labels of $v'$ have been fixed by $f$, then $P = (v = w_i', w_{i-1}', \\ldots, w_{1}'=r, v')$ is a desired path.\nOtherwise, similar to the analysis in the previous case, we can find a path $P' = (v', \\ldots, w)$ connecting $v'$ to a vertex $w$ whose half-edge labels have been fixed by $f$. All vertices in $P'$ are of layer-3, and the number of vertices in $P'$ is within $[2, {\\ell_{\\operatorname{pump}}} + 1]$. \nCombining $P'$ with the path $(v = w_i', w_{i-1}', \\ldots, w_{1}'=r, v')$, we obtain the desired path $P$ whose number of vertices satisfies $2 \\leq k \\leq 2 {\\ell_{\\operatorname{pump}}} + 1$.\n \n \n\\end{proof}\n\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width= .75\\textwidth]{fig\/y2.pdf}\n \\caption{An illustration for the proof of \\cref{lem:find_path_1}.}\n \\label{fig:ell_full_construction1}\n\\end{figure}\n\nFor \\cref{lem:find_path_2}, note that if $\\mathtt{a}$ appears more than once in the multiset $x$, then we still have $\\mathtt{a} \\in x \\setminus \\{\\mathtt{a}\\}$.\n\n\n\\begin{lemma}\\label{lem:find_path_2}\nFor any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$, $\\mathtt{a}' \\in x' \\in \\mathcal{V}^\\ast$, and $0 \\leq t \\leq {\\ell_{\\operatorname{pump}}} - 1$, we have $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ for some $\\mathtt{b} \\in x \\setminus \\{\\mathtt{a}\\}$ and $\\mathtt{b}' \\in x' \\setminus \\{\\mathtt{a}'\\}$\n with $k = 2 {\\ell_{\\operatorname{pump}}} + 4 + t$.\n\\end{lemma}\n\\begin{proof}\nFor any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$, we can find a rooted tree $T_{x, \\mathtt{a}} \\in W_2^\\ast$ such that the path $(v_1$, $v_2$, $\\ldots$, $v_{ {\\ell_{\\operatorname{pump}}} +1})$ of the layer-1 vertices in $T_{x, \\mathtt{a}}$ where $v_1 = r$ is the root of $T_{x, \\mathtt{a}}$ satisfies the property that the half-edge labels of $v_{ {\\ell_{\\operatorname{pump}}} +1}$ have been fixed to $x$ by $f$ where the half-edge label for $\\{v_{ {\\ell_{\\operatorname{pump}}} }, v_{ {\\ell_{\\operatorname{pump}}} +1}\\}$ is in $x \\setminus \\{\\mathtt{a}\\}$.\n\nNow, observe that for any choice of $(\\Delta-2)( {\\ell_{\\operatorname{pump}}} +1)$ rooted trees \\[\\{T_{i,j}\\}_{1 \\leq i \\leq {\\ell_{\\operatorname{pump}}} +1, 1 \\leq j \\leq \\Delta-2}\\] in $W_2^\\ast$, there is a rooted tree $T' \\in W_3^\\ast = Z_f(W_2^\\ast)$ formed by appending $T_{i, 1}$, $T_{i,2}$, $\\ldots$, $T_{i, \\Delta-2}$ to $u_i$ in the path $(u_1$, $u_2$, $\\ldots$, $u_{ {\\ell_{\\operatorname{pump}}} +1})$ and fixing the half-edge labeling of $u_{ {\\ell_{\\operatorname{pump}}} +1}$ by $f$. All vertices in $(u_1, u_2, \\ldots, u_{ {\\ell_{\\operatorname{pump}}} +1})$ are of layer-2.\n\nWe choose any $T' \\in W_2^\\ast$ with $T_{i,j} = T_{x, \\mathtt{a}}$ and $T_{i',j'} = T_{x', \\mathtt{a}'}$ such that $i' - i = t + 1$.\nThe possible range of $t$ is $[0, {\\ell_{\\operatorname{pump}}} - 1]$.\nConsider any $T \\in W_L^\\ast$ that contains $T'$ as its subtree, and consider any correct labeling of $T$.\nThen the path $P$ resulting from concatenating the path $(v_{ {\\ell_{\\operatorname{pump}}} +1}, v_{ {\\ell_{\\operatorname{pump}}} }, \\ldots, v_1)$ in $T_{x, \\mathtt{a}}$, the path $(u_i, u_{i+1}, \\ldots, u_{i'})$, and the path $(v_1', v_2', \\ldots, v_{ {\\ell_{\\operatorname{pump}}} +1}')$ in $T_{x', \\mathtt{a}'}$\nshows that $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ \nfor $k = ( {\\ell_{\\operatorname{pump}}} +1) + (t+2) + ( {\\ell_{\\operatorname{pump}}} +1) = 2 {\\ell_{\\operatorname{pump}}} + 4 + t$. See \\cref{fig:ell_full_construction2} for an illustration.\n\nFor the rest of the proof, we show that the desired rooted tree $T_{x, \\mathtt{a}} \\in W_2^\\ast$ exists for any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$.\nFor any $x \\in \\mathcal{V}^\\ast$, we can find a bipolar tree\n\\[H_\\ast = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T_\\ast^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r) \\in Y_1^\\ast = Y_f(W_1^\\ast)\\]\nsuch that the vertex configuration of the root $r^m$ of $T_\\ast^m$ is fixed to be $x$ by the labeling function $f$.\nThen, for any $\\mathtt{a} \\in x$, at least one of the two rooted trees in $W_2^\\ast = Z_f(W_1^\\ast)$ resulting from cutting $H_\\ast$ satisfies the desired requirement. \n\\end{proof}\n\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width= .65\\textwidth]{fig\/y3.pdf}\n \\caption{An illustration for the proof of \\cref{lem:find_path_2}.}\n \\label{fig:ell_full_construction2}\n\\end{figure} \n\nIn the subsequent discussion, the length of a path refers to the number of edges in a path. In particular, the length of a $k$-vertex path is $k-1$.\n\nThe following lemma is proved by iteratively applying \\cref{lem:find_path_2} via intermediate vertex configurations $\\tilde{x} \\in \\mathcal{V}^\\ast$. \n\n\\begin{lemma}\\label{lem:find_path_3}\nFor any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$ and $\\mathtt{a}' \\in x' \\in \\mathcal{V}^\\ast$, we have $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ for some $\\mathtt{b} \\in x \\setminus \\{\\mathtt{a}\\}$ and $\\mathtt{b}' \\in x' \\setminus \\{\\mathtt{a}'\\}$\n for all $k \\geq 6 {\\ell_{\\operatorname{pump}}} + 10$.\n\\end{lemma}\n\\begin{proof}\nBy applying \\cref{lem:find_path_2} for three times,\nwe infer that \\cref{lem:find_path_2} also works for any path length $k - 1 = (k_1 - 1) + (k_2 - 1) + (k_3 - 1)$, where $k_i - 1 = 2 {\\ell_{\\operatorname{pump}}} + 3 + t_i$ for $0 \\leq t_i \\leq {\\ell_{\\operatorname{pump}}} -1$. \n\nSpecifically, we first apply \\cref{lem:find_path_2} to find a path realizing $(x,\\mathtt{b}) \\overset{k_1}{\\leftrightarrow} \\tilde{x}$ for some $\\tilde{x} \\in \\mathcal{V}^\\ast$, and for some $\\mathtt{b} \\in x \\setminus \\{\\mathtt{a}\\}$.\nLet $\\tilde{\\mathtt{a}} \\in \\tilde{x}$ be the real half-edge label used in the last vertex of the path. \nWe extend the length of this path by $k_2 - 1$ by attaching to it the path realizing $(\\tilde{x},\\tilde{\\mathtt{b}}) \\overset{k_2}{\\leftrightarrow} \\tilde{x}$, for some $\\tilde{\\mathtt{b}} \\in \\tilde{x} \\setminus \\{\\tilde{\\mathtt{a}}\\}$.\nFinally, let $\\tilde{\\mathtt{c}} \\in \\tilde{x}$ be the real half-edge label used in the last vertex of the current path. \nWe extend the length of the current path by $k_3$ by attaching to it the path realizing $(\\tilde{x},\\tilde{\\mathtt{d}}) \\overset{k_3}{\\leftrightarrow} (x', \\mathtt{b}')$, for some $\\tilde{\\mathtt{d}} \\in \\tilde{x} \\setminus \\{\\tilde{\\mathtt{x}}\\}$, and for some $\\mathtt{b}' \\in x' \\setminus \\{\\mathtt{a}'\\}$.\nThe resulting path realizes $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ for some $\\mathtt{b} \\in x \\setminus \\{\\mathtt{a}\\}$ and $\\mathtt{b}' \\in x' \\setminus \\{\\mathtt{a}'\\}$.\n\n\nTherefore, \\cref{lem:find_path_2} also works with path length of the form $k-1 = 6 {\\ell_{\\operatorname{pump}}} + 9 + t$, for any $t \\in [0, 3 {\\ell_{\\operatorname{pump}}} - 3]$. \n\nAny $k-1 \\geq 6 {\\ell_{\\operatorname{pump}}} + 9$ can be written as $k-1 = b(2 {\\ell_{\\operatorname{pump}}} +3) + (6 {\\ell_{\\operatorname{pump}}} + 9 + t)$, for some $t \\in [0, 3 {\\ell_{\\operatorname{pump}}} - 3]$ and some integer $b \\geq 0$.\nTherefore, similar to the above, we can find a path showing $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ by first applying \\cref{lem:find_path_2} with path length\n$2 {\\ell_{\\operatorname{pump}}} +3$ for $b$ times, and then applying the above variant of \\cref{lem:find_path_2} to extend the path length by $6 {\\ell_{\\operatorname{pump}}} + 9 + t$.\n\\end{proof}\n\nWe show that \\cref{lem:find_path_1,lem:find_path_3} imply that $\\mathcal{V}'$ is $\\ell$-full for some $\\ell$. \n\n\\begin{lemma}[$\\mathcal{V}'$ is $\\ell$-full] \\label{lem:find_path_4}\nThe set $\\mathcal{V}'$ is $\\ell$-full for $\\ell = 10 {\\ell_{\\operatorname{pump}}} + 10$. \n\\end{lemma}\n\\begin{proof}\nWe show that for any target path length $k-1 \\geq 10 {\\ell_{\\operatorname{pump}}} + 9$, and for any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$ and $\\mathtt{a}' \\in x' \\in \\mathcal{V}^\\ast$, we have $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} (x',\\mathtt{a}')$.\n\n\nBy \\cref{lem:find_path_1}, there exists a vertex configuration $\\tilde{x} \\in \\mathcal{V}^\\ast$ so that we can find a path $P$ realizing $(x,\\mathtt{a}) \\overset{\\ell_1}{\\leftrightarrow} \\tilde{x}$ for some path length $1 \\leq (\\ell_1-1) \\leq 2 {\\ell_{\\operatorname{pump}}} $.\nSimilarly, there exists a vertex configuration $\\tilde{x}' \\in \\mathcal{V}^\\ast$ so that we can find a path $P'$ realizing $(x',\\mathtt{a}') \\overset{\\ell_2}{\\leftrightarrow} \\tilde{x}'$ for some path length $1 \\leq (\\ell_2-1) \\leq 2 {\\ell_{\\operatorname{pump}}} $. \n\n\nLet $\\tilde{\\mathtt{a}}$ be the real half-edge label for $\\tilde{x}$ in $P$, and let $\\tilde{\\mathtt{a}}'$ be the real half-edge label for $\\tilde{x}'$ in $P'$.\nWe apply \\cref{lem:find_path_3} to find a path $\\tilde{P}$ realizing $(x,\\mathtt{b}) \\overset{\\tilde{\\ell}}{\\leftrightarrow} (x', \\mathtt{b}')$ for some $\\mathtt{b} \\in \\tilde{x} \\setminus \\{\\tilde{\\mathtt{a}}\\}$ and $\\mathtt{b}' \\in \\tilde{x}' \\setminus \\{\\tilde{\\mathtt{a}}'\\}$ with path length \n\\[\\tilde{\\ell}-1 = (k-1) - (\\ell_1-1) - (\\ell_2-1) \\geq 6 {\\ell_{\\operatorname{pump}}} + 9.\\] \nThe path formed by concatenating $P_1$, $\\tilde{P}$, and $P_2$ shows that $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} (x',\\mathtt{a}')$.\n\\end{proof}\n\n\n\n\n\n\n \n\n\n\n\\paragraph{A Feasible Labeling Function $f$ exists} We show that a feasible labeling function $f$ exists given that $\\Pi$ can be solved in $\\mathsf{LOCAL}(n^{o(1)})$ rounds. We will construct a labeling function $f$ in such a way each equivalence class in $\\textsf{Class}_1(W^\\ast)$ contains only rooted trees that admit legal labeling. That is, for each $c \\in \\textsf{Class}_1(W^\\ast)$, the mapping $h$ associated with $c$ satisfies $h(X) = \\textsf{YES}$ for some $X$.\n\n\n\nWe will consider a series of modifications in the construction \\[W \\rightarrow X_f(W) \\rightarrow Y_f(W) \\rightarrow Z_f(W)\\] that do not alter the sets of equivalence classes in these sets, conditioning on the assumption that all trees that we have processed admit correct labelings. That is, whether $f$ is feasible is invariant of the modifications.\n\n\n\\paragraph{Applying the Pumping Lemma} Let $w > 0$ be some target length for the pumping lemma. In the construction of $Y_f(W)$, when we process a bipolar tree \n\\[H = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r),\\] we apply \\cref{lem:pump} to the two subtrees $H^l = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l)$ and $H^r = (T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)$ to obtain two new bipolar trees $H_+^l$ and $H_+^r$.\nThe $s$-$t$ path in the new trees $H_+^l$ and $H_+^r$ contains $w + x$ vertices, for some $0 \\leq x < {\\ell_{\\operatorname{pump}}} $. The equivalence classes do not change, that is, $\\texttt{type}(H^l) = \\texttt{type}(H_+^l)$ and $\\texttt{type}(H^r) = \\texttt{type}(H_+^r)$.\n\n\nWe replace $H^l$ by $H_+^l$ and replace $H^r$ by $H_+^r$ in the bipolar tree $H$.\nRecall that the outcome of applying the labeling function $f$ to the root $r$ of the rooted tree $T_{ {\\ell_{\\operatorname{pump}}} +1}$ depends only on \\[\\texttt{type}(H^l), \\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m), \\texttt{type}(H^r),\\] so applying the pumping lemma to $H^l$ and $H^r$ during the construction of $Y_f(W)$ does not alter $\\textsf{Class}_1(T_\\ast^m)$ and $\\texttt{type}(H^\\ast)$ for the resulting bipolar tree $H^\\ast$ and its middle rooted tree $T_\\ast^m$, by \\cref{lem:labeling_function}.\n\n\n\n\\paragraph{Reusing Previous Trees} During the construction of the fixed point $W^\\ast$, we remember all bipolar trees to which we have applied the feasible function $f$.\n\nDuring the construction of $Y_f(W)$. Suppose that we are about to process a bipolar tree $H$, and there is already some other bipolar tree $\\tilde{H}$ to which we have applied $f$ before so that \n$\\texttt{type}(H^l)=\\texttt{type}(\\tilde{H}^l)$, $\\textsf{Class}_1(T_{i}^m) = \\textsf{Class}_1(\\tilde{T}_{i}^m)$ for each $1 \\leq i \\leq \\Delta-2$, and $\\texttt{type}(H^r)=\\texttt{type}(\\tilde{H}^r)$.\nThen we replace $H$ by $\\tilde{H}$, and then we process $\\tilde{H}$ instead.\n\nBy \\cref{lem:labeling_function}, this modification does not alter $\\textsf{Class}_1(T_\\ast^m)$ and $\\texttt{type}(H^\\ast)$ for the resulting bipolar tree $H^\\ast$.\n\n\\paragraph{Not Cutting Bipolar Trees} We consider the following different construction of $Z_f(W)$ from $Y_f(W)$. For each $H^\\ast \\in Y_f(W)$, we simply add two copies of $H^\\ast$ to $Z_f(W)$, one of them has $r = s$ and the other one has $r = t$.\nThat is, we do not cut the bipolar tree $H^\\ast$, as in the original construction of $Z_f(W)$.\n\nIn general, this modification might alter $\\textsf{Class}_1(Z_f(W))$.\nHowever, it is not hard to see that if all trees in $Y_f(W)$ already admit correct labelings, then $\\textsf{Class}_1(Z_f(W))$ does not alter after the modification.\n\nSpecifically, let $T$ be a rooted tree in $Z_f(W)$ with the above modification. \nThen $T$ is identical to a bipolar tree $H^\\ast=(T_1, T_2, \\ldots, T_k) \\in Y_f(W)$, where there exists some $1 < i < k$ such that the root $r_i$ of $T_i$ has its half-edge labels fixed by $f$.\nSuppose that the root of $T$ is the root $r_1$ of $T_1$.\nIf we do not have the above modification, then the rooted tree $T'$ added to $Z_f(W)$ corresponding to $T$ is $(T_1, T_2, \\ldots, T_i)$, where the root of $T'$ is also $r_1$. \n\nGiven the assumption that all bipolar trees in $Y_f(W)$ admit correct labelings, it is not hard to see that $\\textsf{Class}_1(T') = \\textsf{Class}_1(T)$. For any correct labeling $\\mathcal{L}'$ of $T'$, we can extend the labeling to a correct labeling $\\mathcal{L}$ of $T$ by labeling the remaining vertices according to any arbitrary correct labeling of $H^\\ast$. This is possible because the labeling of $r_i$ has been fixed. For any correct labeling $\\mathcal{L}$ of $T$, we can obtain a correct labeling $\\mathcal{L}'$ of $T'$ by restricting $\\mathcal{L}$ to $T'$.\n\n\n\n\n\n\n\\paragraph{Simulating a $\\mathsf{LOCAL}$ Algorithm} We will show that there is a labeling function $f$ that makes all the rooted trees in $W^\\ast$ to admit correct solutions, where we apply the above three modifications in the construction of $W^\\ast$. In view of the above discussion, such a function $f$ is feasible.\n\nThe construction of $W^\\ast$ involves only a finite number $k$ of iterations of $Z_f$ applied to some previously constructed rooted trees. If we view the target length $w$ of the pumping lemma as a variable, then the size of a tree in $W^\\ast$ can be upper bounded by $O(w^{k})$. \n\n\nSuppose that we are given an arbitrary $n^{o(1)}$-round $\\mathsf{LOCAL}$ algorithm $\\mathcal{A}$ that solves $\\Pi$. It is known~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup,chang2020n1k_speedups} that randomness does not help for LCL problems on bounded-degree trees with round complexity $\\Omega(\\log n)$, so we assume that \n$\\mathcal{A}$ is deterministic.\n\nThe runtime of $\\mathcal{A}$ on a tree of size \n$O(w^{k})$ can be made to be at most $t = w\/10$ if $w$ is chosen to be sufficiently large. \n\nWe pick the labeling function $f$ as follows. \nWhenever we encounter a new bipolar tree $H$ to which we need to apply $f$, we simulate $\\mathcal{A}$ locally at the root $r^m$ of the middle rooted tree $T^m$ in $H$. Here we assume that the pumping lemma was applied before simulating $\\mathcal{A}$. Moreover, when we do the simulation, we assume that the number of vertices is $n = O(w^{k})$. \n\nTo make the simulation possible, we just locally generate arbitrary distinct identifiers in the radius-$t$ neighborhood $S$ of $r^m$. This is possible because $t = w\/10$ is so small that for any vertex $v$ in any tree $T$ constructed by recursively applying $Z_f$ for at most $k$ iterations, the radius-$(t+1)$ neighborhood of $v$ intersects at most one such $S$-set. Therefore, it is possible to complete the identifier assignment to the rest of the vertices in $T$ so that for any edge $\\{u,v\\}$, the set of identifiers seen by $u$ and $v$ are distinct in an execution of a $t$-round algorithm. \n\nThus, by the correctness of $\\mathcal{A}$, the labeling of all edges $\\{u,v\\}$ are configurations in $\\mathcal{E}$ and the labeling of all vertices $v$ are configurations in $\\mathcal{V}$.\nOur choice of $f$ implies that all trees in $W^\\ast$ admit correct labeling, and so $f$ is feasible. Combined with \\cref{lem:find_path_4}, we have proved that $\\mathsf{LOCAL}(n^{o(1)})$ $\\Rightarrow$ $\\ell$-full.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Concluding Remarks}\\label{sec:concludingRemarksTrees}\n\nHere we collect some open problems we find interesting. \n\n\n\\begin{enumerate}\n \\item What is the relation of $\\mathsf{ffiid}, \\mathsf{MEASURE}, \\mathsf{fiid}$? In particular is perfect matching in the class $\\mathsf{ffiid}$ or $\\mathsf{MEASURE}$? \n \\item Are there LCL problems whose uniform complexity is nontrivial and slower than $\\operatorname{poly}\\log 1\/\\varepsilon$?\n \n \n \n \n \n \n \\item Suppose that $\\Pi_G$ is a homomorphism LCL, is it true that $\\Pi_G\\in\\mathsf{BOREL}$ if and only if $\\Pi_G\\in O(\\log^*(n))$?\n \\item Are there finite graphs of chromatic number bigger than $2\\Delta-2$ such that the corresponding homomorphism LCL is not in $\\mathsf{BOREL}$? See also Remark \\ref{r:hedet}.\n\\end{enumerate}\n\n\\subsection*{Acknowledgements} \nWe would like to thank Anton Bernshteyn, Endre Cs\u00f3ka, Mohsen Ghaffari, Jan Hladk\u00fd, Steve Jackson, Alexander Kechris, Edward Krohne, Oleg Pikhurko, Brandon Seward, Jukka Suomela, and Yufan Zheng for insightful discussions. \nWe especially thank Yufan Zheng for suggesting potentially hard instances of homomorphism LCL problems.\n\n\n\\bibliographystyle{alpha}\n\n\\section{Introduction}\n\nIn this paper, we study local problems on grid graphs from three different perspectives: (a) the perspective of the theory of distributed algorithms, (b) the perspective of the theory of random processes, (c) the perspective of the field of descriptive combinatorics. \nAlthough different communities in these three fields ask very similar questions, no systematic connections between them were known, until the insightful work of Bernshteyn \\cite{Bernshteyn2021LLL} who applied results from the field (a) to the field (c). In this work, our most important results follow by applying techniques from the field (a) to (b), but perhaps more importantly, we present all three different perspectives as a part of one common theory (see \\cref{fig:big_picture_grids}). \n\nIn this introductory section, we first quickly present the three fields and define the object of interest, local problems, a class of problems that contains basic problems like vertex or edge coloring, matchings, and others. \nThen, we briefly present our results, most of which can be seen as inclusions between different classes of local problems shown in \\cref{fig:big_picture_grids}. \n\n\\paragraph{Distributed Computing}\n\nThe $\\mathsf{LOCAL}$ model of computing \\cite{linial1987LOCAL} was motivated by understanding distributed algorithms in huge networks. \nAs an example, consider the network of all wifi routers, where two routers are connected if they are close enough to exchange messages. In this case, they should better communicate with user devices on different channels, so as to avoid interference. In the graph-theoretic language, we want to color the network. Even if the maximum degree is $\\Delta$ and we want to color it with $\\Delta+1$ colors, the problem remains nontrivial, because the decision of each node should be done after few communication rounds with its neighbors. \n\nThe $\\mathsf{LOCAL}$ model formalizes this setup: we have a large network, with each node knowing its size, $n$, and perhaps some other parameter like the maximum degree $\\Delta$. In case of randomized algorithms, each node has an access to a random string, while in case of deterministic algorithms, each node starts with a unique identifier from a range of size polynomial in $n$. \nIn one round, each node can exchange any message with its neighbors and can perform an arbitrary computation. We want to find a solution to a problem in as few communication rounds as possible. \nImportantly, there is an equivalent view of $t$-rounds $\\mathsf{LOCAL}$ algorithms: such an algorithm is simply a function that maps $t$-hop neighbourhoods to the final output. An algorithm is correct if and only if applying this function to each node solves the problem. \n\nThere is a rich theory of distributed algorithms and the local complexity of many problems is understood. One example: if the input graph is the $d$-dimensional torus (think of the grid $\\mathbb{Z}^d$ and identify nodes with the same remainder after dividing the coordinate with $n^{1\/d}$), a simple picture emerges. Any local problem (we define those formally later) is solvable with local complexity $O(1)$, $\\Theta(\\log^* n)$, or we need to see the whole graph to solve it, i.e., the local complexity is $\\Theta(n^{1\/d})$ (see \\cref{thm:classification_of_local_problems_on_grids}). For example, a proper coloring of nodes with $4$ colors has local complexity $\\Theta(\\log^* n)$, while coloring with $3$ colors has already complexity $\\Theta(n^{1\/d})$. \n\n\\paragraph{Finitary factors of iid processes}\n\\todo{zeptat se Spinky}\nConsider the following example of the so-called anti-ferromagnetic Ising model. \nImagine atoms arranged in a grid $\\mathbb{Z}^3$. Each atom can be in one of two states and there is a preference between neighboring atoms to be in a different state. We can formalize this by defining a potential function $V(x,y)$ indicating whether $x$ and $y$ are equal. \nThe Ising model models the behavior of atoms in the grid as follows: for the potential $V$ and inverse temperature $\\beta$, if we look at the atoms, we find them in a state $\\Lambda$ with probability proportional to $\\exp(-\\beta \\sum_{u \\sim v \\in E(\\mathbb{Z}^d)} V(\\Lambda(u), \\Lambda(v)))$. If $\\beta$ is close to zero, we sample essentially from the uniform distribution. If $\\beta$ is infinite, we sample one of the two $2$-colorings of the grid. \n\nTo understand better the behavior of phase transitions, one can ask the following question. For what values of $\\beta$ can we think of the distribution $\\mu$ over states from the Ising model as follows: there is some underlying random variable $X(u)$ at every node $u$, with these variables being independent and identically distributed. We can think of $\\mu$ as being generated by a procedure, where each node explores the random variables in its neighborhood until it decides on its state, which happens with probability $1$. If this view of the underlying distribution is possible, we say it is a finitary factor of an iid process (ffiid). \nIntuitively, being an ffiid justifies the intuition that there is a correlation decay between different sites of the grid. To understand ffiids, we try to analyze solutions to which local problems can be sampled from an ffiid distribution (2-coloring of the grid is not among them). \n\nAlthough this definition was raised in a different context then distributed computing, it is actually almost identical to the definition of a distributed algorithm! The differences are two. First, the ffiid procedure does not know the value of $n$ since, in fact, it makes most sense to consider it in infinite graphs (like the infinite grid $\\mathbb{Z}^d$). Second, the natural measure of the complexity is different: it is the speed of the decay of the random variable corresponding to the time until the local algorithm finds the solution. \nNext, note that the algorithm has to return a correct solution with probability $1$, while the usual randomized local algorithm can err with a small probability. We call ffiids \\emph{oblivious local algorithms} to keep the language of distributed computing (see \\cref{sec:preliminaries}).\nWe define the class $\\mathsf{TAIL}(f(r))$ as the class of problems solvable by an oblivious local algorithm if the probability of the needed radius being bigger than $r$ is bounded by $1\/f(r)$ (see \\cref{fig:big_picture_grids}). \n\n\\paragraph{Descriptive Combinatorics}\n\nA famous Banach-Tarski theorem states that one can split a three-dimensional unit volume ball into five pieces, translate and rotate each piece and end up with two unit volume balls. \nThis theorem serves as an introductory warning in the beginner measure theory courses: not every subset of $\\mathbb{R}^3$ can be measurable.\nSimilarly surprising was when in 1990 Laczkovich answered the famous Circle Squaring Problem of Tarski\\todo{cite laczkovich}: A square of unit area can be decomposed into finitely many pieces such that one can translate each of them to get a circle of unit area.\nUnlike in Banach-Tarski paradox, this result was improved in recent years to make the pieces measurable in various sense \\cite{OlegCircle, Circle, JordanCircle}.\n\n\nThis theorem and its subsequent strengthenings are the highlights of a field nowadays called descriptive combinatorics \\cite{kechris_marks2016descriptive_comb_survey,pikhurko2021descriptive_comb_survey} that has close connections to distributed computing as was shown in an insightful paper by Bernshteyn \\cite{Bernshteyn2021LLL}. \nGoing back to the Circle Squaring Problem, the proofs of its versions first choose a clever set of constantly many translations $\\tau_1, \\dots, \\tau_k$. Then, one constructs a bipartite graph between the vertices of the square and the cycle and connects two vertices $x, y$ if there is a translation $\\tau_i$ mapping the point $x$ in the square to the point $y$ in the cycle. The problem can now be formulated as a perfect matching problem and this is how combinatorics enters the picture. In fact, versions of a matching or flow problems are then solved on the grid graph $\\mathbb{Z}^d$ using a trick called \\emph{toast construction} also used in the area of the study of ffiids \\cite{HolroydSchrammWilson2017FinitaryColoring,ArankaFeriLaci,Spinka}. \n\nWe discuss the descriptive combinatorics classes in a more detail in the expository \\cref{sec:OtherClasses} but we will not need them in the main body. Let us now give an intuitive definition of the class $\\mathsf{BOREL}$. \nBy a result of Chang and Pettie \\cite{chang2016exp_separation}, we know that the maximal independent set problem is complete for the class $\\mathsf{LOCAL}(O(\\log^* n))$ in the sense that this problem can be solved in $O(\\log^* n)$ rounds and any problem in that class can be solved in constant number of rounds if we are given access to an oracle computing MIS (we make this precise in \\cref{rem:chang_pettie_decomposition}). Imagine now that the input graph is infinite and our local algorithm has access to MIS oracle and local computations but now both can be used countably many times instead of just once or, equivalently, fixed number of times as is the case with the class $\\mathsf{LOCAL}(O(\\log^* n))$. That is, we can think of an algorithm that constructs MIS of larger and larger powers of the input graph and uses this infinite sequence to solve an LCL. Those constructions are in the class $\\mathsf{BOREL}$ from \\cref{fig:big_picture_grids}. In fact, the exact relationship between this definition and full power of $\\mathsf{BOREL}$ is not clear.\nHowever, it should give an idea about the difference between descriptive combinatorics and distributed computing -- from the descriptive combinatorics perspective, reasonable classes are closed under repeating basic operations countably many times.\n\n\n\n\n\\paragraph{Locally Checkable Problems on Grids as a ``Rosetta Stone''}\n\nLocally checkable problems (LCLs) are graph problems where a correctness of a solution can be checked locally. That is, there is a set of local constraints such that if they are satisfied at every node, the solution to a problem is correct. \nAn examples of such constraints include vertex coloring, edge coloring, or perfect matching. We also allow inputs on each node, so the list coloring is also an example of LCL (given that the total number of colors in all lists is finite). One should think of this class as an analogue of the class NP from complexity theory. \n\nIn this paper, we study locally checkable problems on grid graphs in the three settings introduced above. This approach reveals number of connections. For example, in \\cref{fig:big_picture_grids}, we can see that the two classes of local complexities $\\mathsf{LOCAL}(O(1))$ and $\\mathsf{LOCAL}(O(\\log^* n))$ are equivalent to different natural classes from the perspective of factors of iid and descriptive combinatorics. On the other hand, those two worlds are richer. Many of our discussions in fact apply to graphs of bounded growth, but the grids serve as the simplest, but still quite nontrivial model in which one can try to understand all three settings together (the case of lines is understood quite well \\cite{grebik_rozhon2021LCL_on_paths} and we study aspects of the connection for trees in a parallel paper \\cite{brandt_chang_grebik_grunau_rozhon_vidnyaszky2021LCLs_on_trees_descriptive}). \n\n\\begin{landscape}\n\\begin{figure}\n \\centering\n \\includegraphics[width = 1.5\\textwidth]{fig\/big_picture_grids.png}\n \\caption{The big picture of local complexity classes on grids from the perspective of distributed algorithms, factors of iid, and descriptive combinatorics. Canonical example of each class is on the left side. }\n \\label{fig:big_picture_grids}\n\\end{figure}\n\\end{landscape}\n\n\n\\subsection{Our Contribution }\n\nOur contribution can be mostly summed up as several new arrows in \\cref{fig:big_picture_grids}. In general, this clarifies several basic questions regarding the connections of the three fields and helps to view all three fields as part of a general theory of locality. \n \n\n\nFor example, consider the following question from the paper of Brandt et al. \\cite{brandt_grids} that considered LCL problems on grids: \n\\begin{quote}\n\\emph{\nHowever, despite the fact that techniques seem to translate between finitary colorings and distributed complexity, it remains unclear how to directly translate results from one setting to the other in a black-box manner; for instance, can we derive the lower bound for 3-colouring from\nthe results of Holroyd et al., and does our complexity classification imply answers to the open questions they pose?\n}\n\\end{quote}\nWe show how to 'directly translate' results between the setting of distributed algorithms and finitary factors of iid in \\cref{subsec:preliminaries_tail,subsec:speedup_warmup}. This indeed answers two open questions of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n\nTo understand the connection between the world of factors of iids and descriptive combinatorics, we formalize the notion of toast construction used in both setups. We prove general results connecting the toast constructions with oblivious local algorithms and this enables us to understand the $\\mathsf{TAIL}$ complexity of a number of concrete problems like vertex $3$-coloring, edge $4$-coloring, perfect matchings and others. \n\nIn general, we hope that the concept of $\\mathsf{TAIL}$ complexities will find other usecases. We discuss in \\cref{subsec:congestLCA} how it relates to studied notions of oblivious algorithms, energy complexity, or its relevance to the $\\mathsf{LCA}$ model.\\todo{Oleg se ptal jestli uz ma tady vedet co je LCA a jestli to nekde neprehlid}\nIn fact, the $\\mathsf{TAIL}$ complexity approach was already considered e.g. in the celebrated MIS paper of \\cite{ghaffari2016MIS} (see the introduction of his paper) and the $\\mathsf{TAIL}$ bound on the MIS algorithm there may be the reason of its wide applicability. \n\nWe now list concrete results that we prove (or they follow from the work of others, see \\cref{fig:big_picture_grids}). \n\n\\paragraph{Classification Theorems for $\\mathsf{TAIL}$}\n\nWe prove several results that one can view as arrows in \\cref{fig:big_picture_grids}. Those can be viewed as an analogy to the line of work aiming for the classification of $\\mathsf{LOCAL}$ complexities discussed in \\cref{subsec:preliminaries_local}. \n\n\\begin{itemize}\n \\item $\\mathsf{TAIL}(\\mathrm{tower}(\\omega(r))) = \\mathsf{LOCAL}(O(1))$\\hspace*{1cm} We show that any oblivious local algorithm with tail decay faster than tower function can be in fact solved in $O(1)$ local complexity. This answers a question of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n \\item $\\mathsf{LOCAL}(O(\\log^* n)) = \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r))) = \\mathsf{CONTINUOUS}$\\hspace*{1cm} There are three natural definitions of classes of basic symmetry breaking problems in all three considered fields and they are the same \\cite{GJKS,HolroydSchrammWilson2017FinitaryColoring}. \n As a direct application one gets that from the three proofs that $3$-coloring of grids is not in $\\mathsf{LOCAL}(O(\\log^* n))$ \\cite{brandt_grids}, not in $ \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$ \\cite{HolroydSchrammWilson2017FinitaryColoring} and not in $ \\mathsf{CONTINUOUS}$ \\cite{GJKS} any one implies the others. These three classes are the same in a much more general setup (see \\cref{subsec:CONT}). \n \\item $\\mathsf{TAIL}(\\omega(r^{d-1})) = \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$\\hspace*{1cm} We prove that any problem that can be solved with an oblivious local algorithm with tail decay faster than $r^{d-1}$, can in fact be solved with a tower tail decay. This answers a question of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n \\item Similarly to the time hierarchy theorems of the $\\mathsf{LOCAL}$ model \\cite{chang2017time_hierarchy}, we study the range of possible $\\mathsf{TAIL}$ complexities and give examples of problems LCL problems with $\\mathsf{TAIL}$-complexities $\\Theta(r^{j - o(1)})$ for any $1 \\le j \\le d-1$. \n\\end{itemize}\n\n\\paragraph{$\\TOAST$ constructions}\nWe formally define $\\TOAST$ algorithms (we mentioned earlier that these constructions are used in descriptive combinatorics and the theory of random processes) and prove results about them. \nLet us first explain the toast constructions in a more detail. A $q$-toast in the infinite graph $\\mathbb{Z}^d$ is a collection $\\mathcal{D}$ of finite connected subsets of $\\mathbb{Z}^d$ that we call \\emph{pieces} such that (a) for any two nodes of the graph we can find a piece that contains both,\\todo{it is enough that every vertex is contained in a piece} (b) boundaries of any two different pieces are at least distance $q$ apart. \nThat is, imagine a collection of bigger and bigger pieces that form a laminar family (only inclusions, not intersections) that, in the limit, covers the whole infinite graph.\n\nThis structure can be built in the descriptive combinatorics setups \\cite{FirstToast,GJKS,Circle} but also with oblivious local algorithms \\cite{HolroydSchrammWilson2017FinitaryColoring,Spinka}. In fact, we show that one can construct the structure with $1\/r^{1-o(1)}$ $\\mathsf{TAIL}$ decay in \\cref{thm:toast_in_tail}. This is best possible up to the $o(1)$ loss in the exponent. \nIf one has an inductive procedure building the toast, we can try to solve an LCL problem during the construction. That is, whenever a new piece of the toast is constructed, we define how the solution of the problem looks like at this piece. This is what we call a $\\TOAST$ algorithm. \nThere is a randomized variant of the $\\TOAST$ algorithm and a lazy variant. In the lazy variant one can postpone the decision to color given piece, but it needs to be colored at some point in the future. \nNext, we prove several relations between the $\\TOAST$ classes (see \\cref{fig:big_picture_grids}). \n\n\\begin{itemize}\n \\item $\\TOAST \\subseteq \\mathsf{TAIL}(r^{1-o(1)})$\\hspace*{1cm} This result allows us to translate toast-algorithm constructions used in descriptive combinatorics to the $\\mathsf{TAIL}$-setting. This includes problems like vertex $3$-coloring (this answers a question of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}), edge $2d$-coloring, or perfect matching. \n \\item $\\RTOAST = \\TOAST$\\hspace*{1cm} This derandomization result for example implies on the first glance surprising inclusion $\\mathsf{FINDEP} \\subseteq \\mathsf{BOREL}$ -- these two classes are discussed in \\cref{sec:OtherClasses}. \n \\item $\\mathsf{TAIL} = \\RLTOAST$\\hspace*{1cm} Here, $\\mathsf{TAIL}$ is defined as the union of $\\mathsf{TAIL}(f)$ over all functions $f \\in \\omega(1)$. This result shows how the language of toast constructions can in fact characterize the whole hierarchy of constructions by oblivious local algorithms. \n \\item We show that the problem of $2d$ edge coloring of $\\mathbb{Z}^d$ is in $\\TOAST$, hence in $\\mathsf{BOREL}$ which answers the question of Gao et al \\cite{GJKS}. Independently, this was proven by \\cite{ArankaFeriLaci}. \n\\end{itemize}\n\n\\subsection{Roadmap}\n\nIn \\cref{sec:preliminaries} we recall the basic definitions regarding the $\\mathsf{LOCAL}$ model and explain the formal connection between finitary factors of iids. \nNext, in \\cref{sec:speedup_theorems} we prove the basic speedup theorems for $\\mathsf{TAIL}$. \nIn \\cref{sec:Toast} we define $\\TOAST$ algorithms and relate it to the $\\mathsf{TAIL}$ model. \nIn \\cref{sec:SpecificProblems} we discuss what our results mean for specific local problems and in \\cref{sec:Inter} we construct problems with $r^{j-o(1)}$ $\\mathsf{TAIL}$ complexities for $1 \\le j \\le d-1$. Finally, in \\cref{sec:OtherClasses} we discuss connection of our work to other classes of problems and finish with open questions in \\cref{sec:open_problems}. \n\n\n\n\n\n\\begin{comment}\nHonza - Introduction\n\nA \\emph{locally checkable problems (LCL) $\\Pi$ on a family of graphs $\\mathbf{G}$} is a procedure that checks whether a given ``candidate'' vertex coloring of $G\\in \\mathbf{G}$ satisfies a local constraints given by $\\Pi$\\footnote{\nFormally, $\\Pi$ is a triplet $\\Pi=(b,t,\\mathcal{P})$, where $b$ is a set of colors, $t\\in \\mathbb{N}$ is the locality of $\\Pi$ and $\\mathcal{P}$ is a set of allowed configurations around each vertex of locality at most $t$.}.\nMore concretely, given a graph $G=(V(G),E(G))\\in \\mathbf{G}$ and a ``candidate'' coloring $c:V(G)\\to b$, we say that $c$ is a \\emph{$\\Pi$-coloring} if for every $v\\in V(G)$ the restriction of $c$ to $t$-neighborhood of $v$ is in $\\mathcal{P}$, where $t$ is the locality of $\\Pi$ and $\\mathcal{P}$ is the set of constraints of $\\Pi$.\n\nIn this paper the family $\\mathbf{G}$ always refers to graphs that look locally like the standard Cayley graph of $\\mathbb{Z}^d$, $\\mathbb{Z}^d$, for some $d\\ge 1$.\nThis includes $\\mathbb{Z}^d$ as well as various finite $d$-dimensional tori with oriented edges that are labeled by the standard generators of $\\mathbb{Z}^d$.\nThe dimension $d$ is always assumed to be bigger than $1$, since for $d=1$ there is a full classification of LCL problems.\nWe omit mentioning ${\\bf G}$ and refer to all LCL problems on the family ${\\bf G}$ simply as \\emph{LCL problems (on $\n\\mathbb{Z}^d$)}.\nWhen we want to stress that a given LCL problem is on $\\mathbb{Z}^d$ we write $d$ in the superscript, i.e., $\\Pi^d$.\n\nWe study a complexity of LCL problems in various settings.\nIn the most abstract terms, a complexity of a given LCL problem $\\Pi$ is measured in terms of how difficult it is to \\emph{transform} a \\emph{given labeling} of a graph into a $\\Pi$-coloring.\nThe different contexts varies in what is considered to be an allowed starting labeling and what is allowed transformation.\nMost relevant for our investigation is the following probability notion.\nStart with a graph $G$ and assign to each vertex $v\\in V(G)$ a real number from $[0,1]$ uniformly and independently, i.e., \\emph{iid labeling} of $G$.\nA function $F$ that transforms these iid labels in a measurable and equivaraint way into a different coloring is called \\emph{factor of iid (fiid)}.\nHere equivariant refers to the group of all symmetries of $G$.\nIt is not hard to see that the group of all symmetries of $\\mathbb{Z}^d$ is the group $\\mathbb{Z}^d$ itself that acts by shift-translations.\nA fiid $F$ on $\\mathbb{Z}^d$ is called \\emph{finitary fiid (ffiid)} if for almost every iid labeling of $\\mathbb{Z}^d$ the color $F(x)({\\bf 0}_d)$ only depends on the labels of $x$ on some finite neighborhood of ${\\bf 0}_d$.\nThat is to say, if $x$ and $x'$ agrees on this neighborhood, then $F(x)({\\bf 0}_d)=F(x')({\\bf 0}_d)$.\nGiven ffiid $F$, we write $R_\\mathcal{A}$ for the random variable that codes minimum diameter of such neighborhood and we refer to it as the coding radius of $F$.\n\nWe say that a ffiid $F$ \\emph{solves an LCL problem $\\Pi$} if it produces a $\\Pi$-coloring for almost every iid labeling of $\\mathbb{Z}^d$.\nan LCL problem $\\Pi$ is in the class $\\mathsf{TAIL}$ if $\\Pi$ admits a ffiid solution.\nFollowing Holroyd, Schramm and Wilson \\cite{HolroydSchrammWilson2017FinitaryColoring}, the complexity of LCL problems in $\\mathsf{TAIL}$ can be measured on a finer scale by the properties of the coding radius $R_\\mathcal{A}$, where $F$ runs over all ffiid's that solve the given problem.\nSimilarly, we define when fiid $F$ solves $\\Pi$ and the class $\\mathsf{fiid}$.\n\nBefore we move to distributed algorithm we recall what is known about the notorious class of LCL problems that are the \\emph{proper vertex $k$-colorings of $\\mathbb{Z}^d$}, $\\Pi^d_{\\chi,k}$, where $k>1$, in the context of ffiid.\nIt is known that if $d>1$ and $k>3$, then $\\Pi^d_{\\chi,k}$ admits ffiid solution with a tower tail decay, see Holroyd et.al~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\nIf $d>0$ and $k=2$, then $\\Pi^d_{\\chi,2}$ is not even in $\\mathsf{fiid}$.\nThe most interesting case is when $d>1$ and $k=3$.\nIt was shown by Holroyd et al~\\cite{HolroydSchrammWilson2017FinitaryColoring} that in that case $\\Pi^d_{\\chi,3}$ is in the class $\\mathsf{TAIL}$ and every ffiid that solves $\\Pi^d_{\\chi,3}$ must have infinite the $2$-moment of its coding radius. \nMoreover, they found a ffiid that solve $\\Pi^d_{\\chi,3}$ and has finite some moments of its coding radius.\nHere we strengthen their result by showing that ${\\bf M}(\\Pi^d_{\\chi,3})=1$ and every ffiid solution of $\\Pi^d_{\\chi,3}$ has infinite the $1$-moment of its coding radius.\n\nIn the context of deterministic distributed computing the setting is the following.\nStart with finite graph $G$ on $n$-vertices (in our case $n$ is big enough and $G$ is a torus) and assign to every vertex of $G$ its unique identifier from $[n]$.\\footnote{In generality, the identifier could be assigned from some domain that has size polynomial in $n$.}\nA \\emph{local algorithm $\\mathcal{A}_n$ of locality $t_n$} is a rule that computes color of a vertex $v$ based only on its $t$-neighborhood and the unique identifiers in this neighborhood.\nWe say that $\\mathcal{A}_n$ \\emph{solves} $\\Pi$ if for every (admissible)\\footnote{As a torus $G$ needs to be wider than the locality of $\\mathcal{A}_n$} graph $G$ on $n$-vertices and any assignment of unique identifiers to $G$ the rule $\\mathcal{A}_n$ produces a $\\Pi$-coloring.\nWe say that $\\Pi$ is in the class $\\mathsf{LOCAL}(O(\\log^*(n)))$ if there is a sequence of local rules \\todo{co je to local rule?}$\\{\\mathcal{A}_n,t_n\\}_{n\\in \\mathbb{N}}$ such that $\\mathcal{A}_n$ has locality $t_n$, solves $\\Pi$ for every $n\\in\\mathbb{N}$ and $n\\mapsto t_n$ is in the class $O(\\log^*(n))$.\nThe reason why we picked $\\log^*(n)$ is that there is a complete classification of LCL problems on oriented labeled grids.\nRoughly speaking, an LCL problem $\\Pi$ is either \\emph{trivial}, i.e., there is a single color that produces $\\Pi$-coloring, \\emph{local}, i.e., there is an algorithm that needs locality $\\Theta(\\log^*(n))$, or \\emph{global}, i.e., every local algorithm that solves $\\Pi$ needs to see essentially the whole graph.\\footnote{Same classification holds in the context of randomized distributed algorithms.}\n\nWe use this classification to provide several speed-up results in the context of ffiid solutions.\nWe note that the connection between distributed computing and probability setting was previously observed by several authors.\nFor example, it follows easily from the result of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring} that local coloring problems that are not global in the distributed setting admit ffiid solution with tower tail decay.\nIn fact, Brandt et al. \\cite{brandt_grids} asked:\n\\begin{quote}\n\\emph{\nHowever, despite the fact that techniques seem to translate between finitary colourings\nand distributed complexity, it remains unclear how to directly translate results from one setting to\nthe other in a black-box manner; for instance, can we derive the lower bound for 3-colouring from\nthe results of Holroyd et al. [24], and does our complexity classification imply answers to the open\nquestions they pose?\n}\n\\end{quote}\nAs one of our main result we address their question.\n\nThe most abstract setting that we consider is Borel combinatorics.\nIn this context one studies graphs that additionally posses a completely metrizable and separable topological structure (or a standard Borel structure) on its vertex set.\nIt follows from the work of Seward and Tucker-Drob \\cite{ST} and Gao, Jackson, Krohne and Seward \\cite{GJKS} that the canonical object to investigate is the graph structure on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$, the free part of the shift action $\\mathbb{Z}^d\\curvearrowright 2^{\\mathbb{Z}^d}$, induced by the standard generating set of $\\mathbb{Z}^d$. \nThe topological and Borel structure on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ is the one inherited from the product topology on $2^{\\mathbb{Z}^d}$.\nWe say that an LCL problem $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ if there is a continuous and equivariant map from $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ into the space of $\\Pi$-colorings.\nThe class $\\mathsf{BOREL}$ is defined similarly, only the requirement on the map to be continuous is relaxed to be merely Borel measurable.\nBernshteyn \\cite{Bernshteyn2021LLL} proved that LCL problems that are not global in the distributed setting admit continuous and therefore Borel solution.\nIn another words, he showed that the class $\\mathsf{LOCAL}(O(\\log^*(n)))$ is contained in $\\mathsf{CONTINUOUS}$.\nIt is not hard to show that the other inclusion follows from the Twelve Tile Theorem of Gao et al \\cite{GJKS}, i.e., the class $\\mathsf{CONTINUOUS}$ in fact coincide with the class $\\mathsf{LOCAL}(O(\\log^*(n)))$.\nHere we investigate in greater detail what is the connection between existence of a Borel solution and a tail behavior of ffiid.\nIn fact all known Borel constructions uses the following structure that can be rephrased without mentioning topology or Borel measurability.\n\nA \\emph{$q$-toast on $\\mathbb{Z}^d$} is a collection $\\mathcal{D}$ of finite connected subsets of $\\mathbb{Z}^d$ such that\n\\begin{itemize}\n \\item for every $g,h\\in \\mathbb{Z}^d$ there is $D\\in\\mathcal{D}$ such that $g,h\\in D$,\n \\item if $D,E\\in \\mathcal{D}$ and $d(\\partial D,\\partial E)\\le q$, then $D=E$.\n\\end{itemize}\nWe say that an LCL problem $\\Pi$ is in the class $\\TOAST$ if there is $q\\in \\mathbb{N}$ and an equivariant rule $\\Omega$ that extends partial colorings such that if we apply $\\Omega$ on every $q$-toast $\\mathcal{D}$ inductively on the depth, then we produce a $\\Pi$-coloring.\nIt is not known whether the classes $\\mathsf{BOREL}$ and $\\TOAST$ coincide, however, it follows from the work of Gao et al.~\\cite{GJKS2} that $\\TOAST\\subseteq \\mathsf{BOREL}$.\nThis is proved by constructing a toast structure in a Borel way on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$, see Marks and Unger~\\cite{Circle} for application and proof of this fact.\nThe specific problems that belong to the class $\\TOAST$ include $3$-vertex coloring, perfect matching, $2d$-edge coloring, Schreier decoration (for non-oriented grids), some rectangular tilings, space filling curve etc.\nThese results were proved by many people in slightly different language, \\cite{GJKS2, ArankaFeriLaci, spencer_personal}.\nWe utilize here all the constructions and show that these all belong to $\\TOAST$.\n\nWe connect the classes $\\TOAST$ and $\\mathsf{TAIL}$ as follows.\nWe show that for every $q\\in \\mathbb{N}$ and $\\epsilon>0$ there is a ffiid $F$ that produces $q$-toast that has the $(1-\\epsilon)$-moment finite.\nA slightly stronger result implies that, in fact, ${\\bf M}(\\Pi)\\ge 1$, whenever an LCL problem $\\Pi$ is in the class $\\TOAST$.\nThis shows that ${\\bf M}(\\Pi)\\ge 1$ for all the problems mentioned above.\nWe note that in the special case when $d=2$ and the corresponding LCL problem $\\Pi$ does not admit ffiid solution with a tower tail decay, we deduce that ${\\bf M}(\\Pi)=1$.\nThis follows from our speed-up result in the context of ffiid.\n\nTraditionally there are two more classes that are investigated in Borel combinatorics, $\\mathsf{BAIRE}$ and $\\mathsf{MEASURE}$.\nTo define the class $\\mathsf{BAIRE}$ we replace the topological requirement on continuity by \\emph{Baire measurability}, i.e., we require a solution to be continuous only on a \\emph{comeager} set, see the work of Bernshteyn \\cite{Bernshteyn2021LLL}.\nThe class $\\mathsf{MEASURE}$ generalizes the setting of fiid.\nThe goal here is to find a Borel solution on a conull set for a given Borel graph that is endowed with a Borel probability measure.\nThis direction has been recently investigated by Bernshteyn \\cite{Bernshteyn2021LLL} for general graphs, where he proved a measurable version of Lovasz's Local Lemma.\nSee \\cref{fig:big_picture_grids} for all the relations that are known to us.\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{fig\/big_picture_grids.png}\n \\caption{Big picture of lcl for $\\mathbb{Z}^d$}\n \n\\end{figure}\n\n\n\n\n\n\n\n\\subsection{The Results}\n\nIn this subsection we formulate our main results.\nThe formulation uses mostly the language of ffiid because the tail behavior of the coding radius allows to compare problems on a quantitative and, as it turns out, rich scale.\nOne should always re-interpret the result with the help of Figure~\\ref{fig:big_picture_grids} to different contexts.\nIronically enough, the techniques that are most valuable when investigating ffiid solutions come either from local algorithms or Borel combinatorics, where such a rich and quantitative distinction is not possible.\nRecall that in the distributed algorithms there is a trichotomy of LCL problems, i.e., trivial, local or global, and in Borel combinatorics we distinguish mainly between trivial, continuous, Borel, and global problems.\n\n\nWe start with results that use ideas from distributed algorithms.\nFirst, we prove that LCL problems that are not global in distributed algorithms are precisely problems that admit ffiid solution with a tower tail decay.\n\n\\begin{theorem*}\nLet $d\\in \\mathbb{N}$ and $\\Pi$ be an LCL problem on $\\mathbb{Z}^d$.\nThen $\\Pi$ is not global in the local world, i.e., there is a deterministic algorithm in the local model that runs in $O(\\log^*(n))$ rounds, if and only if there is an ffiid solution $F$ of $\\Pi$ such that\n$$\\P(R_\\mathcal{A}>r)\\le \\frac{1}{tower(Cr)}$$\nfor some $C>0$ and large enough $r\\in \\mathbb{N}$.\n\\end{theorem*}\n\nThe translation between ffiid and distributed algorithms allows to deduce the following result that answers problem (ii) of Holroyd at el.~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\n\n\\begin{theorem*}\nLet $d\\in \\mathbb{N}$ and $\\Pi$ be an LCL problem on $\\mathbb{Z}^d$.\nIf $\\Pi$ admits an ffiid solution $F$ that has faster tail decay than any tower function, then $\\Pi$ is trivial, i.e., there is a color such that a constant coloring with this color is a correct solution.\n\\end{theorem*}\n\nAs a last application of ideas from the distributed algorithms, we show the following speed-up theorem for ffiid.\n\n\\begin{theorem*}\nLet $d,t\\in \\mathbb{N}$.\nThere is $C_{d,t}>0$ such that if an LCL problem $\\Pi$ admits an ffiid solution $F$ such that\n$$\\P(R_\\mathcal{A}>r)\\le C_{d,t}r^{d-1},$$\nthen there is a deterministic algorithm in the local model that runs in $O(\\log^*(n))$ rounds and solves $\\Pi$.\n\\end{theorem*}\n\nTo understand better LCL problems that are global in the distributed algorithms or admit no continuous solution in the context of Borel combinatorics we introduce the following numerical invariant.\nThe inspiration comes from the paper of Holroyd et al.~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\n\n\\begin{definition*}\nLet $\\Pi$ be an LCL on $\\mathbb{Z}^d$.\nWe define ${\\bf M}(\\Pi)\\in [0,+\\infty]$ to be the supremum over all $\\alpha\\in [0,+\\infty)$ such that there is an ffiid solution $F$ of $\\Pi$ such that $R_F$ has finite the $\\alpha$-moment.\n\\end{definition*}\n\nCombination of previous results immediately implies the following strengthening of the upper bound on the necessary infinite moment of ffiid solution of the proper vertex $3$-coloring problem, $\\Pi^d_{\\chi,3}$, from Holroyd et al.~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\nMoreover, this answers their problem (iii).\n\n\\begin{theorem*}\nLet $d\\in \\mathbb{N}$ and $\\Pi$ be an LCL problem on $\\mathbb{Z}^d$.\nIf ${\\bf M}(\\Pi)>d-1$, then $\\Pi$ admits an ffiid solution with tower tail decay.\n\\end{theorem*}\n\nAs a first application of ideas from Borel combinatorics, i.e., the toast structure, we compute the optimal tail decay for $\\Pi^d_{\\chi,3}$, thus answering a problem (i) of Holroyd et al.~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\n\n\\begin{theorem*}\nLet $d>1$ and $\\Pi^d_{\\chi,3}$ be the proper vertex $3$-coloring problem.\nThen ${\\bf M}(\\Pi^d_{\\chi,3})=1$.\n\\end{theorem*}\n\nA high level explanation of a toast construction in the context of ffiid says that if $r\\in \\mathbb{N}$ is given, then the fraction of vertices not covered by first $r$-layers of the toast is at most $\\frac{1}{r}$.\nSimilar construction that uses only $1$-dimensional skeleton of some rectangular tiling of $\\mathbb{Z}^d$ improve this upper bound to $\\frac{1}{r^{d-1}}$.\nWe show that this type of structure is enough to find a perfect matching in a bigger graph then $\\mathbb{Z}^d$, a graph where we moreover add diagonal edges in every $2$-dimensional hyperplane.\nThis yields an existence of intermediate problems from the point of view of ${\\bf M}$.\n\n\\begin{theorem*}\nLet $2\\le \\ell\\le d\\in \\mathbb{N}$.\nThen there is an LCL problem $\\Pi$ on $\\mathbb{Z}^d$ that satisfies ${\\bf M}(\\Pi)=\\ell-1$.\n\\end{theorem*}\n\nAs a last result we summarize what is known about specific problems.\nSince we work mainly with oriented grids we state everything in this context, however, some of the results can be deduced for non-oriented grids as well, see Section~\\ref{sec:SpecificProblems}.\nIn particular, we answer Question~6.5 of Gao et al.~\\cite{GJKS} about Borel edge chroamtic number of $\\operatorname{Free}(2^{\\mathbb{Z}^d})$.\n\n\\begin{theorem*}\nLet $d\\ge 2$ and $\\Pi$ be any of the following LCL problems:\n\\begin{enumerate}\n \\item proper vertex $3$-coloring, \\cite{GJKS}, \\cite{spencer_personal},\n \\item proper edge $2d$-coloring,\n \\item rectangular tiling with rectangles from some set $\\mathcal{R}$ that satisfies \\cref{eq:GCD}, \\cite{Unger},\n \\item perfect matching, \\cite{GJKS}.\n\\end{enumerate}\nThen $\\Pi$ is in the class $\\TOAST$.\nConsequently, $\\Pi$ is in the class $\\mathsf{BOREL}$ and satisfies $1\\le {\\bf M}(\\Pi)$.\n\\end{theorem*}\n\n\n\nThe paper is organized as follows...\n\n\n\\end{comment}\n\n\n\n\n\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\n\nLet $d\\in \\mathbb{N}$ be fixed.\nThe class of graphs that we consider in this work contains mostly the infinite grid $\\mathbb{Z}^d$, that is $\\mathbb{Z}^d$ with the Cayley graph structure given by the standard set of generators \n$$S=\\{(1,0,\\dots, 0),(0,1,\\dots, 0), \\dots (0,\\dots,0,1)\\}. $$\nWe also used the finite variant of the graph: the torus of side length $\\Theta(n^{1\/d})$ with edges oriented and labeled by $S$.\nUsually, when we talk about a graph $G$ we mean one of the graphs above and it follows from the context whether we mean finite or infinite graph. \n\nGiven a graph $G$, we denote as $V(G)$ the set of vertices, as $E(G)$ the set of edges and as $d_G$ the \\emph{graph distance} on $G$, i.e., $d_G(v,w)$ is the minimal number of edges that connects $v$ and $w$.\nWe write $\\mathcal{B}_G(v,r)$ for the \\emph{ball of radius $r\\in \\mathbb{N}$} around $v\\in V(G)$. When $G$ is the infinite grid $\\mathbb{Z}^d$, then we leave the subscripts blank.\nA \\emph{rooted graph} is a pair $(G,v)$ where $G$ is a graph and $v\\in V(G)$.\nWe denote as $[G,v]$ the \\emph{isomorphism type} of the rooted graph $(G,v)$, where $(G,v)$ and $(G',v')$ are isomorphic if there is a bijection $\\varphi:V(G)\\to V(G')$ that sends $v$ to $v'$ and preserves edge orientation and labeling.\n\nLet $D$ be a set, $G$ a graph and $c:V(G)\\to D$ be a labeling of vertices.\nThen we write $G_c=(G,c)$ for the pair graph + labeling.\nSimilarly we denote as $(G_c,v)$ the rooted graph $(G,v)$ with the labeling $c$, $[G_c,v]$ its isomorphism class and $\\mathcal{B}_{G_c}(v,r)$ the labeled ball of radius $r\\in \\mathbb{N}$ around $v$, where $v\\in V(G)$.\nOften we do not mention the labels and tacitly assume that they are understood from the context.\nAlso, the label set might depend on the size of of the vertex set, e.g., unique labels from $[|V(G)|]=\\{1,\\dots,|V(G)|\\}$ etc.\n\n\\begin{comment}\n\nWe always consider the group $\\mathbb{Z}^d$ with the standard set of generators $S$, i.e.,\n$$S=\\{(1,0,\\dots, 0),(0,1,\\dots, 0), \\dots (0,\\dots,0,1)\\}.$$\nThe \\emph{Cayley graph} $\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right):=\\operatorname{Cay}\\left(\\mathbb{Z}^d,S\\right)$ of $\\mathbb{Z}^d$ is then defined on the vertex set $\\mathbb{Z}^d$ with oriented and $S$-labeled edges \n$$\\left\\{(g,s+g):s\\in S, g\\in \\mathbb{Z}^d\\right\\}=E\\left(\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)\\right).$$\nWe denote the identity in $\\mathbb{Z}^d$ as ${\\bf 0}_d$. \nWhen we talk about $\\mathbb{Z}^d$ next, we mean formally the appropriate Cayley graph with labelled edges, that is, we think of the graph together with its consistent orientation. \n\n\nA \\emph{graph} $G$ is a pair $(V(G),E(G))$, where $V(G)$ is a vertex set and $E(G)$ consists of oriented and $S$-labeled edges.\nGiven a graph $G$, we denote as $d_G$ the \\emph{graph distance} on $G$, i.e., $d_G(v,w)$ is the minimal number of edges that connects $v$ and $w$.\nWe write $\\mathcal{B}_G(v,r)$ for the \\emph{ball of radius $r\\in \\mathbb{N}$} around $v\\in V(G)$. Usually, $G$ is just the grid $\\mathbb{Z}^d$, then we leave the subscript blank. \nA \\emph{rooted graph} is a pair $(G,v)$ where $G$ is a graph and $v\\in V(G)$.\nWe denote as $[G,v]$ the isomorphism type of the rooted graph $(G,v)$, where $(G,v)$ and $(G',v')$ are isomorphic if there is a bijection $\\varphi:V(G)\\to V(G')$ that sends $v$ to $v'$ and preserves edge orientation and labeling.\n\n\\todo{vsechny strany toru maji delku n na 1\/d; zjevne vsechny problemy maji lokalitu nejvyse n na 1\/d}\n\n\\end{comment}\n\n\n\\begin{comment}\nWe write $\\operatorname{\\bf FG}^{S}$ for the set of all graphs that look in their $2$-neighborhoods as $\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)$, i.e., a graph $G=(V(G),E(G))$ is in $\\operatorname{\\bf FG}^{S}$ if $[\\mathcal{B}_G(v,2)]=\\left[\\mathcal{B}_{\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)}({\\bf 0}_d,2)\\right]$ for every $v\\in V(G)$.\nNote that $\\mathbb{Z}^d\\in \\operatorname{\\bf FG}^{S}$ and there are finite graphs in $\\operatorname{\\bf FG}^{S}$, as an example take any torus that is big enough.\nWrite $\\operatorname{\\bf FG}_{\\bullet}^{S}$ for the rooted graphs of the form\n$$(\\mathcal{B}_G(v,r),v)$$\nwhere $G\\in \\operatorname{\\bf FG}^{S}$, $v\\in V(G)$ and $r\\in \\mathbb{N}$.\nWe say that a rooted graph $(G,v)\\in \\operatorname{\\bf FG}_{\\bullet}^{S}$ has \\emph{diameter at most $m\\in \\mathbb{N}$} if $(\\mathcal{B}_G(v,m),v)=(G,v)$.\nNote that, in particular, we have $\\mathcal{B}_{\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)}({\\bf 0}_d,r)\\in \\operatorname{\\bf FG}_{\\bullet}^{S}$ for every $r\\in \\mathbb{N}$.\n\nLet $D$ be a set, $G$ a graph and $c:V(G)\\to D$ be a labeling of vertices.\nThen we write $G_c=(G,c)$ for the pair graph + labeling.\nSimilarly we denote as $(G_c,v)$ the rooted graph $(G,v)$ with the labeling $c$, $[G_c,v]$ its isomorphism class and $\\mathcal{B}_{G_c}(v,r)$ the labeled ball of radius $r\\in \\mathbb{N}$ around $v$, where $v\\in V(G)$.\nOften the label set $D$ depends on the size of of the vertex set, e.g., unique labels from $[|V(G)|]=\\{1,\\dots,|V(G)|\\}$ etc.\n\\end{comment}\n\n\n\n\\subsection{LCLs on $\\mathbb{Z}^d$}\n\nOur goal is to understand locally checkable problems (LCL) on $\\mathbb{Z}^d$. \n\n\\begin{definition}[Locally checkable problems \\cite{naorstockmeyer}]\nA \\emph{locally checkable problems (LCL)} on $\\mathbb{Z}^d$ is a quadruple $\\Pi=(\\Sigma_{in}, \\Sigma_{out},t,\\mathcal{P})$ where $\\Sigma_{in}$ and $\\Sigma_{out}$ are finite sets, $t\\in \\mathbb{N}$ and $\\mathcal{P}$ is an assignment\n$$[H_c,w]\\mapsto \\mathcal{P}([H_c,w])\\in \\{0,1\\},$$\nwhere $(H,w)$ is a rooted grid of radius at most $t$ together with an input and output coloring $c: V(H)\\to \\Sigma_{in}\\times\\Sigma_{out}$. \n \nGiven a graph $G$ together with a coloring $c:V(G)\\to \\Sigma_{in}\\times\\Sigma_{out}$, we say that $c$ is a \\emph{$\\Pi$-coloring} if\n$$\\mathcal{P}([\\mathcal{B}_{G_c}(v,t),v])=1$$\nfor every $v\\in V(G)$.\n\\end{definition}\n\n\n\n\\begin{comment}\n\nClasses of functions\n\nBefore we proceed to definition of local algorithms and their complexity we recall here the relevant notions concerning functions from $\\mathbb{N}$ to $\\mathbb{N}$. \nImportantly, we often use asymptotic notation as usual in Computer Science, e.g., we define the class of functions $O(f(n))$ as the union over all functions $g:\\mathbb{N} \\rightarrow \\mathbb{N}$ such that there exists $C > 0$ such that for all $n$ we have $g(n) \\le C f(n)$. Other symbols $\\Omega, \\Theta, o, \\omega$ are defined analogously and their usage is analogous (see e.g. \\cite{?}). \n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{LOCAL model}\n\\label{subsec:preliminaries_local}\n\nHere we formally introduce the $\\mathsf{LOCAL}$ model for both deterministic and randomized algorithms and state the classification of LCLs on $\\mathbb{Z}^d$ from the $\\mathsf{LOCAL}$ point of view.\nInformally, a local algorithm $\\mathcal{A}$ of radius $r\\in \\mathbb{N}$ is a rule that takes a labeled graph $G_c$ and then re-labels each vertex $v\\in V(G)$ depending only on the isomorphism type of its neighborhood $[\\mathcal{B}_{G_c}(v,r),v]$ with a new label from some finite set $b$.\n\n\\paragraph{Local algorithms}\n\nHere, we formally define the complexity classes $\\mathsf{LOCAL}(\\mathcal{F})$ of deterministic local algorithms, where $\\mathcal{F}$ is a class of functions from $\\mathbb{N}$ to $\\mathbb{N}$. \nThe somewhat unintuitive part of the definition is that the algorithm takes as input \\emph{unique identifiers}. Usually, they are taken from some range of values that is polynomial in $n$, the number of vertices of the input graph. However, in most scenarios this is equivalent to the assumption that they are from range $\\{1,\\dots n\\}$, i.e., every identifier is used exactly once. We also use this definition here, for simplicity.\nIn all forthcoming definition we have some fixed LCL $\\Pi$ in mind.\n\n\\begin{definition}[A deterministic local algorithm for $\\mathbb{Z}^d$]\n\\label{def:local_algorithm\nLet $\\Sigma_{out}$ be a finite set.\nA \\emph{local (deterministic) algorithm $\\mathcal{A}$} is a sequence of pairs $\\{\\mathcal{A}_n,\\ell_n\\}_{n\\in\\mathbb{N}}$, where $\\ell_n\\in \\mathbb{N}$ and $\\mathcal{A}_n$ is an assignment, such that\n$$[\\mathcal{B}_{G_\\ell}(v,\\ell_n),v]\\mapsto \\mathcal{A}_n([\\mathcal{B}_{G_\\ell}(v,\\ell_n),v])\\in \\Sigma_{out}$$\nfor every $\\Sigma_{in}$-labeled graph $G$ (torus) on $n$ vertices, $v\\in V(G)$ and a bijection $\\ell:V(G)\\to [n]$.\nIn another words, $\\mathcal{A}_n$ is a function that is defined on isomorphism classes of rooted graphs of diameter at most $\\ell_n$ with injectively labeled vertices from the set $\\{1,\\dots, n\\}$ and with range $b$. \n\nGiven such a $G$ and a bijection $\\ell:V(G)\\to |V(G)|$, we define\n$$\\mathcal{A}(G_\\ell)(v):=\\mathcal{A}_{|V(G)|}([\\mathcal{B}_{G_\\ell}(v,\\ell_n),v])$$\nfor every $v\\in V(G)$.\n\\end{definition}\n\nNext we define the deterministic complexity of a given local coloring problem $\\Pi$.\n\n\\begin{definition}[Complexity of deterministic local algorithm for $\\mathbb{Z}^d$]\nWe say that a local algorithm $\\mathcal{A}=\\{\\mathcal{A}_n,\\ell_n\\}_{n\\in \\mathbb{N}}$ \\emph{solves} an LCL $\\Pi=(\\Sigma_{in},\\Sigma_{out},t,\\mathcal{P})$ if $\\mathcal{A}(G_l)$ is a $\\Pi$-coloring of $G$ for every finite $\\Sigma_{in}$-labeled graph $G$ and any bijection $\\ell:V(G)\\to |V(G)|$.\n\nWe say that $\\Pi$ is in the class $\\mathsf{LOCAL}(\\mathcal{F})$, where $\\mathcal{F}$ is a class of functions, if $n\\mapsto \\ell_n$ is in $\\mathcal{F}$.\n\\end{definition}\n\nA randomized local algorithm and its complexity is defined analogously but instead of labeling with unique identifiers each node has an iid random string and the algorithm succeeds with probability $1 - 1\/n$.\n\nCrucial for our investigation are so-called \\emph{oblivious local algorithms}.\nRoughly speaking an algorithm is \\emph{oblivious} if it does not have knowledge of $n$ (see \\cite{Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring} for details). \nSince we do not consider deterministic oblivious algorithms, by an oblivious local algorithm we mean an oblivious randomized local algorithm.\n\n\\begin{definition}[Oblivious local algorithm]\n\\label{def:uniform_local}\nA \\emph{oblivious local algorithm} $\\mathcal{A}$ is a pair $(\\mathcal{C},f)$, where $\\mathcal{C}$ is a (measurable) collection of rooted graphs labeled with $\\Sigma_{in}$ and real numbers from $[0,1]$, and $f: \\mathcal{C} \\rightarrow \\Sigma_{out}$ is a measurable assignment.\n\nWe define $\\mathcal{A}(G)$ to be the labeling computed by $\\mathcal{A}$ on $G$, where for a given node $u$ we have $\\mathcal{A}(G)(u)=g([B_G(u, t), u])$ where $t$ is the smallest integer such that $[B_G(u, t),u] \\in \\mathcal{C}$.\n\\end{definition}\n\nNote that since there is no $n$ in the definition of oblivious local algorithm, the algorithm is correct if it always finishes with probability $1$, while randomized algorithms can have small probability of error dependent on $n$. Hence, one should think about such algorithms as an analogue of Las Vegas algorithms, while the classical randomized local algorithms correspond to Monte Carlo algorithms. \nAn advantage of not knowing $n$ is that it is possible to run $\\mathcal{A}$ on, e.g., the infinite grid $\\mathbb{Z}^d$ instead of just a torus. Usually, these two models give the same results but for toast algorithms that we consider later, there is a difference: While we define the algorithms for an infinite grid, for example the $3$-coloring algorithm makes sense also for a finite torus, while the perfect matching algorithm does not make sense in the finite world for the simple reason that the input graph can have an odd number of vertices. \n\n\n\n\n\n\n\n\n\\begin{comment}\n\\paragraph{Randomized local algorithms}\\todo{merge with previous paragraph}\n\nThe main difference between deterministic and randomized model is that the input of the local rule in the randomized case are graphs with vertices labeled uniformly independently at random by elements of the interval $[0,1]$ and we require that the output coloring is correct everywhere with high probability.\nThe complexity of local algorithm is then defined as the locality it needs.\n\n\\begin{definition}[Randomized local algorithm for $\\mathbb{Z}^d$]\nLet $b$ be a set.\nWe say that $\\mathbb{A}=\\{\\mathbb{A}_n,\\ell_n\\}_{n\\in\\mathbb{N}}$ is a randomized local algorithm for the class $\\operatorname{\\bf FG}^{S}$ if $\\ell_n\\in \\mathbb{N}$ and \n$$[\\mathcal{B}_{G_l}(v,\\ell_n),v]\\mapsto \\mathbb{A}_n([\\mathcal{B}_{G_l}(v,\\ell_n),v])\\in b$$\nis a measurable assignment, where $G\\in \\operatorname{\\bf FG}^{S}$, $|V(G)|=n$, $v\\in V(G)$ and $\\ell:V(G)\\to [0,1]$.\nIn another words, $\\mathbb{A}_n$ is a measurable function that is defined on isomorphism classes of rooted graphs of diameter at most $\\ell_n$ with vertices labeled by $[0,1]$ and with range $b$. \n\nGiven $G\\in \\operatorname{\\bf FG}^{S}$ such that $n=|V(G)|$ and a map $\\ell:V(G)\\to [0,1]$, we define\n$$\\mathbb{A}(G_l)(v):=\\mathbb{A}_n([\\mathcal{B}_{G_l}(v,\\ell_n),v])$$\nfor every $v\\in V(G)$.\n\\end{definition}\n\nNext we define when a randomized algorithm solves an LCL $\\Pi$ and the randomized complexity classes.\n\n\\begin{definition}[Complexity of randomized local algorithm for $\\mathbb{Z}^d$]\nLet $\\Pi=(b,t,\\mathcal{P})$ be an LCL.\nWe say that a randomized algorithm $\\mathbb{A}=\\{\\mathbb{A}_n,\\ell_n\\}_{n\\in \\mathbb{N}}$ solves $\\Pi$ if \n$$\\lambda^{\\oplus n}\\left(\\left\\{\\ell\\in [0,1]^{V(G)}:\\mathbb{A}(G_l) \\ \\operatorname{is \\ a} \\ \\Pi\\operatorname{-coloring} \\right\\}\\right)>1-\\frac{1}{n}$$\nholds for every $n\\in \\mathbb{N}$ and $G\\in \\operatorname{\\bf FG}^{S}$ such that $|V(G)|=n$ and\n$$[\\mathcal{B}_{G}(v,\\ell_n+t),v]=\\left[\\mathcal{B}({\\bf 0}_d,\\ell_n+t),{\\bf 0}_d\\right]$$\nfor every $v\\in V(G)$.\n\nWe say that $\\Pi$ is in the class $\\mathsf{R-LOCAL}(\\mathcal{F})$, where $\\mathcal{F}$ is a class of functions, if $n\\mapsto \\ell_n$ is in $\\mathcal{F}$.\n\\end{definition}\n\\end{comment}\n\n\\begin{comment}\n\\label{sec:clocal_definition}\n\nThe last class of local algorithm that we introduce serves on one hand as a main technical tool for deterministic speed up tricks and on the other hand allows a smooth translation of local algorithm into the measurable setting.\nIntuitively, an algorithm is in the class $\\mathsf{C-LOCAL}$ if it only needs a fixed finite locality (not depending on the size of a graph) given some coloring of the vertices with fixed number of colors (not depending on the size of a graph).\n\nLet $s\\in \\mathbb{N}$ and $G$ be a graph.\nDenote as $G^s$ the $s$-power graph of $G$, i.e., $V(G^s)=V(G)$ and $(x,y)\\in E(G^s)$ if and only if $0 r),$$\nwhat we mean is the following. \nFirst, for a given rooted neighborhood $(G,v)$ with a fixed $\\Sigma_{in}$-labeling we ask what is the probability that, after labeling $G$ with real numbers uniformly at random, that $(G,v)$ is in $\\mathcal{C}$. \nSecond, we take the supremum over all such values over all $\\Sigma_{in}$-labels.\n\n\\begin{definition}[$\\mathsf{TAIL}(f)$]\n\\label{def:tail}\nWe say that an LCL $\\Pi$ is in the class $\\mathsf{TAIL}(f)$ for some function $f:\\mathbb{N}\\to \\mathbb{N}$ if there is an oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi$ such that\n$$\\P(R_\\mathcal{A} > r)\\le \\frac{1}{f(r)}$$\nfor every large enough $r\\in \\mathbb{N}$. \n\nWe say that $\\Pi \\in \\mathsf{TAIL}$ if there exists $f \\in \\omega(1)$ such that $\\Pi \\in \\mathsf{TAIL}(f)$. \nWe say that $\\Pi$ is solved by a \\emph{block factor} if there is an oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi$ such that for some $r_0$ we have $\\P(R_\\mathcal{A} > r_0) = 0$.\n\\end{definition}\n\nHere things are made simple because we think of the algorithm as being run on an infinite graph so nothing depends on $n$ in this complexity measure (while it is a crucial parameter in the classical local complexity definition). \n\nWe also use the asymptotic notation when talking about $\\mathsf{TAIL}$ complexities. For example, an important class $\\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$ is defined as the class of LCLs $\\Pi$ such that there is a constant $C\\in \\mathbb{N}$ and oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi$ that satisfies\n$$\\P(R_\\mathcal{A}> r)\\le \\frac{1}{\\mathrm{tower}(Cr)}$$\nwhere $\\mathrm{tower}(x)=\\exp\\circ\\dots\\circ \\exp(1)$ is $\\lfloor x\\rfloor$-many times iterated exponential.\n\nA coarser way to measure the complexity of a problem $\\Pi$ is to ask what moments of $R_\\mathcal{A}$ could be finite, when $\\mathcal{A}$ is an oblivious local algorithm.\nFor example, the first moment finite corresponds to the fact that average time until an algorithm finishes is defined (for finite graphs this is independent on the size of the graph). \nIn particular, we investigate the question what is the least infinite moment for any oblivious local algorithm to solve a given problem $\\Pi$, as this language was used in \\cite{HolroydSchrammWilson2017FinitaryColoring}. \nRecall that $R_\\mathcal{A}$ has a \\emph{finite $\\alpha$-moment}, where $\\alpha\\in [0,+\\infty)$, if\n$$\\mathbb{E}\\left(\\left(R_\\mathcal{A}\\right)^\\alpha\\right)=\\sum_{r \\in \\mathbb{N}} r^\\alpha \\cdot \\P(R_\\mathcal{A} = r) <+\\infty.$$\n\n\\begin{definition}\nLet $\\Pi$ be an LCL.\nWe define ${\\bf M}(\\Pi)\\in [0,+\\infty]$ to be the supremum over all $\\alpha\\in [0,+\\infty)$ such that there is an oblivious $\\mathcal{A}$ that solves $\\Pi$ and has the $\\alpha$-moment of $R_\\mathcal{A}$ finite.\n\\end{definition}\n\nThe $\\alpha$ moment being finite roughly corresponds to $\\Omega(r^{\\alpha})$ $\\mathsf{TAIL}$ complexity. In fact we now recall the following observation.\n\n\\begin{fact}\n\\label{fact:moment_implies_tail}\nSuppose that the coding radius of an oblivious local algorithm $\\mathcal{A}$ has the $\\alpha$-moment finite.\nThen for every $C>0$ we have\n$$\\P(R_\\mathcal{A} > r) \\le C\\frac{1}{r^\\alpha}$$\nfor large enough $r\\in \\mathbb{N}$.\n\\end{fact}\n\\begin{proof}\nWe prove the contrapositive. Suppose the tail of the coding radius $R$ satisfies $\\P(R > r_i) > C \\cdot \\frac{1}{r^\\alpha_i}$ for some $C > 0$ and an infinite sequence $(r_i)_{i\\in \\mathbb{N}}$. \nAs the sequence $\\P(R> r_i)$ goes to zero, we may pass to a subsequence such that for each $i\\in \\mathbb{N}$ we have $\\P(R > r_{i+1}) \\le \\frac{1}{2}\\P(R > r_i)$. We call the new subsequence still $(r_i)_{i\\in \\mathbb{N}}$. \nWe have \n\\begin{align*}\n\\mathbb{E}(R^\\alpha) &\\ge \\sum_{i \\in \\mathbb{N}} r_i^\\alpha \\cdot (\\P(R> r_i) - \\P(R > r_{i+1}))\\\\\n&\\ge \\sum_{i \\in \\mathbb{N}} r_i^\\alpha \\cdot \n\\frac{1}{2}\\P(R > r_i)\n\\ge \\sum_{i \\in \\mathbb{N}} r_i^\\alpha \\cdot \n\\frac{C}{2} \\cdot \\frac{1}{r_i^\\alpha}\n= +\\infty\n\\end{align*}\nwhich finishes the proof. \n\\end{proof}\n\nFinally, an important fact is that a small $\\mathsf{TAIL}$ complexity of an oblivious randomized local algorithm implies that it has small randomized complexity (on finite graphs). This is because if we run an oblivious algorithm with tail decay $1\/f(r)$ on an $n$-node graph and mark its run as a failure if the coding radius exceeds $f^{-1}(n^2)$, the failure at some node happens only with probability $1\/n$. This was formally proven in \\cite{brandt_grebik_grunau_rozhon2021classification_of_lcls_trees_and_grids} and here we only state the result. \n\n\\begin{lemma}\n\\label{lem:tail->local}\nLet $\\mathcal{A}$ be an oblivious local randomized algorithm with $\\mathsf{TAIL}$ complexity $f(r)$. Then the randomized complexity of $\\mathcal{A}$ is $O(f^{-1}(n^2))$. \n\\end{lemma}\n\nIn fact, one can prove that for $o(\\log n)$ local complexities the two notions are the same. We refer to the reader to the discussion in \\cite{brandt_grebik_grunau_rozhon2021classification_of_lcls_trees_and_grids} for more details. \n\n\\begin{theorem}[\\cite{brandt_grebik_grunau_rozhon2021classification_of_lcls_trees_and_grids}]\n\\label{thm:tail=uniform}\nLet $\\mathcal{A}$ be an oblivious local randomized algorithm with randomized complexity $f(n) = o(\\log n)$. Then this algorithm in fact never returns an incorrect solution and its tail complexity is $\\Omega(f^{-1}(r))$. \n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\\subsection{Subshifts of finite type}\n\\label{subsec:subshifts}\n\nHere we connect our definition of LCLs and oblivious local algorithms with the well-studied notions of subshifts of finite type and finitary factors of iid labelings.\n\nTo define subshifts of finite type we restrict ourselves to LCLs without input.\nThat is $\\Sigma_{in}$ is empty or contains only one element.\nMoreover, for convenience we set $b:=\\Sigma_{out}$.\nThen an LCL $\\Pi$ is of the form $\\Pi=(b,t,\\mathcal{P})$.\nWe define $X_\\Pi\\subseteq b^{\\mathbb{Z}^d}$ to be the space of all $\\Pi$-colorings.\nThat is we consider all possible labelings of $\\mathbb{Z}^d$ by elements from $b$ and then include in $X_{\\Pi}$ only those that satisfies the finite constraints $\\mathcal{P}$.\nViewing $\\mathbb{Z}^d$ as a group, there is a natural shift action of $\\mathbb{Z}^d$ on (the Cayley graph) $\\mathbb{Z}^d$.\nNamely, $g\\in \\mathbb{Z}^d$ shifts $h$ to $g+h$.\nThis action naturally ascend to an action $\\cdot $ on $b^{\\mathbb{Z}^d}$ as\n$$(g\\cdot x)(h)=x(g+h)$$\nwhere $x\\in b^{\\mathbb{Z}^d}$.\nIt is not hard to see (since $\\mathcal{P}$ is defined on isomorphism types and not rooted in any particular vertex) that $X_\\Pi$ is invariant under the shift map.\nThen we say that $X_\\Pi$ together with the shift structure is a \\emph{subshift of finite type}.\n\nA \\emph{finitary factor of iid process (ffiid) $F$} is a function that takes as an input $\\mathbb{Z}^d$ labeled with real numbers from $[0,1]$, independently according to Lebesgue measure, and outputs a different labeling of $\\mathbb{Z}^d$, say with elements of some set $b$. \nThis labeling satisfies the following: (a) $F$ is equivariant, that is for every $g\\in \\mathbb{Z}^d$ shifting the real labels with $g$ and applying $F$ is the same as applying $F$ and then shifting with $g$, (b) the value of $F(x)({\\bf 0})$ depends almost surely on the real labels on some random but finite neighborhood of ${\\bf 0}$.\nSimilarly as for oblivious local algorithms one defines the random variable $R_F$ that encodes minimal such neighborhood.\nWe say that ffiid $F$ solves $\\Pi$ if $F(x)\\in X_{\\Pi}$ with probability $1$.\n\nThe difference between ffiid and oblivious local algorithms is just formal.\nNamely, given an oblivious local algorithm $\\mathcal{A}$, we can turn it into an ffiid $F_\\mathcal{A}$ simply by running it simultaneously in every vertex of $\\mathbb{Z}^d$.\nWith probability $1$ this produces a labeling of $\\mathbb{Z}^d$ and it is easy to see that both conditions from the definition of ffiid are satisfied.\nOn the other hand, if $F$ is ffiid, then we collect the finite neighborhoods that determine the value at ${\\bf 0}$ together with the decision about the label.\nThis defines a pair $\\mathcal{A}_F=(\\mathcal{C},f)$ that satisfies the definition of an oblivious local algorithm.\nIt follows that the coding radius $R_\\mathcal{A}$, resp $R_F$, is the same after the translations.\nWe chose to work with the language of distributed algorithms as we borrow the techniques mainly from that area. \n\n\\begin{remark}\nIf in (b), in the definition of ffiid, we only demand that the value at ${\\bf 0}$ depends measurably on the labeling, i.e., $F$ is allowed to see the labels at the whole grid, then we say that $F$ is a \\emph{factor of iid labels (fiid)}.\n\nGiven an LCL $\\Pi$, we say that $\\Pi$ is in the class $\\mathsf{fiid}$, if there is a fiid $F$ that produces element of $X_\\Pi$ almost surely.\nIt is easy to see that $\\mathsf{TAIL}\\subseteq \\mathsf{fiid}$.\nWe remark that in the setting without inputs we have $\\mathsf{TAIL}=\\mathsf{fiid}$ on $\\mathbb{Z}$ and it is unknown, however, still plausible, whether this holds for $d>1$.\nFor general LCLs with inputs (where the definition of $\\mathsf{fiid}$ is extended appropriately) we have $\\mathsf{TAIL}\\not=\\mathsf{fiid}$ even for $d=1$, see \\cite{grebik_rozhon2021LCL_on_paths}.\n\\end{remark}\n\n\\begin{comment}\nGiven an LCL $\\Pi=(b,t,\\mathcal{P})$, we define $X_\\Pi\\subseteq b^{\\mathbb{Z}^d}$ to be the space of all $\\Pi$-colorings.\nWe say that an LCL $\\Pi$ is \\emph{satisfiable} if $\\emptyset\\not=X_\\Pi$, i.e., if there is a $\\Pi$-coloring of $\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)$.\nWe say that an LCL $\\Pi$ is \\emph{trivial} if there is $\\alpha\\in b$ such that the constant function $\\alpha$ is a $\\Pi$-coloring.\n\nThere is a canonical (right-)shift action $\\cdot$ of $\\mathbb{Z}^d$ on $b^{\\mathbb{Z}^d}$ (and in general on any $X^{\\mathbb{Z}^d}$, where $X$ is a set) that is defined as\n$$(g\\cdot x)(h)=x(h + g)$$\nfor every $g,h\\in \\mathbb{Z}^d$ and $x\\in b^{\\mathbb{Z}^d}$.\nIt is easy to see that $X_\\Pi$ is invariant under this action, i.e., $x\\in X_\\Pi$ implies $g\\cdot x\\in X_\\Pi$ for every $g\\in \\mathbb{Z}^d$.\nMoreover, since $b$ is a finite set, the product topology turns $b^\\Gamma$ into a compact metric space and it is easy to see $X_\\Pi$ is closed subspace.\nThe set $X_\\Pi$ together with the shift action is called a \\emph{sub-shift of finite type}.\n\nSuppose that $\\mathbb{Z}^d$ has an action $\\star_Y$ and $\\star_Z$ on spaces $Y$ and $Z$, respectively. \nWe say that a function $f:Y\\to Z$ is \\emph{equivariant} if $f(g\\star_Y x)=g\\star_Z f(x)$ for every $g\\in \\mathbb{Z}^d$ and $x\\in Y$.\n\nGiven an LCL $\\Pi$, our aim is to study complexity of finitary factors of iid that produce elements in $X_\\Pi$ almost surely.\nWe say that $\\Pi$ admits a \\emph{fiid solution} if there is a measurable equivariant map $F:[0,1]^{\\mathbb{Z}^d}\\to X_\\Pi$ that is defined on a $\\mu$-conull set, where $\\mu$ is the product measure on $[0,1]^{\\mathbb{Z}^d}$.\n\nWrite $\\varphi_F:[0,1]^{\\mathbb{Z}^d}\\to b$ for the composition of $F$ with the projection to ${\\bf 0}_d$-coordinate, i.e., \n$$x\\mapsto \\varphi_F(x)=F(x)({\\bf 0}_d).$$\nWe say that $F$ is a \\emph{finitary fiid (ffiid)} if for $\\mu$-almost every $x\\in [0,1]^{\\mathbb{Z}^d}$ the value of $\\varphi_F$ depends only on values of $x$ on some finite neighborhood of ${\\bf 0}_d$.\nThat is to say, there is a random variable $R_F$ that takes values in $\\mathbb{N}$, is defined $\\mu$-almost everywhere and satisfies for $\\mu$-almost every $x\\in [0,1]^{\\mathbb{Z}^d}$ the following: if $y\\in [0,1]^{\\mathbb{Z}^d}$ is such that\n$$x\\upharpoonright \\mathcal{B}_{\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)}\\left({\\bf 0}_d,R_\\varphi(x)\\right)=y\\upharpoonright \\mathcal{B}\\left({\\bf 0}_d,R_\\varphi(x)\\right),$$\nthen $\\varphi_F(x)=\\varphi_F(y)$.\n\\todo{nejak to zakoncit tim ze to nebudeme potrebovat}\n\\end{comment}\n\n\n\n\n\\begin{comment}\n\nWarm-up: Equivalence of $\\mathsf{LOCAL}(\\log^* n)$, $\\mathsf{TAIL}(\\mathrm{tower})$, $\\mathsf{CONTINUOUS}$\n\nRecall that $\\mathsf{LOCAL}(O(\\log^* n))$ denotes the class of satisfiable LCLs that satisfy (1) or (2) in Theorem~\\ref{thm:classification_of_local_problems_on_grids}.\nIn this section we show that every problem from $\\mathsf{LOCAL}(O(\\log^* n))$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\nIn the proof we use the fact that $\\mathsf{LOCAL}=\\mathsf{C-LOCAL}(O(1))$, see~Theorem~\\ref{thm:classification_of_local_problems_on_grids}.\n\nWe let $\\Delta_{s}:=|\\mathcal{B}({\\bf 0}_d,s)|$, where $s\\in \\mathbb{N}$.\nDenote as $\\mathcal{M}_s$ the subspace of $(\\Delta_s+1)^{\\mathbb{Z}^d}$ that consists of all proper vertex $(\\Delta_s+1)$-colorings of the $s$-power graph of $\\mathbb{Z}^d$.\nIt is easy to see that $\\mathcal{M}_s$ is nonempty, $\\mathbb{Z}^d$-invariant and compact.\nIn fact it is a subshift of finite type.\nThe following result holds for any graph of bounded degree, as was proved in \\cite{HolroydSchrammWilson2017FinitaryColoring}.\n\n\\begin{theorem}\\cite[Theorem~6]{HolroydSchrammWilson2017FinitaryColoring}\\label{th:ColorTower}\nLet $s\\in\\mathbb{N}$.\nThen there is an ffiid $F:[0,1]^{\\mathbb{Z}^d}\\to \\mathcal{M}_s$ and a constant $C_s>0$ such that\n$$\\mu(R_F>r)<\\frac{1}{tower(C_sr)}.$$\nIn particular, the LCL of coloring the $s$-power graph of $\\mathbb{Z}^d$ with $(\\Delta_s+1)$-many colors is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\n\\end{theorem}\n\n\\begin{proposition}\\label{pr:LOCALandCOLORING}\nLet $\\Pi=(b,t,\\mathcal{P})$ be a satisfiable LCL problem that is in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\nThen there is a continuous equivariant function $\\varphi:\\mathcal{M}_s\\to \\mathcal{C}_{\\Pi}$ for some $s\\in \\mathbb{N}$.\n\\end{proposition}\n\\begin{proof}\nSuppose that $\\Pi$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\nBy Theorem~\\ref{thm:classification_of_local_problems_on_grids}, we have that $\\Pi$ is in the class $\\mathsf{C-LOCAL}(O(1))$.\nFix $r,\\ell\\in \\mathbb{N}$ and $\\mathcal{A}$ that witnesses this fact.\nLet $x\\in \\mathcal{M}_{r+t}$.\nThen define\n$$\\varphi(x)(h):=\\mathcal{A}([\\mathcal{B}_{\\mathbb{Z}^d_x}(h,\\ell),h]).$$\nIt is clear that $\\varphi$ is well defined, because $\\ellr)<\\frac{\\Delta_{\\ell}}{tower(C_s(r-\\ell))}\\le \\frac{1}{tower(Dr)}$$\nfor $r\\in \\mathbb{N}$ large enough and some constant $D\\in (0,+\\infty)$.\n\\end{proof}\n\n\n\n\n\ngives that\n\n\n\n\n\n\n\n\n\n$\\mathsf{TAIL}(\\tow(\\Omega(r)))$ $\\Rightarrow$ $\\mathsf{LOCAL}(O(\\log^* n))$\n\n\nFirst we show how to get randomized local algorithm from ffiid in a qualitative way.\nWe note that similar results holds for any bounded degree graph.\n\n\\begin{theorem}\\label{th:TailToRandom}\nLet $\\Pi$ be an LCL problem that is in the class $\\mathsf{TAIL}(f)$ for some $f:\\mathbb{N}\\to \\mathbb{N}$.\nSet $g(n)=\\min\\{m\\in \\mathbb{N}:f(m)\\ge n^2\\}$ and suppose that $g\\in o(n^{\\frac{1}{d}})$.\nThen $\\Pi$ is in the class $\\mathsf{R-LOCAL}(O(g(n)))$.\n\\end{theorem}\n\\begin{proof}\nFix an ffiid $F:[0,1]^{\\mathbb{Z}^d}\\to X_\\Pi$ such that\n$$\\mu(R_F>r)\\le \\frac{1}{f(r)}$$\nfor large enough $r\\in \\mathbb{N}$.\nDenote as $\\varphi_F$ the finitary rule and as $\\operatorname{dom}(\\varphi_F)$ its domain.\nThat is $\\alpha\\in \\operatorname{dom}(\\varphi_F)$ if $\\alpha\\in [0,1]^{V(\\mathcal{B}({\\bf 0}_d,s))}$ for some $s\\in \\mathbb{N}$ and satisfies\n$$F(x)=F(y)$$\nfor every extensions $\\alpha\\sqsubseteq x,y\\in [0,1]^{\\mathbb{Z}^d}$.\nIn that case we set $\\varphi_F(\\alpha)=F(x)$ for any such extension $\\alpha\\sqsubseteq x$.\n\nDefine $\\ell_n:=g(n)$.\nLet $(G,v)\\in \\operatorname{\\bf FG}_{\\bullet}^{S}$ be of a radius at most $\\ell_n$.\nConsider any map $\\ell:V(G)\\to [0,1]$.\nDefine\n$$\\mathbb{A}_n([G_l,v])=\\varphi_F(\\ell)$$\nin the case when $\\ell\\in \\operatorname{dom}(\\varphi_F)$.\nOtherwise let $\\mathbb{A}_n([G_l,v])$ to be any element in $b$.\nIt is easy to verify that this defines a randomized local algorithm.\n\nLet $G\\in \\operatorname{\\bf FG}^{S}$ be such that $|V(G)|=n$ and\n$$[\\mathcal{B}_{G}(v,\\ell_n+t),v]=\\left[\\mathcal{B}({\\bf 0}_d,\\ell_n+t),{\\bf 0}_d\\right]$$\nholds for every $v\\in V(G)$.\nNote that by the assumption on $g$ such graphs do exist, e.g., large enough torus.\nFor every $v\\in V(G)$ define the event\n$$A_v=\\left\\{\\ell\\in [0,1]^{V(G)}:\\ell\\upharpoonright \\mathcal{B}_G(v,\\ell_n)\\not \\in \\operatorname{dom}(\\varphi_F)\\right\\}.$$\nNote that if $\\ell\\not \\in \\bigcup_{v\\in V(G)} A_v$, then $\\mathbb{A}(G_l)$ is a $\\Pi$-coloring.\nTo see this pick $v\\in V(G)$ and any extension $x\\in [0,1]^{\\mathbb{Z}^d}$ of $\\ell\\upharpoonright \\mathcal{B}_G(v,\\ell_n+t)$ that is in the domain of $F$, where we interpret $v$ as the origin ${\\bf 0}_d$.\nThen $F(x)\\in X_\\Pi$ and\n$$F(x)\\upharpoonright \\mathcal{B}({\\bf 0}_d,t)\\simeq\\mathbb{A}(G_l)\\upharpoonright \\mathcal{B}_G(v,t).$$\nConsequently, we have $\\mathcal{P}([\\mathcal{B}_{G_{\\mathbb{A}(G_l)}}(v,t),v])=1$.\n\nLet $v\\in V(G)$.\nWe have $\\ell\\in A_v$ if and only if $R_F(x)>\\ell_n$ for every $x\\in [0,1]^\\Gamma$ that extends $\\ell$, i.e., every $x\\in [0,1]^{\\mathbb{Z}^d}$ that satisfies\n$$[B_{G_l}(v,\\ell_n),v]=[B_{\\mathbb{Z}^d_x}({\\bf 0}_d,\\ell_n),{\\bf 0}_d].$$ \nThis implies that\n$$\\lambda^{\\oplus n}(A_v)=\\mu\\left(\\left\\{x\\in [0,1]^{\\mathbb{Z}^d}:R_F(x)>\\ell_n\\right\\}\\right)\\le \\frac{1}{f(\\ell_n)}\\le \\frac{1}{n^2}.$$\nA union bound then shows that $\\lambda^{\\oplus n}\\left(\\bigcup_{v\\in V(G)}A_v\\right)\\le \\frac{1}{n}$ and the proof is finished.\n\\end{proof}\n\n\n\n\\begin{corollary}\\label{cor:tower=local}\n$\\mathsf{TAIL}(\\tow(\\Omega(r)))=\\mathsf{LOCAL}(O(\\log^* n))$.\n\\end{corollary}\n\\begin{proof}\nIt follows from Corollary~\\ref{cor:locallocal} itself has the following implications: \n\n\\begin{corollary}\n\\label{cor:block_factor}\n$\\Pi$ can be solved by a block factor (recall \\cref{def:tail}) if and only if $\\Pi \\in \\mathsf{LOCAL}(O(1))$. \n\\end{corollary}\n\\begin{proof}\nThis follows from \\cref{thm:classification_of_local_problems_on_grids,rem:chang_pettie_decomposition}. \n\\end{proof}\n\n\\begin{corollary}\n\\label{cor:BelowTOWERtrivial}\nIf $\\Pi \\in \\mathsf{TAIL}(\\mathrm{tower}(\\omega(r)))$, then $\\Pi \\in \\mathsf{LOCAL}(O(1))$ and if $\\Sigma_{in} = \\emptyset$ then $\\Pi$ is trivial. \n\\end{corollary}\n\\begin{proof}\n\\cref{lem:tail->local} gives that $\\Pi \\in\\mathsf{LOCAL}(o(\\log^* n))$ and \\cref{thm:classification_of_local_problems_on_grids} then implies the rest. \n\\end{proof}\n\nIn particular, Corollary~\\ref{cor:BelowTOWERtrivial} answers in negative Problem~(ii) in \\cite{HolroydSchrammWilson2017FinitaryColoring} that asked about the existence of a nontrivial local problem (without input labels) with super-tower function tail. \n\nThe second corollary is the following:\n\n\\begin{corollary}\n\\label{cor:local=tower}\n$\\Pi \\in \\mathsf{TAIL}(\\tow(\\Omega(r)))$ if and only if $\\Pi \\in \\mathsf{LOCAL}(O(\\log^* n))$. \n\\end{corollary}\n\\begin{proof}\nIf $\\Pi \\in \\mathsf{TAIL}(\\tow(\\Omega(r)))$, \\cref{lem:tail->local} yields $\\Pi \\in \\mathsf{RLOCAL}(O(\\log^* n))$. The other direction (special case of \\cref{thm:tail=uniform}) follows from \\cref{rem:chang_pettie_decomposition} that essentially states that it suffices to solve the independent set problem in the power graph. But this follows from a version of Linial's algorithm from \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n\\end{proof}\n\nFinally, we get the following. \n\\begin{corollary}\n\\label{cor:Below_d_local}\nIf $\\Pi \\in \\mathsf{TAIL}(\\omega(r^{2d}))$, then $\\Pi \\in \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$. \n\\end{corollary}\n\\begin{proof}\n\\cref{lem:tail->local} gives that $\\Pi \\in \\mathsf{LOCAL}(o(n^{1\/d}))$ and \\cref{thm:classification_of_local_problems_on_grids} then implies that $\\Pi \\in \\mathsf{LOCAL}(O(\\log^* n))$ which is equivalent to $\\Pi \\in \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$ by \\cref{cor:local=tower}. \n\\end{proof}\n\nCorollary~\\ref{cor:Below_d_local} answers in negative Problem~(iii) in \\cite{HolroydSchrammWilson2017FinitaryColoring} that asked about the existence of a problem with all its moments finite but with sub-tower function decay. Recall that \\cref{fact:moment_implies_tail} implies that if the $2d$-th moment is finite, then the tail is $\\omega(r^{2d})$ and, hence, the problem can in fact be solved with a tower tail decay. \nHowever, the $2d$ is not the right dependency and in the next section we prove a tight version of this result.\n\\subsection{Tight Grid Speedup for $\\mathsf{TAIL}$}\n\\label{subsec:tail_d-1_speedup}\n\nWhile \\cref{cor:Below_d_local} already answers the Problem of \\cite{HolroydSchrammWilson2017FinitaryColoring}, we now improve the speedup from the tail of $\\omega(r^{d-1})$. This result is tight (up to $o(1)$ in the exponent, see the discussion in \\cref{subsec:toast_construction} and \\cref{cor:toast_lower_bound}) and it is the first nontrivial difference between the $\\mathsf{TAIL}$ and $\\mathsf{LOCAL}$ model that we encountered so far. \n\n\n\n\n\\begin{theorem}\n\\label{thm:grid_speedup}\nLet $d,t\\in \\mathbb{N}$.\nFor $d \\ge 2$ there is $C_{d,t}>0$ such that if an LCL $\\Pi$ is in $\\mathsf{TAIL}(C_{d,t}r^{d-1})$, then $\\Pi \\in \\mathsf{TAIL}(\\tow(\\Omega(r)))$.\n\nSimilarly, for $d=1$ we have that a local coloring problem $\\Pi$ admits a solution by an oblivious random algorithm (it is in $\\mathsf{TAIL}(f(r))$ for any function $f = \\omega(1)$) if and only if $\\Pi$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\n\\end{theorem}\nNote that \\cref{fact:moment_implies_tail} hence shows that ${\\bf M}(\\Pi) > d-1$ already implies that $\\Pi$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$. \n\nThe proof is a simple adaptation of Lov\u00e1sz local lemma speedup results such as \\cite{chang2017time_hierarchy,fischer2017sublogarithmic}. \nThe proof goes as follows. Split the grid into boxes of side length roughly $r_0$ for some constant $r_0$. This can be done with $O(\\log^* (n))$ local complexity (or, equivalently, with tower tail decay). \n\nNext, consider a box $B$ of side length $r_0$ and volume $r_0^d$. In classical speedup argument we would like to argue that we can deterministically set the randomness in nodes of $B$ and its immediate neighborhood so that we can simulate the original oblivious local algorithm $\\mathcal{A}$ with that randomness and it finishes in at most $r_0$ steps. \nIn fact, ``all nodes of $B$ finish after at most $r_0$ steps'' is an event that happens with constant probability if the tail decay of $\\mathcal{A}$ is $r^d$, since then each node fails with probability $1\/r_0^d$ and there are $r_0^d$ nodes in $B$. \n\nHowever, we observe that it is enough to consider only the boundary of $B$ with $O(r_0^{d-1})$ nodes and argue as above only for the boundary nodes, hence the speedup from $r^{d-1}$ tail decay. \nThe reason for this is that although the coding radius of $\\mathcal{A}$ inside $B$ may be very large, just the fact that there \\emph{exists} a way of filling in the solution to the inside of $B$, given the solution on its boundary, is enough for our purposes. The new algorithm simply simulates $\\mathcal{A}$ only on the boundaries and then fills in the rest. \n\n\\begin{proof}\n\n\nWe will prove the result for $d \\ge 2$, the case $d=1$ is completely analogous and, in fact, simpler. \nSet $C=C_{d,t}:=d\\cdot t\\cdot(2\\Delta)^{2\\Delta}$, where $\\Delta=20^d$.\nSuppose that there is an oblivious local algorithm $\\mathcal{A}$ that explores the neighborhood of a given vertex $u$ and outputs a color of $u$ after seeing a radius $R$ such that \\begin{equation}\\label{eq:tail}\n\\P(R > r)\\le \\frac{1}{C r^{d-1}}\n\\end{equation}\nfor $r\\in \\mathbb{N}$ large enough.\nLet $r_0\\in \\mathbb{N}$ be large enough such that \\eqref{eq:tail} is satisfied.\nWrite $\\Box_{r_0}$ for the local coloring problem of tiling the input graph (which is either a large enough torus or an infinite grid $\\mathbb{Z}^d$) into boxes with length sizes in the set $\\{r_0,r_0+1\\}$ such that, additionally, each box $B$ is colored with a color from $[\\Delta+1]$ which is unique among all boxes $B'$ with distance at most $3(r_0+t)$. \n\nThe problem of tiling with boxes of side lengths in $\\{r_0, r_0+1\\}$ can be solved in $O(\\log^* n)$ local complexity \\cite[Theorem~3.1]{GaoJackson} but we only sketch the reason here: Find a maximal independent set of a large enough power of the input graph $\\mathbb{Z}^d$. Then, each node $u$ in the MIS collects a Voronoi cell of all non-MIS nodes such that $u$ is their closest MIS node. As the shape of the Voronoi cell of $u$ depends only on other MIS vertices in a constant distance from $u$, and their number can be bounded as a function of $d$ (see the ``volume'' argument given next), one can discretize each Voronoi cell into a number of large enough rectangular boxes. Finally, every rectangular box with all side lengths of length at least $r^2$ can be decomposed into rectangular boxes with all side lengths either $r$ or $r+1$. \n\nNote that in additional $O(\\log^* n)$ local steps, the boxes can then be colored with $(\\Delta+1)$-many colors in such a way that boxes with distance at most $3(r_0+t)$ must have different colors.\nTo see this, we need to argue that there are at most $\\Delta$ many boxes at distance at most $3(r_0+t)$ from a given box $B$.\nThis holds because if $B,B'$ are two boxes and $d(B,B')\\le 3(r_0+t)$, then $B'$ is contained in the cube centered around the center of $B$ of side length $10(r_0+t)$.\nThen a volume argument shows that there are at most $\\Delta\\ge \\frac{(10(r_0+t))^d}{r_0^d}$ many such boxes $B'$ (when $B$ is fixed).\n\nWe show now that there is a local algorithm $\\mathcal{A'}$ that solves $\\Pi$ when given the input graph $G$ together with a solution of $\\Box_{r_0}$ as an input.\nThis clearly implies that $\\Pi$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\n\n\n\nLet $\\{B_i\\}_{i\\in I}$ be the input solution of $\\Box_{r_0}$, together with the coloring $B_i\\mapsto c(B_i)\\in [\\Delta+1]$.\nWrite $V(B)$ for the vertices of $B$ and $\\partial B$ for the vertices of $B$ that have distance at most $t$ from the boundary of $B$.\nWe have $|\\partial B|< d\\cdot t \\cdot (r_0+1)^{d-1}$.\nDefine $N(B)$ to be the set of rectangles of distance at most $(r_0+t)$ from $B$.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width= .9\\textwidth]{fig\/tail_LLL.eps}\n \\caption{The picture shows one block $B$ with a red boundary $\\partial B$ of length $t$. Boxes in $N(B)$, that is with distance at most $(r_0+t)$ from $B$ are dark grey. }\n \\label{fig:tail_LLL}\n\\end{figure}\n\nWe are now going to describe how $\\mathcal{A}'$ iterates over the boxes in the order of their colors and for each node $u$ in the box $B$ it carefully fixes a random string $\\mathfrak{r}(u)$. \nThis is done in such a way that after fixing the randomness in all nodes, we can apply the original algorithm $\\mathcal{A}$ to each vertex in $\\partial B$ for each $B$ with the randomness given by $\\mathfrak{r}$ and the local algorithm at $u \\in \\partial B$ will not need to see more than its $r$-hop neighborhood. \n\nGiven a partial function $\\mathfrak{r_\\ell}:V(G)\\to [0,1]$ corresponding to nodes such that their randomness was already fixed, every box $B$ can compute a probability $\\mathbb{P}(\\fE_{\\ell}(B))$, where $\\fE_{\\ell}(B)$ is the (bad) event that there is a vertex in $\\partial B$ that needs to look further than $r_0$ to apply the local algorithm $\\mathcal{A}$.\nNote that for computing $\\mathbb{P}(\\fE_{\\ell}(B))$ it is enough to know the $r_0$-neighborhood of $B$.\n\nThe algorithm $\\mathcal{A}$ has $\\Delta+1$ rounds.\nIn the $0$-th round, we set $\\mathfrak{r}_0$ to be the empty function.\nNote that the probability that a vertex needs to look further than $r_0$ in the algorithm $\\mathcal{A}$ is smaller than $\\frac{1}{C r_0^{d-1}}$ by \\eqref{eq:tail}.\nThe union bound gives $\\mathbb{P}(\\fE_0(B))<\\frac{1}{(2\\Delta)^{2\\Delta}}$ for every $B$.\n\nIn each step $\\ell\\in [\\Delta+1]$ we define $\\mathfrak{r}_B:V(B)\\to [0,1]$ for every $B$ such that $c(B)=\\ell$ and put $\\mathfrak{r}_l$ to be the union of all $\\mathfrak{r}_D$ that have been defined so far, i.e., for $D$ such that $c(D)\\le \\ell$.\nWe make sure that the following is satisfied for every $B$:\n\\begin{enumerate}\n\t\\item [(a)] the assignment $\\mathfrak{r}_B$ depends only on $2(r_0+t)$-neighborhood of $B$ (together with previously defined $\\mathfrak{r}_{\\ell-1}$ on that neighborhood) whenever $c(B)=\\ell$,\n\t\\item [(b)] $\\mathbb{P}(\\fE_l(B))<\\frac{1}{(2\\Delta)^{2\\Delta-\\ell}}$,\n\t\\item [(c)] restriction of $\\mathfrak{r}_l$ to $N(B)$ can be extend to the whole $\\mathbb{Z}^d$ such that $\\mathcal{A}$ is defined at every vertex, where we view $N(B)$ embedded into $\\mathbb{Z}^d$.\n\\end{enumerate}\nWe show that this gives the required algorithm.\nIt is easy to see that by (a) this defines a local construction that runs in constantly many rounds and produces a function $\\mathfrak{r}:=\\mathfrak{r}_{\\Delta+1}:V(G)\\to [0,1]$.\nBy (b) every vertex in $\\partial B$ for some $B$ looks into its $r_0$ neighborhood and use $\\mathcal{A}$ to choose its color according to $\\mathfrak{r}$.\nFinally, by (c) it is possible to assign colors in $B\\setminus \\partial B$ for every $B$ to satisfy the condition of $\\Pi$ within each $B$.\nPicking minimal such solution for each $B\\setminus \\partial B$ finishes the description of the algorithm.\n\nWe show how to proceed from $0\\le \\ell<\\Delta+1$ to $\\ell+1$.\nSuppose that $\\mathfrak{r}_l$ satisfies the conditions above.\nPick $B$ such that $c(B)=\\ell+1$.\nNote that $\\mathfrak{r}_B$ that we want to find can influence rectangles $D\\in N(B)$ and these are influenced by the function $\\mathfrak{r}_l$ at distance at most $r_0$ from them.\nAlso given any $D$ there is at most one $B\\in N(D)$ such that $c(B)=\\ell+1$. \nWe make the decision $\\mathfrak{r}_B$ based on $2r_0$ neighborhood around $B$ and the restriction of $\\mathfrak{r}_l$ there, this guarantees that condition (a) is satisfied.\n\nFix $D\\in N(B)$ and pick $\\mathfrak{r}_B$ uniformly at random.\nWrite $\\mathfrak{S}_{B,D}$ for the random variable that computes the probability of $\\fE_{\\mathfrak{r}_B\\cup \\mathfrak{r}_l}(D)$.\nWe have $\\mathbb{E}(\\mathfrak{S}_{B,D})=\\mathbb{P}(\\fE_l(D))<\\frac{1}{(2\\Delta)^{2\\Delta-\\ell}}$.\nBy Markov inequality we have\n$$\\mathbb{P}\\left(\\mathfrak{S}_{B,D}\\ge 2\\Delta \\frac{1}{(2\\Delta)^{2\\Delta-\\ell}}\\right)<\\frac{1}{2\\Delta}.$$\nSince $|N(B)|\\le \\Delta$ we have that with non-zero probability $\\mathfrak{r}_B$ satisfies $\\mathfrak{S}_{B,D}<\\frac{1}{(2\\Delta)^{2\\Delta-\\ell-1}}$ for every $D\\in N(B)$.\nSince $\\mathcal{A}$ is defined almost everywhere we find $\\mathfrak{r}_B$ that satisfies (b) and (c) above.\nThis finishes the proof. \n\\end{proof}\n\nAs we already mentioned, the proof is in fact an instance of the famous Lov\u00e1sz Local Lemma problem. This problem is extensively studied in the distributed computing literature \\cite{chang2016exp_separation, fischer2017sublogarithmic, brandt_maus_uitto2019tightLLL, brandt_grunau_rozhon2020tightLLL} as well as descriptive combinatorics \\cite{KunLLL,OLegLLL,MarksLLL,Bernshteyn2021LLL}.\nOur usage corresponds to a simple LLL algorithm for the so-called exponential criteria setup (see \\cite{fischer2017sublogarithmic} and \\cite{Bernshteyn2021local=cont}). \n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{$\\TOAST$: General Results}\n\\label{sec:Toast}\n\nIn this section we introduce the toast construction. This construction is very standard in descriptive combinatorics \\cite{FirstToast,GJKS,Circle} and the theory of random processes (there it is often called \\emph{the hierarchy}) \\cite{HolroydSchrammWilson2017FinitaryColoring,ArankaFeriLaci,Spinka}.\n\nWhile the construction itself is well-known, in this paper we study it as a class of problems by itself. An analogy to our work is perhaps the paper of \\cite{ghaffari2018derandomizing} that defines the class $\\mathsf{S-LOCAL}$ and shows how to derandomize it.\n\n\n\nInformally, the basic toast construction works as follows. There is an adversary that has access to the (infinite) input grid $\\mathbb{Z}^d$. The adversary is sequentially choosing some connecting pieces of the grid and gives them to our algorithm $\\mathcal{A}$. The algorithm $\\mathcal{A}$ needs to label all vertices in the piece $D$ once it is given $D$. \nNote that $\\mathcal{A}$ does not know where exactly $D$ lies in $\\mathbb{Z}^d$ and its decision cannot depend on the precise position of $D$ in the input graph. In general, the piece $D$ can be a superset of some smaller pieces that were already colored by $\\mathcal{A}$. In this case, $\\mathcal{A}$ needs to extend their coloring to the whole $D$. This game is played for countable number of steps until the whole grid is colored. The algorithm solves $\\Pi$ if it never happens during the procedure that it cannot extend the labeling to $D$ in a valid manner. \n\n\n\n\n\nThe construction sketched above is what we call a spontaneous deterministic toast or just $\\TOAST$ algorithm. \nIn the next subsection we give a formal definition of $\\TOAST$ algorithms and we follow by proving relations between $\\TOAST$ algorithms and $\\mathsf{TAIL}$ complexities. \n\n\n\\subsection{Formal Definition of $\\TOAST$ classes}\n\\label{subsec:toast_definition}\nWe now give a formal definition of our four $\\TOAST$ classes. First we define the hereditary structure on which the $\\TOAST$ algorithm is interpreted. This is also called a toast (we use lowercase letters) and is parameterized by the minimum distance of boundaries of different pieces of the toast. \nThis structure also corresponds to the adversary that asks our algorithm to extend a partial solution to $\\Pi$ to bigger and bigger pieces used in the informal introduction. \n\n\\begin{definition}[$q$-toast on $\\mathbb{Z}^d$]\nFor a $q >0$, a \\emph{$q$-toast on $\\mathbb{Z}^d$} is a collection $\\mathcal{D}$ of finite connected subsets of $\\mathbb{Z}^d$ that we call \\emph{pieces} such that\n\\begin{itemize}\n \\item for every $g,h\\in \\mathbb{Z}^d$ there is a piece $D\\in\\mathcal{D}$ such that $g,h\\in D$,\n \\item if $D,E\\in \\mathcal{D}$ and $d(\\partial D,\\partial E)\\le q$, then $D=E$, where $\\partial D$ is a boundary of a set $D$.\n\\end{itemize}\n\\end{definition}\n\nGiven a $q$-toast $\\mathcal{D}$, the \\emph{rank} of its piece $D \\in \\mathcal{D}$ is the level in the toast hierarchy that $D$ belongs to in $\\mathcal{D}$. Formally, we define a \\emph{rank function $\\mathfrak{r}_\\mathcal{D}$ for the toast $\\mathcal{D}$} as $\\mathfrak{r}_\\mathcal{D}:\\mathcal{D}\\to \\mathbb{N}$, inductively as\n$$\\mathfrak{r}_\\mathcal{D}(D)=\\max\\{\\mathfrak{r}_{\\mathcal{D}}(E)+1:E\\subseteq D, \\ E\\in \\mathcal{D}\\}$$ \nfor every $D\\in \\mathcal{D}$.\n\nNext, $\\TOAST$ algorithms are described in the following language of extending functions. \n\n\\begin{definition}[$\\TOAST$ algorithm]\n\\label{def:extending_function}\nWe say that a function $\\mathcal{A}$ is a \\emph{$\\TOAST$ algorithm} for a local problem $\\Pi = (\\Sigma_{in},\\Sigma_{out}, t, \\mathcal{P})$ if \n\\begin{enumerate}\n \\item [(a)] the domain of $\\mathcal{A}$ consists of pairs $(A, c)$, where $A$ is a finite connected $\\Sigma_{in}$-labeled subset of $\\mathbb{Z}^d$ and $c$ is a partial map from $A$ to $\\Sigma_{out}$,\n \\item [(b)] $\\mathcal{A}(A,c)$ is a partial coloring from $A$ to $\\Sigma_{out}$ that extends $c$, i.e., $\\mathcal{A}(A,c)\\upharpoonright \\operatorname{dom}(c)=c$, and that depends only on the isomorphism type of $(A,c)$ (not on the position in $\\mathbb{Z}^d$). \n \n\\end{enumerate}\nA \\emph{randomized $\\TOAST$ algorithm} is moreover given as an additional input a labeling of $A$ with real numbers from $[0,1]$ (corresponding to a stream of random bits). \nFinally, if $\\mathcal{A}$ has the property that $\\operatorname{dom}(\\mathcal{A}(A,c)) =A$ for every pair $(A,c)$, then we say that $\\mathcal{A}$ is a \\emph{spontaneous (randomized) $\\TOAST$ algorithm}.\n\\end{definition}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{fig\/3-col.eps}\n \\caption{A $q$-toast and an example of an extending function for $3$-coloring. Observe that our extending function does not give a correct algorithm for $3$-coloring since the piece on the right is contained in an even bigger piece and the $3$-coloring constructed on the right cannot be extended to the node marked with an arrow}\n \\label{fig:toast_example}\n\\end{figure}\n\n\nIt is easy to see that a $\\TOAST$ algorithm $\\mathcal{A}$ can be used to color ($\\Sigma_{in}$-labeled) $\\mathbb{Z}^d$ inductively on the rank $\\mathfrak{r}_\\mathcal{D}$ of any $q$-toast $\\mathcal{D}$ for every $q\\in \\mathbb{N}$. \nNamely, define $\\mathcal{A}(\\mathcal{D})$ to be the partial coloring of $\\mathbb{Z}^d$ that is the union of $\\mathcal{A}(D)$, where $D\\in \\mathcal{D}$ and $\\mathcal{A}(D)$ is defined inductively on $\\mathfrak{r}_\\mathcal{D}$.\nFirst put $\\mathcal{A}(D):=\\mathcal{A}(D,\\emptyset)$ whenever $\\mathfrak{r}_\\mathcal{D}(D)=0$.\nHaving defined $\\mathcal{A}(D)$ for every $D\\in \\mathcal{D}$ such that $\\mathfrak{r}_\\mathcal{D}(D)=k$, we define\n$$\\mathcal{A}(D'):=\\mathcal{A}\\left(\\bigcup\\{E\\in \\mathcal{D}:E\\subseteq D'\\},\\bigcup\\{\\mathcal{A}(E):E\\subseteq D', \\ E\\in \\mathcal{D}\\}\\right).$$\nfor every $D'\\in \\mathcal{D}$ such that $\\mathfrak{r}_\\mathcal{D}(D')=k+1$. That is, we let $\\mathcal{A}$ to extend the partial solution to $\\Pi$ to bigger and bigger pieces of the toast $\\mathcal{D}$. \nObserve that if $\\mathcal{A}$ is a spontaneous $\\TOAST$ function, then $\\operatorname{dom}(\\mathcal{A}(\\mathcal{D}))=\\mathbb{Z}^d$ for every $q$-toast $\\mathcal{D}$ and every $q\\in \\mathbb{N}$.\n\n\\paragraph{$\\TOAST$ Algorithms}\nWe are now ready to define the four types of toast constructions that we are going to use (cf. \\cref{fig:big_picture_grids}). \n\n\\begin{definition}[$\\LTOAST$]\nLet $\\Pi$ be an LCL.\nWe say that $\\Pi$ is in the class $\\LTOAST$ if there is $q>0$ and an $\\TOAST$ algorithm $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is a $\\Pi$-coloring for every $q$-toast $\\mathcal{D}$.\n\nMoreover, we say that $\\Pi$ is in the class $\\TOAST$ if there is $q\\in \\mathbb{N}$ and a spontaneous $\\TOAST$ algorithm $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is a $\\Pi$-coloring for every $q$-toast $\\mathcal{D}$.\n\\end{definition}\n\nBefore we define a randomized version of these notions, we make several comments.\nFirst note that $\\TOAST\\subseteq \\LTOAST$.\nBy (c), if we construct a ($q$-)toast using an oblivious local algorithm, then we get instantly that every problem from the class $\\TOAST$ is in the class $\\mathsf{TAIL}$, see Section~\\ref{sec:Toast}.\nSimilar reasoning applies to the class $\\mathsf{BOREL}$, see Section~\\ref{sec:OtherClasses}.\nWe remark that we are not aware of any LCL that is not in the class $\\TOAST$.\n\n\\begin{comment}\n\\begin{definition}[Randomized toast algorithm]\n\\label{def:extending_function}\nLet $b$ be a set.\nWe say that a function $\\mathcal{A}$ is a randomized $\\TOAST$ algorithm if \n\\begin{enumerate}\n \\item [(a)] the domain of $\\mathcal{A}$ consists of triplets $(A, c, \\ell)$, where $A$ is a finite connected subset of $\\mathbb{Z}^d$, $c$ is a partial coloring of $A$ and $\\ell:A\\to [0,1]$. \n \\item [(b)] $\\mathcal{A}(A,c,\\ell)$ is a partial coloring of $A$ that extends $c$, i.e., $\\mathcal{A}(A,c,\\ell)\\upharpoonright \\operatorname{dom}(c)=c$. \n \\item [(c)] the function $\\mathcal{A}$ is invariant with respect to the shift action of $\\mathbb{Z}^d$ on $[0,1]^{\\mathbb{Z}^d)}$. \n\\end{enumerate}\nMoreover, if $\\mathcal{A}$ has the property that $\\operatorname{dom}(\\mathcal{A}(A,c,\\ell)) =A$ for every triplet $(A,c,\\ell)$, then we say that $\\mathcal{A}$ is a \\emph{spontaneous randomized toast algorithm}.\n\\end{definition}\n\nSimilarly as above, we define $\\mathcal{A}(\\mathcal{D},\\ell)$ for every $q$-toast $\\mathcal{D}$, $q\\in \\mathbb{N}$ and $\\ell\\in [0,1]^{\\mathbb{Z}^d}$.\nObserve that if $\\mathcal{A}$ is a spontaneous randomized toast algorithm, then $\\operatorname{dom}(\\mathcal{A}(\\mathcal{D},\\ell))=\\mathbb{Z}^d$.\n\\end{comment}\n\n\\begin{definition}[$\\RLTOAST$]\nLet $\\Pi$ be an LCL.\nWe say that $\\Pi$ is in the class $\\RLTOAST$ if there is $q>0$ and a randomized $\\TOAST$ algorithm $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is a $\\Pi$-coloring for every $q$-toast $\\mathcal{D}$ with probability $1$.\nThat is, after $\\Sigma_{in}$-labeling is fixed, with probability $1$ an assignment of real numbers from $[0,1]$ to $\\mathbb{Z}^d$ successfully produces a $\\Pi$-coloring against all possible toasts $\\mathcal{D}$ when $\\mathcal{A}$ is applied.\n\nMoreover, we say that $\\Pi$ is in the class $\\RTOAST$ if there is $q\\in \\mathbb{N}$ and a spontaneous randomized $\\TOAST$ algorithm $\\mathcal{A}$ as above.\n\\end{definition}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{$\\mathsf{TAIL}$ Construction of $\\TOAST$}\n\\label{subsec:toast_construction_in_tail_sketch}\n\n\nThe class $\\TOAST$ (or $\\RTOAST$) is especially useful since in \\cref{sec:SpecificProblems} we discuss how many local problems that are not in class $O(\\log^* n)$ but on the other hand are not ``clearly hard'' like vertex $2$-coloring are in the class $\\TOAST$. Importantly, in \\cref{subsec:toast_construction} we show how one can construct the toast decomposition with an oblivious local algorithm in $\\mathsf{TAIL}(r^{1 - O(1\/\\sqrt{\\log r})})$, which yields the following theorem.\n\n\\begin{restatable}{theorem}{toastconstruction}\n\\label{thm:toast_in_tail}\nLet $\\Pi$ be an LCL that is in the class $\\TOAST$.\nThen \\[\n\\Pi \\in \\mathsf{TAIL}\\left(\\frac{r}{2^{O(\\sqrt{\\log r})}}\\right) = \\mathsf{TAIL}\\left(r^{1-o(1)}\\right)\n\\]\nand, hence, ${\\bf M}(\\Pi)\\ge 1$.\n\\end{restatable}\n\nWe observe in \\cref{cor:toast_lower_bound} that faster then linear tail decay is not possible. \n\n\\paragraph{Proof Sketch}\n\nHere we provide an intuition for the proof that is otherwise technical. We first check that maximal independent sets of power graphs of the grid $G = \\mathbb{Z}^d$ can be constructed effectively, e.g. because of the MIS algorithm of Ghaffari \\cite{ghaffari2016MIS}. The slowdown by this construction can be neglected in this high-level overview. \n\nWe then take a sequence $r_k = 2^{k^2}$ and construct sparser and sparser MIS in graphs $G^{r_k}, k = 1, 2, \\dots$. After constructing MIS of $G^{r_k}$, we construct Voronoi cells induced by this MIS. These Voronoi cells $V_1, V_2, \\dots$ will be, roughly, the pieces of rank $k$, but we need to ensure that all the boundaries are at least $q$ hops apart. \n\nTo do so, we first shrink each Voronoi cells by $t_k = O(\\sum_{j < k} r_j) \\ll r_k$ hops (grey boundary area in \\cref{fig:toast_construction}) and get new, smaller, cells $C_1, C_2, \\dots$. The final sequence of pieces $H_1, H_2, \\dots$ is then defined inductively, namely to define $H_i$ we start with $C_1$, then add all Voronoi cells from $k-1$-th iteration that $C_1$ intersects, then add all Voronoi cells from $k_2$-th iteration that the new, bigger, cluster intersects, \\dots\nThe definition of $t_k$ ensures that the final pieces constructed after all $k$ iterations are sufficiently far from each other, and also compatible with the smaller rank pieces. In fact, we always have $H_i \\subseteq V_i$. \n\nIt remains to argue that after $k$ steps of this process, only a small fraction of nodes remains uncovered. This is done roughly as follows. The vertices from $V_i \\setminus C_i$ that are not covered by $C_i$s are roughly $t_k \/ r_k \\approx A r_{k-1} \/ r_k$ fraction of vertices where $A$ is some (large) constant. By induction, $1 - A r_{k-2} \/ r_{k-1}$ fraction of those vertices were covered in step $k-1$ and so on. The fraction of remaining vertices is then $\\frac{A r_{k-1}}{r_{k}} \\cdot \\frac{A r_{k-2}}{r_{k-1}} \\dots \\frac{A }{r_1} = \\frac{A^k}{r_k}$. This fraction is asymptotically almost equal to $1\/r_k$, which is the desired tail behavior. However, the additional term $A^k$ is the reason why the final tail decay for $r$ such that $r_k \\le r \\le r_{k+1} = 2^{(k+1)^2}$ is bounded by $2^{O(k)}\/2^{k^2} \\le \\left( \\frac{1}{r} \\right)^{\\frac{k^2 - O(k)}{(k+1)^2}} = \\left( \\frac{1}{r} \\right)^{1 - O(1\/k)} = \\left( \\frac{1}{r} \\right)^{1 - O(1\/\\sqrt{\\log r})} $, as needed. \n\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width = .95\\textwidth]{fig\/toast_construction.eps}\n \\caption{A step during the toast construction. Voronoi cell $V$ is first shrinked to $C$ and we get the final piece $H$ by taking the union of smaller-level Voronoi cells intersecting $C$ (in this picture, there is only one lower level but the construction is recursive in general). }\n \\label{fig:toast_construction}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\\subsection{$\\RTOAST = \\TOAST$}\n\\label{subsec:rtoast=toast}\n\n\n\nIn this section we show that randomness does not help spontaneous toast algorithms.\n\n\\begin{restatable}{theorem}{toastrtoast}\\label{thm:derandomOfToast}\nLet $\\Pi$ be an LCL.\nThen $\\Pi$ is in the class $\\RTOAST$ if and only if $\\Pi$ is in the class $\\TOAST$.\n\\end{restatable}\n\nThe proof goes as follows. We start with a random toast algorithm $\\mathcal{A}$ and make it deterministic as follows. When the deterministic toast algorithm gets a piece $D$ of rank $1$, it considers all possible outputs that the algorithm $\\mathcal{A}$ could produce given the input random string and it uses any one of them that $\\mathcal{A}$ would produce with nonzero probability. \nBy induction, whenever $\\mathcal{A}'$ later gets a piece $D$ containing some pieces of smaller rank with a partial solution that $\\mathcal{A}'$ already produced, the original algorithm $\\mathcal{A}$ could have produced the same partial solution with nonzero probability, hence $\\mathcal{A}$ can extend the coloring in a valid manner and $\\mathcal{A}'$ deterministically choosing any such valid extension that $\\mathcal{A}$ would use with nonzero probability. \nThis shows that $\\mathcal{A}'$ also solves $\\Pi$. \nA formal proof follows. \n\n\\begin{proof}\n\nIt is easy to see that $\\TOAST\\subseteq \\RTOAST$.\nSuppose that $\\Pi$ is in the class $\\RTOAST$ and fix a witnessing $q\\in \\mathbb{N}$ together with a spontaneous randomized $\\TOAST$ algorithm $\\mathcal{A}$.\nWe may assume that $t$, the locality of $\\Pi$, satisfies $t\\le q$.\n\nLet $(A,c)$ be such that $A\\subseteq \\mathbb{Z}^d$ is finite and connected and $\\operatorname{dom}(c)\\subseteq A$.\nWe say that $(A,c)$ comes from a \\emph{partial random $q$-toast} if there is a collection $\\mathcal{D}$ of subsets of $A$ together with an event $\\mathcal{F}$ on $\\operatorname{dom}(c)$ such that\n\\begin{itemize}\n \\item if $D,E\\in \\mathcal{D}$ and $d(\\partial D,\\partial E)\\le q$, then $D=E$,\n \\item $A\\in \\mathcal{D}$,\n \\item $\\bigcup_{E\\in \\mathcal{D}\\setminus \\{A\\}}E=\\operatorname{dom}(c)$,\n \\item $\\mathcal{F}$ has non-zero probability and if we apply inductively $\\mathcal{A}$ on $\\operatorname{dom}(c)$ we obtain $c$.\n\\end{itemize}\nIf $(A,c)$ comes from a partial random $q$-toast, then we let $\\mathcal{E}(A,c)$ to be any event $\\mathcal{F}$ as above.\nOtherwise we put $\\mathcal{E}(A,c)$ to be any event on $\\operatorname{dom}(c)$ of non-zero probability.\nIt is easy to see that $\\mathcal{E}$ does not depend on the position of $A$ in $\\mathbb{Z}^d$.\nThis is because $\\mathcal{A}$ is such.\n\nWe now define a spontaneous toast algorithm $\\mathcal{A}'$.\nGiven $(A,c)$, we let $\\mathcal{F}(A,c)$ to be any event on $A$ that has non-zero probability, satisfies\n$$\\mathcal{F}(A,c)\\upharpoonright \\operatorname{dom}(c)\\subseteq \\mathcal{E}(A,c)$$\nand $\\mathcal{A}(A,c)$ gives the same outcome on $\\mathcal{F}(A,c)$.\nLet $\\mathcal{A}'(A,c)$ be equal to this outcome on $\\mathcal{F}(A,c)$.\nIt is clear that $\\mathcal{A}'$ is a spontaneous $\\TOAST$ algorithm.\nObserve that if $(A,c)$ comes from a partial random $q$-toast, then $(A,\\mathcal{A}'(A,c))$ comes from a partial random $q$-toast.\nNamely, use the same $q$-toast witness $\\mathcal{D}$ as for $(A,c)$ together with the event $\\mathcal{F}(A,c)$.\n\nIt remains to show that $\\mathcal{A}'(\\mathcal{D})$ is a $\\Pi$-coloring for every $q$-toast $\\mathcal{D}$.\nInduction on $\\mathfrak{r}_{\\mathcal{D}}$ shows that $(D,\\mathcal{A}'(D))$ comes from a partial random $q$-toast for every $D\\in \\mathcal{D}$.\nThis is easy when $\\mathfrak{r}_{\\mathcal{D}}(D)=0$ and follows from the observation above in general.\nLet $v\\in \\mathbb{Z}^d$ and pick a minimal $D\\in \\mathcal{D}$ such that there is $D'\\in \\mathcal{D}$ that satisfies $v\\in D'$ and $D'\\subseteq D$.\nSince $t\\le q$ we have that $\\mathcal{B}(v,t)\\subseteq D$.\nSince $(D,\\mathcal{A}'(D))$ comes from a partial random $q$-toast it follows that there is a non-zero event $\\mathcal{F}$ such that \n$$\\mathcal{A}'(D)\\upharpoonright \\mathcal{B}(v,t)=\\mathcal{A}(\\mathcal{D})\\upharpoonright \\mathcal{B}(v,t)$$\non $\\mathcal{F}$.\nThis shows that the constraint of $\\Pi$ at $v$ is satisfied and the proof is finished.\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{$\\mathsf{TAIL} = \\RLTOAST$}\n\\label{subsec:tail=randomlazytoast}\n\nIn this section we prove that the class $\\RLTOAST$ in fact covers the whole class $\\mathsf{TAIL} = \\bigcup_{f = \\omega(1)} \\mathsf{TAIL}(f(r))$. \n\n\\begin{restatable}{theorem}{tailrltoast}\n\\label{thm:tail=rltoast}\nLet $\\Pi$ be an LCL.\nThen $\\Pi$ is in the class $\\RLTOAST$ if and only if $\\Pi$ is in the class $\\mathsf{TAIL}$.\n\\end{restatable}\n\nWe first give an informal proof. Suppose that there is a random lazy $\\TOAST$ algorithm $\\mathcal{A}$ for $\\Pi$. We construct an oblivious algorithm with nontrivial tail behavior $\\mathcal{A}'$ for $\\Pi$ by constructing a toast via \\cref{thm:toast_in_tail} and during the construction, each node moreover simulates the algorithm $\\mathcal{A}$. The algorithm $\\mathcal{A}$ finishes after some number of steps with probability $1$, hence $\\mathcal{A}'$ finishes, too, with probability $1$ for each node. \n\nOn the other hand, suppose that there is oblivious local algorithm $\\mathcal{A}'$ for $\\Pi$. \nThen we can construct a lazy randomized toast algorithm $\\mathcal{A}$ that simply simulates $\\mathcal{A}'$ for every node. This is possible since as the pieces around a given vertex $u$ that has gets are growing bigger and bigger, the radius around $u$ that $\\mathcal{A}$ sees also grows, until it is large enough and $\\mathcal{A}$ sees the whole coding radius around $u$ and applies $\\mathcal{A}'$ then. \n\nA formal proof follows. \n\n\n\\begin{proof}\nSuppose that $\\Pi$ is in the class $\\RLTOAST$, i.e, there is $q\\in\\mathbb{N}$ together with a randomized toast algorithm $\\mathcal{A}$.\nIt follows from Theorem~\\ref{thm:toast moment} that there is an oblivious local algorithm $F$ that produces $q$-toast with probability $1$.\n\nDefine an oblivious local algorithm $\\mathcal{A}'$ as follows.\nLet $\\mathcal{B}(v,r)$ be a $\\Sigma_{in}$-labeled neighborhood of $v$.\nAssign real numbers from $[0,1]$ to $\\mathcal{B}(v,r)$ and run $F$ and $\\mathcal{A}$ on $\\mathcal{B}(v,r)$.\nLet $a_{v,r}$ be the supremum over all $\\Sigma_{in}$-labelings of $\\mathcal{B}(v,r)$ of the probability that $\\mathcal{A}'$ fails.\nIf $a_{v,r}\\to 0$ as $r \\to \\infty$, then $\\mathcal{A}'$ is an oblivious local algorithm.\n\nSuppose not, i.e., $a_{v,r}>a>0$.\nSince $\\Sigma_{in}$ is finite, we find by a compactness argument a labeling of $\\mathbb{Z}^d$ with $\\Sigma_{in}$ such that the event $\\mathcal{F}_r$ that $\\mathcal{A}'$ fails on $\\mathcal{B}(v,r)$ with the restricted labeling has probability bigger than $a$.\nNote that since the labeling is fixed, we have $\\mathcal{F}_r\\supseteq \\mathcal{F}_{r+1}$.\nSet $\\mathcal{F}=\\bigcap_{r\\in \\mathbb{N}} \\mathcal{F}_r$.\nThen by $\\sigma$-additivity we have that the probability of the event $\\mathcal{F}$ is bigger than $a>0$.\nHowever, $F$ produces toast almost surely on $\\mathcal{F}$ and $\\mathcal{A}$ produces $\\Pi$-coloring with probability $1$ for any given toast.\nThis contradicts the fact that probability of $\\mathcal{F}$ is not $0$.\n\n\n\n\\begin{comment}\nIn another words, with probability $1$ any node $v$ can reconstruct bigger and bigger piece of the toast $\\mathcal{D}$ when exploring its neighborhood.\nIn particular, $v$ will eventually became element of some piece of the toast and $\\mathcal{A}$ has to label $v$ with $\\Sigma_{out}$ after finitely many steps.\nThis defines an oblivious local algorithm $\\mathcal{B}$.\n\n\n\nPick a labeling $\\ell\\in [0,1]^{\\mathbb{Z}^d}$ such that $F(\\ell)$ is a $q$-toast and $\\mathcal{A}(\\mathcal{D},\\ell)$ is a $\\Pi$-coloring for $q$-toast $\\mathcal{D}$.\nNote that the picked labeling satisfies these conditions with probability $1$.\n\nDefine $G(\\ell)=\\mathcal{A}(F(\\ell),\\ell)$.\nIt remains to observe that $G$ is an oblivious local algorithm.\nNote that $G(\\ell)({\\bf 0}_d)=\\mathcal{A}(F(\\ell),\\ell)({\\bf 0}_d)$ depends on some $D\\in \\mathcal{D}$ such that ${\\bf 0}_d\\in D$, $\\{E\\subseteq D:E\\in \\mathcal{D}\\}$ and $\\ell\\upharpoonright D$.\nSince $F$ is an oblivious local algorithm, we see that $G$ is too.\nConsequently, $\\Pi$ is in the class $\\mathsf{TAIL}$.\n\\end{comment}\n\nThe idea to prove the converse is to ignore the $q$-toast part and wait for the coding radius.\nSuppose that $\\Pi$ is in the class $\\mathsf{TAIL}$.\nBy the definition there is an oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi$ and an unbounded function $f$ such that\n$$\\P(R_\\mathcal{A}>r)\\le \\frac{1}{f(r)}.$$\nGiven an input labeling of $\\mathbb{Z}^d$ by $\\Sigma_{in}$, we see by the definition of $R_\\mathcal{A}$ that every node $v$ must finish after finitely many steps with probability $1$.\nTherefore it is enough to wait until this finite neighborhood of $v$ becomes a subset of some piece of a given toast $\\mathcal{D}$.\nThis happens with probability $1$ independently of the given toast $\\mathcal{D}$.\n\\end{proof}\n\n\\begin{comment}\nLet $(A,c,\\ell)$ be a triplet as in definition of randomized toast algorithm.\nTake $v\\in A\\setminus \\operatorname{dom}(c)$, and suppose that $v={\\bf 0}_d$.\nOutput $\\mathcal{A}(A,c,\\ell)({\\bf 0}_d)$ if and only if there is $t\\in \\mathbb{N}$ such that $\\mathcal{B}({\\bf 0}_d,t)\\subseteq A$ and $\\ell\\upharpoonright \\mathcal{B}({\\bf 0}_d,t)$ corresponds to some coding neighborhood of $F$.\nThat is to say, any extension $\\ell\\upharpoonright \\mathcal{B}({\\bf 0}_d,t)\\sqsubseteq x\\in [0,1]^{\\mathbb{Z}^d}$ outputs the same color at $F(x)({\\bf 0}_d)$.\nIn that case output $\\mathcal{A}(A,c,\\ell)({\\bf 0}_d)=F(x)({\\bf 0}_d)$.\n\nWe show that $\\mathcal{A}$ produces a $\\Pi$-coloring for $\\mu$-almost every $\\ell\\in [0,1]^{\\mathbb{Z}^d}$ and $1$-toast $\\mathcal{D}$.\nLet $\\ell\\in [0,1]^{\\mathbb{Z}^d}$ be such that $F$ computes a $\\Pi$-coloring an need a finite coding radius for every $v\\in \\mathbb{Z}^d$.\nNote that almost every labeling $\\ell$ satisfies this condition.\nTake a $1$-toast $\\mathcal{D}$.\nIt is easy to see that $\\mathcal{A}(\\mathcal{D},\\ell)$ does not depend on $\\mathcal{D}$ and that $\\mathcal{A}(\\mathcal{D},\\ell)=F(\\ell)$, i.e., $\\mathcal{A}(\\mathcal{D},\\ell)$ is a $\\Pi$-coloring.\nThis finishes the proof.\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\\section{$\\TOAST$: Specific Problems}\\label{sec:SpecificProblems}\n\\todo{explain that some of this is work of others and we show it to substantiate definition of toast}\n\nIn this section we focus on specific problems that are not in the class $\\mathsf{LOCAL}(O(\\log^* n))$ but they admit a solution by a (spontaneous) toast algorithm. Recall, that proving that $\\Pi \\in \\TOAST$ implies that there is an oblivious local algorithm with $1\/r^{1-o(1)}$ tail decay via \\cref{thm:toast_in_tail} and we discuss in \\cref{subsec:descriptive_combinatorics_discussion} that by a result of \\cite{GJKS2} it also implies $\\Pi\\in \\mathsf{BOREL}$. \n\nThe problems we discuss are the following: (a) vertex coloring, (b) edge coloring, (c) tiling the grid with boxes from a prescribed set (this includes, e.g., a perfect matching problem). \n\n\n\n\n\n\nAll these problems were recently studied in the literature from different points of view, see \\cite{ArankaFeriLaci, HolroydSchrammWilson2017FinitaryColoring, GJKS, spencer_personal}.\nIn all cases the main technical tool, sometimes in a slight disguise, is the toast construction.\nOur aim here is to put these problems into a unified context.\nWe note that sometimes the results in the aforementioned literature are stronger or work in a bigger generality (for example, all results discussed here can also be proven to hold on unoriented grids as discussed in \\cite{ArankaFeriLaci}, or the constructed coloring could be aperiodic \\cite{spencer_personal}, etc)\nIn what follows we only give a ``proof by picture'', sketch the description of the corresponding spontaneous toast algorithm and refer the reader to other papers for more details\/stronger results.\n\nThroughout this section we assume that $d>1$.\nWe refer the reader to \\cref{sec:OtherClasses} for definition of classes $\\mathsf{CONTINUOUS}$, $\\mathsf{BOREL}$, etc.\n\n\n\nThe main contribution if this section is the following:\n\\begin{enumerate}\n \\item The fact that the problem of vertex $3$-coloring is in $\\TOAST$ together with the speed-up \\cref{thm:grid_speedup} answers a question (i) from \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n \\item We observe that the problem of edge $2d$-coloring the grid $\\mathbb{Z}^d$ is in $\\TOAST$, thereby answering a question of \\cite{GJKS} coming from the area of descriptive combinatorics. Independently, this was proven in \\cite{ArankaFeriLaci}. \n\\end{enumerate}\n\nIn general, our approach gives that if an LCL $\\Pi$ is in the class $\\TOAST\\setminus \\mathsf{LOCAL}(O(\\log^*(n)))$, then $1\\le {\\bf M}(\\Pi)\\le d-1$ by \\cref{thm:toast_in_tail} and \\cref{thm:grid_speedup}.\nThe examples in Section~\\ref{sec:Inter} show that there are LCLs with ${\\bf M}(\\Pi)=\\ell$ for every natural number $1\\le \\ell\\le d-1$, but for most natural LCLs we do not know the exact value of ${\\bf M}(\\Pi)$, unless $d=2$.\n\n\n\\subsection{Vertex Coloring}\n\\label{subsec:3coloring}\n\nLet $d>1$.\nRecall that $\\Pi^d_{\\chi,3}$ is the \\emph{proper vertex $3$-coloring problem}.\nThis problem was studied extensively in all contexts.\nIt is known that $\\Pi^d_{\\chi,3}\\not\\in \\mathsf{LOCAL}(O(\\log^* n))$ \\cite{brandt_grids}, $\\Pi^d_{\\chi,3}\\not\\in \\mathsf{CONTINUOUS}$ \\cite{GJKS}, $\\Pi^d_{\\chi,3}\\in \\mathsf{BOREL}$ \\cite{GJKS2} and $0<{\\bf M}(\\Pi^d_{\\chi,3})\\le 2$ \\cite{HolroydSchrammWilson2017FinitaryColoring}.\nA version of the following theorem is proven in \\cite{GJKS2}, here we give a simple ``proof by picture''. \n\n\\begin{theorem}[\\cite{GJKS2}]\\label{thm:3in toast}\nLet $d>1$.\nThe LCL $\\Pi^d_{\\chi,3}$ is in the class $\\TOAST$.\n\\end{theorem}\n\n\\begin{proof}[Proof Sketch]\nThe toast algorithm tries to color all nodes in a given piece with two colors, blue and red. However, it can happen that two partial colorings of small pieces that have different parity in a bigger piece have to be connected (see \\cref{fig:3-coloring}). \nIn this case, we take the small pieces colored with incorrect parity and ``cover'' their border with the third color: green. From the inside of a small piece, the green color corresponds to one of the main two colors, while from the outside it corresponds to the other color. This enables us to switch the color parity. \n\\end{proof}\n\nAs a consequence of results from \\cref{sec:Toast} we can now fully determine the moment behavior of the problem which answers a question from \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n\n\\begin{corollary}\nLet $d>1$.\nEvery oblivious local algorithm that solves $\\Pi^d_{\\chi,3}$ has the $1$-moment infinite, but there is an oblivious local algorithm solving $\\Pi^d_{\\chi,3}$ with all $(1-\\varepsilon)$-moments finite. \nIn particular, ${\\bf M}(\\Pi^d_{\\chi,3})=1$. \n\\end{corollary}\n\\begin{proof}\nBy \\cref{thm:3in toast}, $\\Pi^d_{\\chi,3}$ is in the class $\\TOAST$.\nIt follows from \\cref{thm:toast_in_tail} that ${\\bf M}(\\Pi^d_{\\chi,3})\\ge 1$.\nBy the speedup \\cref{thm:grid_speedup} and the fact that $3$-coloring was proven not to be in class $\\mathsf{LOCAL}(O(\\log^* n)) = \\mathsf{TAIL}(\\tow(\\Omega(r))) = \\mathsf{CONTINUOUS}$ \\cite{brandt_grids,HolroydSchrammWilson2017FinitaryColoring,GJKS}, we have that every oblivious local algorithm $\\mathcal{A}$ solving $\\Pi^2_{\\chi,3}$ must have the $1$-moment infinite.\nIn particular, ${\\bf M}(\\Pi^2_{\\chi,3})=1$.\nIt remains to observe that every vertex $3$-coloring of $\\mathbb{Z}^d$ yields a vertex $3$-coloring of $\\mathbb{Z}^2$ by forgetting the last $(d-2)$ dimensions.\nConsequently, every oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi^d_{\\chi,3}$ defines an oblivious local algorithm $\\mathcal{A}'$ that solves $\\Pi^2_{\\chi,3}$ with the same tail behavior.\nThis finishes the proof.\n\\end{proof}\n\nThe $3$-coloring discussion above also implies that the toast itself cannot be constructed faster then with linear tail, otherwise $3$-coloring could be constructed with such a decay, too. \n\\begin{corollary}\n\\label{cor:toast_lower_bound}\nThere is a constant $C > 0$ such that a $q$-toast cannot be constructed in $\\mathsf{TAIL}(C r)$ on any $\\mathbb{Z}^d$.\nHere, by constructing a $q$-toast we mean that the coding radius of each node $u$ is such that $u$ can describe the minimal toast piece $D \\in \\mathcal{D}$ such that $u \\in D$. \n\\end{corollary}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width = .95\\textwidth]{fig\/3-col-real.eps}\n \\caption{The $\\TOAST$ algorithm is given small pieces of a toast that are already vertex-colored. First step is to match the parity of colors with respect to the bigger piece. Then color the bigger piece.}\n \\label{fig:3-coloring}\n\\end{figure}\n\n\n\\subsection{Edge Coloring}\n\\label{subsec:edgecoloring}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width = .95\\textwidth]{fig\/4-edge-col.png}\n \\caption{The $\\TOAST$ algorithm is given small pieces of a toast that are already edge-colored. First step is to match the parity of colors with respect to the bigger piece. This is done by adding a three layered gadget in the problematic directions. After that the coloring is extended to the bigger piece.}\n \\label{fig:4-edge-coloring}\n\\end{figure}\\todo{dopsat}\nLet $d>1$.\nWe denote as $\\Pi^d_{\\chi',2d}$ the \\emph{proper edge $2d$-coloring problem}.\nObserve that a solution to $\\Pi^d_{\\chi',2d}$ yields a solution to the perfect matching problem $\\Pi^d_{pm}$ (because $\\mathbb{Z}^d$ is $2d$-regular).\nThis shows that $\\Pi^d_{\\chi',2d}$ is not in the class $\\mathsf{LOCAL}(O(\\log^* n))$, because the perfect matching problem clearly cannot be solved with a local algorithm: if it was, we could find a perfect matching in every large enough finite torus, which is clearly impossible since it can have an odd number of vertices \\cite{brandt_grids}. \n\n\n\n\\begin{theorem}[See also \\cite{ArankaFeriLaci}]\\label{thm:edge toast}\nLet $d>1$.\nThen $\\Pi^d_{\\chi',2d}$ is in the class $\\TOAST$.\n\\end{theorem}\n\\begin{proof}[Proof Sketch]\nThe spontaneous toast algorithm is illustrated for $d=2$ in \\cref{fig:4-edge-coloring}.\nThe ``interface'' used by the toast algorithm between smaller and bigger pieces is: smaller pieces pretend they are colored such that in the vertical direction violet and green edges alternate, while in the horizontal direction blue and red edges alternate. If all small pieces in a bigger one have the same parity, this interface is naturally extended to the bigger piece. Hence, we need to work out how to fix pieces with bad horizontal\/vertical parity, similarly to \\cref{thm:3in toast}. \n\n\\cref{fig:4-edge-coloring} shows how to fix the vertical parity, adding the highlighted gadget in the middle picture above and below given pieces keeps its interface the same but switches the parity. The same can be done in the horizontal direction. \n\nTo show the result for $d>2$ it is enough to realize that switching parity in one dimension can be done with a help of one extra additional dimension, while fixing the parity in the remaining dimensions.\nConsequently, this allows to iterate the previous argument at most $d$-many times to fix all the wrong parities.\n\\end{proof}\n\n\n\\begin{corollary}\nLet $d>1$.\nThen $1 \\le {\\bf M}(\\Pi^d_{\\chi',2d}) \\le d-1$. \nIn particular, ${\\bf M}(\\Pi^2_{\\chi',4})=1$.\n\\end{corollary}\n\nWe leave as an open problem to determine the exact value of ${\\bf M}(\\Pi^d_{\\chi',2d})$, where $d>2$.\n\nThe following corollary answers a question of \\cite{GJKS}. See \\cref{sec:OtherClasses} for the definition of class $\\mathsf{BOREL}$. \n\\begin{corollary}\nLet $d\\ge 1$.\nThen $\\Pi^d_{\\chi',2d}$ is in the class $\\mathsf{BOREL}$.\nIn particular, if $\\mathbb{Z}^d$ is a free Borel action and $\\mathcal{G}$ is the Borel graph generated by the standard generators of $\\mathbb{Z}^d$, then $\\chi'_\\mathcal{B}(\\mathcal{G})=2d$, i.e., the Borel edge chromatic number is equal to $2d$.\n\\end{corollary}\n\n\n\\begin{remark}[Decomposing grids into finite cycles]\nBencs, Hru\u0161kov\u00e1, T\u00f3th \\cite{ArankaFeriLaci} showed recently that there is a $\\TOAST$ algorithm that decomposes an unoriented grid into $d$-many families each consisting of finite pairwise disjoint cycles.\nHaving this structure immediately gives the existence of edge coloring with $2d$-colors as well as Schreier decoration.\nSince every $\\TOAST$ algorithms imply Borel solution on graphs that look locally like unoriented grids \\cite{AsymptoticDim}, this result implies that all these problems are in the class $\\mathsf{BOREL}$.\nWe refer the reader to \\cite{ArankaFeriLaci} for more details.\n\\end{remark}\n\n\n\n\n\n\n\n\\begin{comment}\n\\paragraph{Decomposing grids into finite cycles} \\todo{Jsem pro zkratit to na jeden odstavec}\n\nWe sketch how to use toast structure to decompose $\\mathbb{Z}^d$ into $d$-many families each consisting of finite pairwise disjoint cycles, formally, a collection $\\{\\mathcal{C}_i\\}_{i\\in [d]}$ such that $\\mathcal{C}_i$ is family of pairwise vertex disjoint cycles for every $i\\in [d]$ and $\\bigcup_{i\\in [d]} \\mathcal{C}_i=\\mathbb{Z}^d$.\nNote that finding such a decomposition is not an LCL because we do not have bound on the length of the cycles.\nOn the other hand, if we allow infinite paths, then such problem is LCL problem and we denote it as $\\Pi^d_{\\operatorname{cyc},d}$.\nMoreover, it is easy to see that since we work with oriented grids $\\Pi^d_{\\operatorname{cyc},d}$ is in the class $\\mathsf{LOCAL}(O(1))$.\n\nWe show that there is a spontaneous extension function $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is always a $\\Pi^d_{\\operatorname{cyc},d}$-coloring with the additional property that all the cycles in $\\mathcal{A}(\\mathcal{D})$ are finite for every $37$-toast $\\mathcal{D}$.\nMoreover, it is easy to see that our toastable algorithm also works for non-oriented grids, where the LCL problem $\\Pi^d_{\\operatorname{cyc},d}$ is not trivial.\nThe combinatorial construction is inspired by result of Bencs, Hruskova and Toth \\cite{ArankaFeriLaci} who originally proved similar result for $d=2$.\nAfter reading their paper we realized that the argument goes through for any dimension $d>1$ and we refer the reader to their paper for a detailed proof of this fact.\nHere we only sketch the main idea.\n\n\\begin{theorem}\nLet $d>1$.\nThen there is a spontaneous extension function $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is always a $\\Pi^d_{\\operatorname{cyc},d}$-coloring for every $37$-toast $\\mathcal{D}$ with the additional property that all the cycles in $\\mathcal{A}(\\mathcal{D})$ are finite.\n\\end{theorem}\n\n\\begin{remark}\nA straightforward use of the toastable algorithm together with the fact that Borel graphs with finite Borel asymptotic dimension admit Borel toast structure (see~\\cite{AsymptoticDim}) gives a decomposition of any Borel graph that looks locally like a $d$-dimensional grid (oriented or non-oriented) into $d$-many Borel families of edges each consisting of pairwise disjoint cycles.\nSince each such cycle is necessarily of even length, we get immediately that such graphs admit Borel Schreier decoration, Borel balanced orientation and their Borel chromatic index is $2d$.\n\\end{remark}\n\n\\begin{proof}[Proof Sketch]\nLet $d>1$.\nIt is easy to see that decomposing $\\mathbb{Z}^d$ into $d$ families of cycles (possibly infinite) is the same as coloring the edges with $d$-colors, such that every vertex $v\\in \\mathbb{Z}^d$ has exactly two adjacent edges of each color.\nLet $\\{\\alpha_1,\\dots,\\alpha_d\\}$ be the set of the colors.\n\nAn \\emph{(edge) coloring} of a connected finite set $A\\subseteq \\mathbb{Z}^d$ is a coloring of all edges that are adjacent to $A$.\nWe say that an edge $e$ is a \\emph{boundary edge}, if only one vertex of $e$ is contained in $A$.\nA coloring $c$ is \\emph{suitable}, if after choosing a reference point and orientation of $A$ we have that parallel boundary edges in the $i$-th direction are assigned the color $\\alpha_i$.\n\nThere is a constant $q\\in \\mathbb{N}$ such that for every $i\\not=j\\in [d]$, finite connected set $A\\subseteq \\mathbb{Z}^d$ and a suitable edge coloring $c$, there is a suitable edge coloring $c'$ of $A':=\\mathcal{B}(A,q)$ that extends $c$ such that \n\\begin{itemize}\n \\item every $c'$-cycle of color $\\alpha_i$ that contains a vertex from $A$ is finite for every $i\\in [d]$,\n \\item the $i$-th and $j$-th directions of $A$ and $A'$ are swapped, e.g., parallel $A'$-boundary edges in the $i$-th direction from the point of view of $A$ are $c'$-colored with $\\alpha_j$,\n \\item other directions remain the same.\n\\end{itemize}\nIt is easy to see that iterating this extension possibly $2d$-many times produces a spontaneous toast algorithm $\\mathcal{A}$ with all the desired properties.\n\nIt remains to sketch how to find such extension $c'$.\nFirst, since $c$ is suitable and we want to swap $i$-th and $j$-th direction, we may assume that $d=2$.\nSecond, we may assume that $A$ is a union of rectangles, such that each side has even number of vertices adjacent to a perpendicular boundary edge.\nThis could be done as follows, pick a reference point and consider decomposition of $\\operatorname{Cay}(\\mathbb{Z}^2)$ into a square of a fixed even number of vertices on a side.\nTake the union of all such squares that intersect $A$.\nThird, see Figure~\\ref{fig:SwapingColors} how to swap colors and close all cycles coming out of $A$.\nThis finishes the sketch.\n\\end{proof}\n\\end{comment}\n\n\n\n\n\\subsection{Tiling with Rectangles}\n\\label{subsec:rectangle_tiling}\n\n\nLet $d>1$ and $\\mathcal{R}$ be a finite collection of $d$-dimensional rectangles.\nWe denote as $\\Pi^d_{\\mathcal{R}}$ the problem of \\emph{tiling $\\mathbb{Z}^d$ with elements of $\\mathcal{R}$}.\nIt is not fully understood when the LCL $\\Pi^d_{\\mathcal{R}}$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$ (or equivalently $\\mathsf{CONTINUOUS}$).\nCurrently the only known condition implying that $\\Pi^d_{\\mathcal{R}}$ is \\emph{not} in $\\mathsf{LOCAL}(O(\\log^* n))$ is when GCD of the volumes of elements of $\\mathcal{R}$ is not $1$.\nMinimal unknown example is when $\\mathcal{R}=\\{2\\times 2, 2\\times 3,3\\times 3\\}$. \nFor the other complexity classes the situation is different.\nGiven a collection $\\mathcal{R}$ we define a condition \\eqref{eq:GCD} as\n\\begin{equation}\n\\label{eq:GCD}\\tag{$\\dagger$}\n \\text{GCD of side lengths in the direction of} \\ i \\text{-th generator is} \\ 1 \\ \\text{for every} \\ i\\in [d].\n\\end{equation}\nIt follows from the same argument as for vertex $2$-coloring, that if $\\mathcal{R}$ does not satisfy \\eqref{eq:GCD}, then $\\Pi^d_{\\mathcal{R}}$ is a global problem, i.e., it is not in any class from \\cref{fig:big_picture_grids}.\nOn the other hand, the condition \\eqref{eq:GCD} is sufficient for a spontaneous toast algorithm. For example, this implies $\\Pi^2_{\\{2\\times 3,3\\times 2\\}}\\in \\mathsf{BOREL} \\setminus \\mathsf{CONTINUOUS}$.\nA version of the following theorem was announced in \\cite{spencer_personal}, here we give a simple ``proof by picture''.\n\n\\begin{theorem}[\\cite{spencer_personal, stephen}]\n\\label{thm:tilings}\nLet $d>1$ and $\\mathcal{R}$ be a finite collection of $d$-dimensional rectangles that satisfy \\cref{eq:GCD}.\nThen $\\Pi^d_{\\mathcal{R}}$ is in the class $\\TOAST$.\nConsequently, ${\\bf M}(\\Pi^d_{\\mathcal{R}})\\ge 1$.\n\\end{theorem}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=.9\\textwidth]{fig\/rectangle_filling.eps}\n \\caption{The $\\TOAST$ algorithm is given small pieces of a toast that are already tiled. First step is to match the parity of the tiling with respect to the bigger piece. Then tile the bigger piece.}\n \\label{fig:Tiling}\n\\end{figure}\n\n\\begin{proof}[Proof Sketch]\nThe two-dimensional version of the construction is shown in \\cref{fig:Tiling}. The interface of pieces this time is that they cover themselves with boxes of side length $L$ which is the least common multiple of all the lengths of basic boxes. As in the previous constructions, we just need to work out how to ``shift'' each piece so that all small pieces agree on the same tiling with boxes of side length $L$ which can then be extended to the whole piece. This can be done as in \\cref{fig:Tiling}. \n\\end{proof}\n\n\n\\begin{corollary}\nLet $d=2$ and $\\mathcal{R}$ be a finite collection of $2$-dimensional rectangles.\nThen $\\Pi^2_{\\mathcal{R}}$ is either in the class $\\mathsf{LOCAL}(O(\\log^* n))$, satisfies ${\\bf M}(\\Pi^2_{\\mathcal{R}})=1$, or is global (that is, in no class in \\cref{fig:big_picture_grids}).\n\\end{corollary}\n\nFor any $d$, \\cref{thm:tilings} implies that if a tiling problem is ``hard'' (in no class in \\cref{fig:big_picture_grids}), it is because of a simple reason: a projection to one dimension gives a periodic problem. It would be interesting to understand whether there are problems that are hard but their solutions do not have any periodicity in them (see \\cref{sec:open_problems}). \n\nWe do not know if there is $\\mathcal{R}$ that would satisfy $1< {\\bf M}(\\Pi^d_{\\mathcal{R}})\\le d-1$.\nLet $d\\ge \\ell>0$ and denote as $\\mathcal{R}_\\ell$ the restriction of rectangles to first $\\ell$-dimensions.\nAs in the case of proper $3$-vertex coloring problem, we have that ${\\bf M}(\\Pi^d_{\\mathcal{R}})\\le {\\bf M}(\\Pi^\\ell_{\\mathcal{R}_{\\ell}})$.\nIn particular, if, e.g., $\\Pi^d_{\\mathcal{R}_2}$ is not in the class $\\mathsf{LOCAL}(O(\\log^* n))$, then ${\\bf M}(\\Pi^d_{\\mathcal{R}})\\le 1$.\n\n\\begin{remark}\nIn the context of non-oriented grids we need to assume that $\\mathcal{R}$ is invariant under swapping coordinates. \nSimilar argument, then shows that $\\Pi^d_\\mathcal{R}\\in \\TOAST$ if and only if $\\mathcal{R}$ satisfies \\eqref{eq:GCD}.\nCombined with the fact that there is a Borel toast on such Borel graphs (see~\\cite{AsymptoticDim}), we deduce that \\eqref{eq:GCD} for $\\mathcal{R}$ is equivalent with existence of Borel tiling with rectangles from $\\mathcal{R}$.\n\\end{remark}\n\n\n\\todo{2x3 a 3x2 se daji poskladat z elek, coz odpovida nejake partition kruznice}\n\n\n\n\n\n\\paragraph{Perfect matching}\n\nLet $d>1$.\nA special case of a tiling problem is the \\emph{perfect matching problem} $\\Pi^d_{pm}$.\nSince there are arbitrarily large tori of odd size we conclude that $\\Pi^d_{pm}\\not \\in \\mathsf{LOCAL}(O(\\log^* n))$.\nConsequently, $\\Pi^d_{pm}\\not \\in \\mathsf{TAIL}(\\tow(\\Omega(r)))$ by \\cref{thm:tail=uniform}, and it follows from the Twelve Tile Theorem of Gao et al \\cite[Theorem 5.5]{GJKS} that $\\Pi^d_{pm}\\not \\in \\mathsf{CONTINUOUS}$.\nSimilarly as with the vertex $3$-coloring problem, the proof of $\\Pi^d_{pm}\\in \\mathsf{BOREL}$, where $d>1$, from \\cite{GJKS2} uses the toast construction and, in our notation, yields the following result. \n\n\\begin{theorem}[\\cite{GJKS2}]\\label{thm:perfect_matching_intoast}\nLet $d>1$.\nThe LCL $\\Pi^d_{pm}$ is in the class $\\TOAST$.\n\\end{theorem}\n\n\\begin{corollary}\nLet $d>1$.\nThen ${\\bf M}(\\Pi^d_{pm})\\ge 1$ and every ffiid $F$ that solves $\\Pi^d_{\\chi,3}$ has the $(d-1)$-moment infinite.\nIn particular, ${\\bf M}(\\Pi^2_{pm})=1$.\n\\end{corollary}\n\nWe leave as an open problem to determine the exact value of ${\\bf M}(\\Pi^d_{pm})$, where $d>2$.\nIn fact, this problem is similar to intermediate problems that we study in \\cref{sec:Inter}. \n\n\n\n\n\n\n\n\n\n\n\n\\renewcommand{\\pm}[2]{\\operatorname{PM}^{#1, #2}_\\boxtimes}\n\\renewcommand{\\pm}[1]{\\operatorname{PM}^{#1}_\\boxtimes}\n\\newcommand{\\mathbb{Z}^d_\\boxtimes}{\\mathbb{Z}^d_\\boxtimes}\n\n\\section{Intermediate Problems}\\label{sec:Inter}\n\nIn this section we construct local problems with $\\mathsf{TAIL}$ complexities $\\Theta(r^{j-o(1)})$ for $1 \\le j \\le d-1$. \nWe first define the problem $\\pm{d}$: this is just a perfect matching problem, but in the graph $\\mathbb{Z}^d_\\boxtimes$ where two different nodes $u = (u_1, \\dots, u_d)$ and $v = (v_1, \\dots, v_d)$ are connected with an edge if $\\sum_{1 \\le i \\le d} |u_i - v_i| \\le 2$. That is, we also add some ``diagonal edges'' to the grid. We do not know whether perfect matching on the original graph (Cayley graph of $\\mathbb{Z}^d$) is also an intermediate problem (that is, not $\\mathsf{LOCAL}$ but solvable faster than $\\TOAST$). \n\nWe will show that the tail complexity of $\\pm{d}$ is $\\Theta(r^{d - 1 - o(1)})$. \nAfter this is shown, the hierarchy of intermediate problems is constructed as follows. \nConsider the problem $\\pm{d,\\ell}$. This is a problem on $\\mathbb{Z}^d$ where one is asked to solve $\\pm{d}$ in every hyperplane $\\mathbb{Z}^\\ell$ given by the first $\\ell$ coordinates in the input graph. \nThe $\\mathsf{TAIL}$ complexity of $\\pm{d,\\ell}$ is clearly the same as the complexity of $\\pm{\\ell}$, that is, $\\Theta(r^{\\ell-1-o(1)})$. \nHence, we can focus just on the problem $\\pm{d}$ from now on. \n\n\\begin{claim}\\label{cl:PM is global}\nLet $2\\le d\\in \\mathbb{N}$.\nThen the local coloring problem $\\pm{d}$ is not in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\nVia \\cref{thm:grid_speedup} this is equivalent to $\\pm{d} \\not\\in \\mathsf{TAIL}(Cr^{d-1})$ for large enough $C$. \n\\end{claim}\n\nThis follows from the fact that if there is a local algorithm, then in particular there exists a solution on every large enough torus. But this is not the case if the torus has an odd number of vertices. \n\nSecond, we need to show how to solve the problem $\\pm{d}$ faster than with a $\\TOAST$ construction algorithm. \n\n\\begin{restatable}{theorem}{intermediate}\n\\label{thm:intermediate}\nThe problem $\\pm{d}$ can be solved with an oblivious local algorithm of tail complexity $r^{d-1-O(\\frac{1}{\\sqrt{\\log r}})}$. \n\\end{restatable}\n\nHere we give only a proof sketch, the formal proof is deferred to \\cref{app:intermediate_problems}. \n\n\\begin{proof}[Proof Sketch]\nWe only discuss the case $d=3$ (and the actual proof differs a bit from this exposition due to technical reasons). As usual, start by tiling the grid with boxes of side length roughly $r_0$. \nThe \\emph{skeleton} $\\mathcal{C}_0$ is the set of nodes we get as the union of the $12$ edges of each box. \nIts \\emph{corner points} $X_0 \\subseteq \\mathcal{C}_0$ is the union of the $8$ corners of each box. \n\n\nWe start by matching vertices inside each box that are not in the skeleton $\\mathcal{C}_0$. It is easy to see that we can match all inside vertices, maybe except of one node per box that we call a \\emph{pivot}. We can ensure that any pivot is a neighbor of a vertex from $\\mathcal{C}_0$. \n\nWe then proceed similarly to the toast construction of \\cref{thm:toast_in_tail} and build a sequence of sparser and sparser independent sets that induce sparser and sparser subskeletons of the original skeleton.\nMore concretely, we choose a sequence $r_k = 2^{O(k^2)}$ and in the $k$-th step we construct a new set of corner points $X_k \\subseteq X_{k-1}$ such that any two corner points are at least $r_k$ hops apart but any point of $\\mathbb{Z}^d$ has a corner point at distance at most $2r_k$ (in the metric induced by $\\mathbb{Z}^d$). \nThis set of corner points $X_k$ induces a subskeleton $\\mathcal{C}_{k} \\subseteq \\mathcal{C}_{k-1}$ by connecting nearby points of $X_k$ with shortest paths between them in $\\mathcal{C}_{k-1}$. \n\nAfter constructing $\\mathcal{C}_k$, the algorithm solves the matching problem on all nodes of $\\mathcal{C}_{k-1} \\setminus \\mathcal{C}_k$. This is done by orienting those nodes towards the closest $\\mathcal{C}_k$ node in the metric induced by the subgraph $\\mathcal{C}_k$. \nThen, the matching is found on connected components that have radius bounded by $r_k$. There is perhaps only one point in each component, a pivot, that neighbors with $\\mathcal{C}_{k}$ and remains unmatched. \n\nAfter the $k$-th step of this algorithm, only nodes in $\\mathcal{C}_k$ (and few pivots) remain unmatched. Hence, it remains to argue that only roughly a $1\/r_k^{d-1}$ fraction of nodes survives to $\\mathcal{C}_k$. This should be very intuitive -- think about the skeleton $\\mathcal{C}_k$ as roughly looking as the skeleton generated by boxes of radius $r_k$. This is in fact imprecise, because in each step the distance between corner points of $X_k$ in $\\mathcal{C}_k$ may be stretched by a constant factor with respect to their distance in the original skeleton $\\mathcal{C}_{k-1}$. This is the reason why the final volume of $\\mathcal{C}_k$ is bounded only by $2^{O(k)} \/ r_k^{d-1}$ and by a similar computation as in \\cref{thm:toast_in_tail} we compute that the tail decay of the construction is $2^{O(\\sqrt{\\log r})}\/{r^{d-1}}$. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Other Classes of Problems}\\label{sec:OtherClasses}\n\nIn this mostly expository section we discuss other classes of problems from \\cref{fig:big_picture_grids} and how they relate to our results from previous sections. \n\n\n\n\\subsection{$\\mathsf{CONTINUOUS}$}\\label{subsec:CONT}\n\nHere we explain the fact that an LCL $\\Pi$ on $\\mathbb{Z}^d$ is in the class $\\mathsf{LOCAL}(O(\\log^*(n)))$ if and only if it is in the class $\\mathsf{CONTINUOUS}$.\nFor $\\mathbb{Z}^d$, the model $\\mathsf{CONTINUOUS}$ was extensively studied by Gao, Jackson, Krohne, Tucker-Drob, Seward and others, see \\cite{GaoJackson,GJKS,ST,ColorinSeward}.\nIn fact, the connection with local algorithms is implicit in \\cite{GJKS} under the name Twelve Tiles Theorem \\cite[Theorem~5.5]{GJKS}.\nRecently, independently Bernshteyn \\cite{Bernshteyn2021local=cont} and Seward \\cite{Seward_personal} showed that the same holds in bigger generality for all Cayley graphs of finitely generated groups.\nOur aim here is to formally introduce the model $\\mathsf{CONTINUOUS}$ and give an informal overview of the argument for $\\mathbb{Z}^d$ that follows the lines of Gao et al \\cite{GJKS}.\nThe reasons not to simply put reference to the aforementioned papers are two.\nFirst, we work with slightly more general LCLs that have inputs.\nSecond, our goal is to introduce all classes in a way accessible for a broader audience. \n\nThe setup of the $\\mathsf{CONTINUOUS}$ model is somewhat similar to the deterministic $\\mathsf{LOCAL}$ model: there is no randomness in the definition and there is a certain symmetry breaking input on the nodes. Specifically, each vertex is labelled with one label from the set $\\{0, 1\\}$; the space of all such labelings is denoted as $2^{\\mathbb{Z}^d}$. \nHowever, for the same reason as why we insist on the uniqueness of the identifiers in the $\\mathsf{LOCAL}$ model, we insist that the input $\\{0,1\\}$ labeling breaks all symmetries. Namely, we require that there is no $g$ such that shifting all elements by $g$ preserves the labeling. We write $\\operatorname{Free}(2^{\\mathbb{Z}^d}) \\subseteq 2^{\\mathbb{Z}^d}$ for this subspace. \nA \\emph{continuous algorithm} is a pair $\\mathcal{A}=(\\mathcal{C},f)$, where $\\mathcal{C}$ is a collection of $\\{0,1\\} \\times \\Sigma_{in}$-labeled neighborhoods and $f$ is a function that assigns to each element of $\\mathcal{C}$ an output label. It needs to satisfy the following. Suppose we fix a node $v$ and the input $\\{0,1\\}$-labeling $\\Lambda$ of $\\mathbb{Z}^d$ satisfies $\\Lambda \\in \\operatorname{Free}(2^{\\mathbb{Z}^d})$. If we let $v$ explore its neighborhood with an oblivious local algorithm, then we encounter an element from $\\mathcal{C}$ after finitely many steps. \nWe write $\\mathcal{A}(\\Lambda)(v)$, or simply $\\mathcal{A}(v)$, when $\\Lambda$ is understood, for the corresponding output.\n\n\\begin{comment}\n\\vspace*{5cm}\nTo break symmetries on $\\mathbb{Z}^d$ in the setting of random processes one considers iid labels on vertices.\nThe event that some symmetries remain has measure $0$ and we simply ignore such labelings.\nIn the continuous settings we label vertices with $2:=\\{0,1\\}$.\nThe space of all such labelings is denoted as $2^{\\mathbb{Z}^d}$ and we write $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ for the space of labelings that break all symmetries.\nA \\emph{continuous algorithm} is a pair $\\mathcal{A}=(\\mathcal{C},g)$, where $\\mathcal{C}$ is a collection of ${0,1},\\Sigma_{in}$-labeled neighborhoods and $g$ is a function that assigns to each element of $\\mathcal{C}$ an output label, such that if we let a node $v$ explore its labeled neighborhood $x$, then it encounters element from $\\mathcal{C}$ after finitely many steps whenever $x\\in \\Sigma_{in}^{\\mathbb{Z}^d}\\times \\operatorname{Free}(2^{\\mathbb{Z}^d})$, i.e., $x$ breaks all symmetries.\nWe write $\\mathcal{A}(x)(v)$ for the corresponding output.\n\\end{comment}\n\n\\begin{remark}\nThis is equivalent with the definition given in e.g. \\cite{GJKS} in the same way as the correspondence between oblivious local algorithms and ffiid, see \\cref{subsec:subshifts}.\nThe fact that the collection $\\mathcal{C}$ is dense, i.e., every node finishes exploring its neighborhoods in finite time, means that $\\mathcal{A}({\\bf 0})$ is \\emph{continuous} with respect to the product topology and defined in the whole space $\\operatorname{Free}(2^{\\mathbb{Z}^d})$.\nMoreover, the value computed at $v$ does not depend on the position of $v\\in \\mathbb{Z}^d$ but only on the isomorphism type of the labels, this is the same as saying that $\\mathcal{A}(\\_)$ is \\emph{equivariant}.\n\\end{remark}\n\nWe say that an LCL $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ if there is a continuous algorithm $\\mathcal{A}$ that outputs a solution for every $\\Lambda \\in \\operatorname{Free}(2^{\\mathbb{Z}^d})$.\nNote that, in general, we cannot hope for the diameters of the neighborhoods in $\\mathcal{C}$ to be uniformly bounded.\nIn fact, if that happens then $\\Pi \\in \\mathsf{LOCAL}(O(1))$. \nThis is because there are elements in $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ with arbitrarily large regions with constant $0$ label.\nApplying $\\mathcal{A}$ on that regions gives an output label that depends only on $\\Sigma_{in}$ after bounded number of steps, hence we get a local algorithm with constant local complexity.\n\nThe main idea to connect this context with distributed computing is to search for a labeling in $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ where any $\\mathcal{A}$ necessarily has a uniformly bounded diameter.\nSuch a labeling does exist!\nIn \\cite{GJKS} these labelings are called \\emph{hyperaperiodic}.\nTo show that hyperaperiodic labelings exist on $\\mathbb{Z}^d$ is not that difficult, unlike in the general case for all countable groups \\cite{ColorinSeward}. \nIn fact there is enough flexibility to encode in a given hyperaperiodic labeling some structure in such a way that the new labeling remains hyperaperiodic.\n\nWe now give a definition of a hyperaperiodic labeling. \nLet $\\Lambda \\in 2^{\\mathbb{Z}^d}$ be a labeling of $\\mathbb{Z}^d$.\nWe say that $\\Lambda$ is \\emph{hyperaperiodic} if for every $g\\in \\mathbb{Z}^d$ there is a finite set $S\\subseteq \\mathbb{Z}^d$ such that \nfor every $t\\in \\mathbb{Z}^d$ we have \n$$\\Lambda(t+S) \\not= \\Lambda(g+t+S)$$\nThis notation means that the $\\Lambda$-labeling of $S$ differs from the $\\Lambda$-labeling of $S$ shifted by $g$ everywhere. \nIn another words, given some translation $g\\in \\mathbb{Z}^d$, there is a uniform witness $S$ for the fact that $\\Lambda$ is nowhere $g$-periodic.\nIt follows from a standard compactness argument that any continuous algorithm $\\mathcal{A}$ used on a hyperaperiodic labeling finishes after number of steps that is uniformly bounded by some constant that depends only on $\\mathcal{A}$. \n\n\n\n\\begin{comment}\n\\vspace*{5cm}\n\nLet $\\Lambda \\in 2^{\\mathbb{Z}^d}$ be a labeling of $\\mathbb{Z}^d$.\nWe say that $\\Lambda$ is \\emph{hyperaperiodic} if for every $g\\in \\mathbb{Z}$ there is a finite set $S\\subseteq \\mathbb{Z}^d$ such that \nfor every $s\\in \\mathbb{Z}^d$ we have \n$$x(s+t)\\not=x(g+s+t)$$\nfor some $t\\in S$.\nIn another words, given a shift $g\\in \\mathbb{Z}^d$ there is a uniform witness $S$ for the fact that $x$ is nowhere $g$-periodic.\nIt follows from some standard compactness argument that every hyperaperiodic element needs uniformly bounded neighborhoods for each continuous algorithm $\\mathcal{A}$, the bound of course depends on the algorithm. \n\\end{comment}\n\n\\begin{theorem}[\\cite{GJKS,Bernshteyn2021local=cont}]\nLet $d>0$.\nThen an LCL $\\Pi$ on $\\mathbb{Z}^d$ is in the class $\\mathsf{CONTINUOUS}$ if and only if it is in the class $\\mathsf{LOCAL}(O(\\log^*(n)))$.\n\\end{theorem}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=.9\\textwidth]{fig\/continuous.eps}\n \\caption{On the left side we have examples of a labeled local tiling configurations compared with a possible continuity window $w$. These are then sparsely embedded into $\\mathbb{Z}^2$.}\n \\label{fig:my_label}\n\\end{figure}\n\n\\begin{proof}[Proof Sketch]\nOne direction was observed by Bernshteyn \\cite{Bernshteyn2021local=cont} and follows from the fact that there is continuous algorithm that produces a maximal $r$-independent set for every $r\\in \\mathbb{N}$ (recall \\cref{thm:classification_of_local_problems_on_grids}).\nThis is implicit, even though in a different language, already in Gao and Jackson \\cite{GaoJackson}.\n\nNext, suppose that $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ and write $\\mathcal{A}$ for the continuous algorithm for $\\Pi$. \nWe will carefully construct a certain hyperaperiodic element $\\Lambda$ and let $\\mathcal{A}$ solve $\\Pi$ on it. \nWe know that there is a certain number $w$ such that $\\mathcal{A}$ solves $\\Pi$ on $\\Lambda$ with local complexity $w$. \nWe note that even though we do not know the value $w$ in advance, it depends on $\\Lambda$ that is due to be constructed, the flexibility of the construction guarantees to ``encode'' in $\\Lambda$ all possible scenarios.\nSee the discussion at the end of the argument. \n\nThe overall approach is as follows. \nOur local algorithm $\\mathcal{A}'$ first tiles the plane with tiles of length much bigger than $w$ with $O(\\log^* n)$ local complexity. We cannot tile the plane in a periodic manner in $O(\\log^* n)$ steps, but it can be done with a constant number of different tiles such that the side length of each tile is either $r$ or $r+1$ for some $r \\gg w$ (this construction is from \\cite[Theorem~3.1]{GaoJackson}, see also the proof of \\cref{thm:grid_speedup}). \nFor each tile $T$, $\\mathcal{A}'$ labels it with certain $\\{0,1\\}$ labeling. \nThis labeling depends only on the shape of the tile.\nWe make sure that not only $T$ with its labeling, but also $T$ together with all possible local configurations of tiles around it and corresponding labelings appears in $\\Lambda$.\nIn particular, as $r\\gg t,w$ ($t$ is the radius necessary to check the solution $\\Pi$), we have that the $(w+t)$-hop neighborhood of any node is labelled in a way that also appears somewhere in the labeling $\\Lambda$.\nAfter labeling each tile, $\\mathcal{A}'$ simulates $\\mathcal{A}$. We know that $\\mathcal{A}$ finishes after $w$ steps on $\\Lambda$ and is correct there, hence $\\mathcal{A}'$ is correct and has local complexity $O(\\log^* n)$. What remains is to describe is how to get the labeling of each tile. \n\nWe use the fact that there is a lot of flexibility when constructing a hyperaperiodic labeling. \nWe define the hyperaperiodic labeling $\\Lambda$ of the input graph $\\mathbb{Z}^d$ in such a way that, after fixing $r$, each one of constantly many local configurations of $\\{r,r+1\\}$-tiles are embedded in $\\Lambda$, each one in fact in many copies that exhaust possible $\\Sigma_{in}$-labelings.\n\nThis is done as follows.\nFix any hyperaperiodic element $\\Lambda'$, these exist by \\cite[Lemma~2.8]{GJKS}.\nGiven a tile $T$, that depends on $r$, $\\{0,1\\}$-label it by any $T$-pattern that appears in $\\Lambda'$.\nConsider the collection of all possible local configurations of such labeled tiles that are moreover labeled by $\\Sigma_{in}$.\nThis collection is finite. Since we do not know $w$ in advance, do this for any value of $r$. This yields a countable collection of possible configurations. \nEmbed this collection sparsely into $\\Lambda'$, \\cref{fig:my_label}. Extend the $\\Sigma_{in}$-labeling arbitrarily. \nThis defines $\\Lambda$.\nIt is routine to check that $\\Lambda$ is hyperaperiodic, see \\cite[Lemma~5.9]{GJKS}.\n\n\n\nIt remains to recall that since $\\Lambda$ is hyperaperiodic, there is $w\\in \\mathbb{N}$ such that $\\mathcal{A}$ has uniformly bounded locality by $w$ on $\\Lambda$.\nPick $r\\gg w,t$.\nThis works as required since any $w+t$-hop neighborhood $\\mathcal{A}'$ can encounter is present in $\\Lambda$.\n\\end{proof}\n\n\\begin{comment}\n\\vspace*{4cm}\nNext, suppose that $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ and write $\\mathcal{A}$ for the continuous algorithm.\nA high-level strategy for $d=2$ is as follows.\nThere are 12 type of tiles that are parametrized by natural numbers $n,p,q\\in \\mathbb{N}$, i.e., for fixed $n,p,q$ there are $12$ tiles.\nIf $n,p,q$ are fixed, then there is a local deterministic algorithm that tiles given, large enough, torus with these tiles in such a way that the boundary parts matches.\\todo{picture here ad reference to the nice pictures in \\cite{GJKS}}\nIt is not clear at this point how large parameters we should choose.\nThe trick is to encode all such possible tiles for all $n,p,q\\in \\mathbb{N}$ in one hyperaperiodic element $x$.\nThe encoding of the tiles in $x$ guarantees that whenever the regions \\todo{picture} are the same for the fixed parameters $n,p,q$, then $\\mathcal{A}$ outputs the same label.\nSince $x$ is hyperaperiodic, we have that $\\mathcal{A}$ stops after $T$-many steps to output $\\mathcal{A}(x)(v)$ for some $T\\in \\mathbb{N}$.\nWe choose $n,p,q$ according to $T$ in such a way that for any vertex at any region there is a tile such that the $T$-neighborhood of this vertex is fully contained in the tile.\nConsequently, this defines local rule once a given, large enough, torus is tiled into these tiles of 12 types.\n\nThis description exactly follows the proof of (1) $\\rightarrow$ (2) in \\cite[Theorem~5.5]{GJKS}, where (1) is the same as saying that there is a continuous algorithm and (2) fixes the coloring of the tiles for some parameters $n,p,q$.\nHowever, we note that we are skipping some technical details about additional parameter $w$ etc.\nThe additional step to produce a local algorithm by tiling a large enough torus into these 12 tiles is standard.\nIn fact, this is proved in \\cite[Theorem~5.10]{GJKS}, again in different language, to show (2) $\\rightarrow$ (1) in \\cite[Theorem~5.5]{GJKS}.\n\nThe main difference that we encounter are the input labels.\nWe explain how to handle them and hint, simultaneously, what we mean by encoding tiles in hyperaperiodic element, again ignoring a technical parameter $w$, see \\cite[Figures~13,14,etc]{GJKS}.\nFix a hyperaperiodic element $y$.\nFor $n,p,q\\in \\mathbb{N}$ and a tile $G$, see \\eqref{fig:???}, assign to each region labels from $\\{0,1\\}$ in such a way that (a) same regions get the same pattern, (b) this pattern is part of $y$.\nConsider moreover any possible input labeling of such $\\{0,1\\}$-labeled tiles.\nNow place these labeled tiles, for every $n,p,q\\in \\mathbb{N}$, sparsely in $y$ (Figure ???) and extend the input labeling on them to the whole $\\mathbb{Z}^2$, this defines $x$.\nIt is not hard to see that $x$ remains hyperaperiodic.\n\nIn higher dimension, i.e., when $d>2$, the same argument works as well with the only difference that there are more than 12 types of tiles.\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\\subsection{Descriptive Combinatorics Classes}\n\\label{subsec:descriptive_combinatorics_discussion}\n\nIn this section we define the classes of problems that are studied in descriptive combinatorics.\nThere are two equivalent ways how the notion of a \\emph{Borel graph} can be defined.\nThe less abstract one is as follows.\nConsider the unit interval $[0,1]$ with the $\\sigma$-algebra of Borel sets, that is the smallest algebra of subsets that contains sub-intervals and is closed under countable unions and complements.\nLet $A,B\\subseteq [0,1]$ be Borel sets and consider a bijection $\\varphi:A \\to B$ between them.\nWe say that $\\varphi$ is \\emph{Borel measurable} if $\\varphi^{-1}(D\\cap B)$ is a Borel set for every Borel set $D$.\nNow, let $\\mathcal{G}$ be a graph with vertex set $[0,1]$ and assume that the degree of every vertex is bounded by some $\\Delta<\\infty$.\nWe say that $\\mathcal{G}$ is a \\emph{Borel graph} if the edge relation is induced by Borel bijections, that is there are sets $A_1,\\dots, A_\\ell$, $B_1,\\dots, B_\\ell$ and Borel bijections $\\varphi_1,\\dots, \\varphi_\\ell$ such that $(x,y)$ is an edge in $\\mathcal{G}$ if and only if there is $1\\le i\\le \\ell$ such that either $\\varphi_i(x)=y$ or $\\varphi_i(y)=x$.\n\nThe more abstract way how to define Borel graphs uses the notion of a \\emph{standard Borel space}.\nFor example $[0,1]$ with the $\\sigma$-algebra of Borel sets, when we forget that it comes from the standard topology, is a standard Borel space.\nFormally, a standard Borel space is a pair $(X,\\mathcal{B})$, where $X$ is a set and $\\mathcal{B}$ is a $\\sigma$-algebra that comes from some separable and complete metric structure on $X$.\nA uniformly bounded degree graph $\\mathcal{G}$ on $(X,\\mathcal{B})$ is Borel if it satisfies the same condition as in the case of the particular space $[0,1]$.\n\nIt is easy to see that with this definition all the basic constructions from graph theory are \\emph{relatively} Borel, e.g. if $A$ is a Borel set of vertices, then $N_\\mathcal{G}(A)$ the neighbors of elements in $A$ form a Borel set.\nAlso note that since we assume that the degree is uniformly bounded, every connected component in $\\mathcal{G}$ is at most countable.\nTypical questions that are studied in this context are Borel versions of standard notions, e.g., \\emph{Borel} chromatic number, \\emph{Borel} chromatic index, \\emph{Borel} perfect matching etc.\nWe refer the reader to \\cite{pikhurko2021descriptive_comb_survey,kechris_marks2016descriptive_comb_survey} for more details.\n\nThe most relevant Borel graphs for us are the ones that are induced by a free Borel action of $\\mathbb{Z}^d$.\nThat is, there are $d$-many aperiodic Borel bijections $\\varphi_i:X\\to X$ that pairwise commute, where \\emph{aperiodic} means that $\\varphi_i^k(x)\\not=x$ for every $k>0$, $i\\in [d]$ and $x\\in X$.\nThen the Borel (labeled) graph $\\mathcal{G}$ that is induced by this action consists of edges of the form $(x,\\varphi_i(x))$, where $x\\in X$ and $i\\in [d]$.\nThe most important result in this context is that $\\mathcal{G}$ admits a \\emph{Borel toast structure}.\nThis follows from work of Gao et al \\cite{GJKS2}, see Marks and Unger \\cite{Circle} for proof.\nWe note that it is not understood in general what Borel graphs admit a Borel toast structure.\nThis is connected with the notoriously difficult question about Borel hyperfiniteness, see \\cite{AsymptoticDim}.\n\nGiven an LCL $\\Pi$, we say that $\\Pi$ is in the class $\\mathsf{BOREL}$ if there is a Borel measurable $\\Pi$-coloring of every Borel graph $\\mathcal{G}$ as above, i.e., induced by a free Borel action of $\\mathbb{Z}^d$.\nEquivalently, it is enough to decide whether $\\Pi$ can be solved on the canonical Borel graph $\\mathcal{G}$ on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$.\nMoreover, if $\\Pi$ has some inputs $\\Sigma_{in}$, then we consider any Borel labeling $X\\to \\Sigma_{in}$ and then ask if there is a Borel solution that satisfies all the constraints on $\\Sigma_{in}\\times \\Sigma_{out}$.\n\n\\begin{theorem}[Basically \\cite{GJKS}]\nLet $\\Pi$ be an LCL that is in the class $\\LTOAST$.\nThen $\\Pi\\in\\mathsf{BOREL}$.\n\\end{theorem}\n\nIn fact, all the LCL problems without input that we are aware of and that are in the class $\\mathsf{BOREL}$ are in fact in the class $\\TOAST$.\nIt is an open problem to decide whether it is always the case.\n\nAnother interesting perspective how to see Borel constructions is as a countably infinite analogue of $\\mathsf{LOCAL}(O(\\log^*(n)))$.\nNamely, suppose that you are allowed to inductively (a) construct Borel MIS with some parameter (b) do local $O(1)$-construction that depends on the previously build structure.\nIt is clear that objects that are build in this way are Borel, however, it is not clear if this uses the whole potential of Borel sets to find Borel solutions of LCLs.\n\nAs a last remark we note that traditionally in descriptive combinatorics, people consider relaxations of Borel constructions.\nThis is usually done via ignoring how a solution behave on some negligible Borel set.\nProminent examples are \\emph{measure $0$ sets} for some Borel probability measure on $(X,\\mathcal{B})$ or \\emph{meager sets} for some compatible complete metric on $(X,\\mathcal{B})$.\nWe denote the classes of problems that admit such a solution for \\emph{every} Borel probability measure or compatible complete metric as $\\mathsf{MEASURE}$ and $\\mathsf{BAIRE}$, respectively.\nWithout going into any details we mention that on general graphs these classes are much richer than $\\mathsf{BOREL}$, see e.g. \\cite{DetMarks,BrooksMeas}.\nHowever, it is still plausible that on $\\mathbb{Z}^d$ we have $\\mathsf{BOREL}=\\mathsf{BAIRE}=\\mathsf{MEASURE}$.\nIt is well-know that this is true in $\\mathbb{Z}$ \\cite{grebik_rozhon2021LCL_on_paths}.\n\n\n\\begin{comment}\nIn both cases we are allowed to neglect some small sets, where we either do not care about the solution or are allowed to use the axiom of choice.\nSuppose that $\\mathcal{G}$ is a Borel graph on a standard Borel space $(X,\\mathcal{B})$ and $\\Pi$ is an LCL.\nSuppose, in addition, that there is a Borel probability measure $\\mu$ on $(X,\\mathcal{B})$, that is a $\\sigma$-additive measure on $\\mathcal{B}$, see \\cite[Section~?]{Kec}.\nWe say that $\\Pi$ admits \n\n\n\n\n\n\n\n\n\n\nand show the arrows that are left to complete Figure~\\ref{fig:big_picture_grids}.\nRecall that a \\emph{Borel graph} $\\mathcal{G}$ is a triplet $\\mathcal{G}=(X,\\mathcal{B},E)$, where $(X,\\mathcal{B})$ is a standard Borel space, $(X,E)$ is a graph and $E\\subseteq X\\times X$ is an element of the product $\\sigma$-algebra.\nFor the purposes of this paper we always assume that $\\mathcal{G}$ is induced by a free Borel action of $\\mathbb{Z}^d$ for some $d>0$, i.e., there is a Borel action $\\mathbb{Z}^d\\curvearrowright X$ with trivial stabilizers such that $\\mathcal{G}$ is induced by the standard generators of $\\mathbb{Z}^d$.\n\nLet $\\Pi$ be an LCL.\nThen $\\Pi$ belongs to the class $\\mathsf{BOREL}$ if there is a Borel $\\Pi$-coloring of every such Borel graph $\\mathcal{G}$.\nMoreover, we say that $\\Pi$ is in the class $\\mathsf{MEASURE}$ if for every such Borel graph $\\mathcal{G}$ and for every \\emph{Borel probability measure} $\\mu$ on $(X,\\mathcal{B})$ there is a Borel $\\Pi$-coloring of $\\mathcal{G}$ on a $\\mu$-conull set.\nSimilarly, we say that $\\Pi$ is in the class $\\mathsf{BAIRE}$ if for every such Borel graph $\\mathcal{G}$ and for every \\emph{compatible Polish topology} $\\tau$ on $(X,\\mathcal{B})$ there is a Borel $\\Pi$-coloring of $\\mathcal{G}$ on a $\\tau$-comeager set.\nIt follows directly from the definition that $\\mathsf{BOREL}\\subseteq \\mathsf{MEASURE},\\mathsf{BAIRE}$.\n\\end{comment}\n\n\n\n\\begin{comment}\nNext we follow Gao et al. \\cite{GJKS}.\nDenote as $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ the free part of the Bernoulli shift $\\mathbb{Z}^d\\curvearrowright 2^{\\mathbb{Z}^d}$ endowed with the inherited product topology.\nThen it is easy to see that $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ is a Polish space and that the action is continuous.\nConsider the Borel graph $\\mathcal{G}$ on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ induced by the standard set of generators.\nThis graph is universal in the sense that an LCL $\\Pi$ is in the class $\\mathsf{BOREL}$ if and only if there is a Borel $\\Pi$-coloring of $\\mathcal{G}$, this follows from result of Seward and Tucker-Drob~\\cite{?}.\nWe say that an LCL $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ if there is a $\\Pi$-coloring of $\\mathcal{G}$ that is continuous.\nThis is the same as saying that that there is a continuous equivariant map from $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ to $X_\\Pi$.\nThe following is a corollary of the Twelve Tiles Theorem of Gao et al. \\cite{GJKS}.\n\n\\begin{theorem}\\cite[Theorem 5.5]{GJKS}\n$\\mathsf{CONTINUOUS}=\\mathsf{LOCAL}(O(\\log^* n))$.\n\\end{theorem}\n\nWe note that in the general case of all groups this equality was shown by Bernshteyn\\cite{?} and, independently, by Seward. \n\nIt remains to describe how toastable algorithms give Borel solutions.\nAs an immediate corollary we get that all the problems from Section~\\ref{sec:SpecificProblems} are in the class $\\mathsf{BOREL}$.\n\n\\begin{theorem}\n$\\LTOAST \\subseteq \\mathsf{BOREL}$.\n\\end{theorem}\n\\begin{proof}\nSuppose that $\\Pi$ is in the class $\\LTOAST$ and let $f$ be an toast algorithm with parameter $k$.\nIt follows from work of Gao et al \\cite{GJKS2}, see Marks and Unger \\cite{Circle} for proof, that every Borel graph $\\mathcal{G}$ (that is induced by a free action of $\\mathbb{Z}^d$) admits a Borel $k$-toast for every $k\\in \\mathbb{N}$.\nIt is a standard fact that using $f$ inductively on the depth of the toast produces a Borel $\\Pi$-coloring of $\\mathcal{G}$.\n\\end{proof}\n\\end{comment}\n\n\n\n\n\n\\subsection{Finitely Dependent Processes}\n\nIn this section we use the notation that was developed in \\cref{subsec:subshifts}, also we restrict our attention to LCLs without input.\nLet $\\Pi=(b,t,\\mathcal{P})$ be such an LCL.\nRecall that $X_\\Pi$ is the space of all $\\Pi$-colorings.\n\nWe say that a distribution $\\mu$ on $b^{\\mathbb{Z}^d}$ is \\emph{finitely dependent} if it is shift-invariant and there is $k\\in \\mathbb{N}$ such that the events $\\mu\\upharpoonright A$ and $\\mu \\upharpoonright B$ are independent for every finite set $A,B\\subseteq \\mathbb{Z}^d$ such that $d(A,B)\\ge k$.\nNote that shift-invariance is the same as saying that the distribution $\\mu\\upharpoonright A$ does not depend on the position of $A$ in $\\mathbb{Z}^d$ but rather on its shape.\n\nHere we study the following situation:\nSuppose that there is a finitely dependent process $\\mu$ that is concentrated on $X_\\Pi$.\nIn that case we say that $\\Pi$ is in the class $\\mathsf{FINDEP}$. \nIn other words, $\\Pi \\in \\mathsf{FINDEP}$ if there is a distribution from which we can sample solutions of $\\Pi$ such that if we restrict the distribution to any two sufficient far sets, the corresponding restrictions of the solution of $\\Pi$ that we sample are independent. \n\nIt is easy to see that all problems in $\\mathsf{LOCAL}(O(1))$ (that is, problems solvable by a block factor) are finitely dependent, but examples of finitely dependent processes outside of this class were rare and unnatural, see the discussion in \\cite{FinDepCol}, until Holroyd and Liggett \\cite{FinDepCol} provided an elegant construction for 3- and 4-coloring of a path and, more generally, for distance coloring of a grid with constantly many colors. This implies the following result.\n\n\\begin{theorem}[\\cite{FinDepCol}]\n$\\mathsf{LOCAL}(O(\\log^* n)) \\subseteq \\mathsf{FINDEP}$ (see \\cref{fig:big_picture_grids}).\n\\end{theorem} \n\nOn the other hand, recently Spinka \\cite{Spinka} proved that any finitely dependent process can be described as a finitary factor of iid process.\\footnote{His result works in a bigger generality for finitely generated amenable groups.}\nUsing \\cref{thm:toast_in_tail}, we sketch how this can be done, we keep the language of LCLs.\n\n\\begin{theorem}[\\cite{Spinka}]\nLet $\\Pi$ be an LCL on $\\mathbb{Z}^d$ and $\\mu$ be finitely dependent process on $X_\\Pi$, i.e., $\\Pi\\in \\mathsf{FINDEP}$.\nThen there is a finitary factor of iid $F$ (equivalently an oblivious local algorithm $\\mathcal{A}$) with tail decay $1\/r^{1- O(\\sqrt{\\log r})}$ such that the distribution induced by $F$ is equal to $\\mu$.\nIn fact, $\\Pi$ is in the class $\\TOAST$. \n\\end{theorem}\n\\begin{proof}[Proof Sketch]\nSuppose that every vertex is assigned uniformly two reals from $[0,1]$.\nUse the first string to produce a $k$-toast as in \\cref{thm:toast moment}, where $k$ is from the definition of $\\mu$.\n\nHaving this use the second random string to sample from $\\mu$ inductively along the constructed $k$-toast.\nIt is easy to see that this produces ffiid $F$ that induces back the distribution $\\mu$.\nIn fact, it could be stated separately that the second part describes a randomized $\\TOAST$ algorithm: always samples from $\\mu$ given the coloring on smaller pieces that was already fixed. The finitely dependent condition is exactly saying that the joint distribution over the solution on the small pieces of $\\mu$ is the same as the distribution we sampled from.\nNow use \\cref{thm:derandomOfToast} to show that $\\Pi\\in \\TOAST$. \n\\end{proof}\n\nIn our formalism, $\\mathsf{FINDEP} \\subseteq \\TOAST$, or equivalently we get $\\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r))) \\subseteq \\mathsf{FINDEP} \\subseteq \\mathsf{TAIL}(r^{1-o(1)})$.\nIt might be tempting to thing that perhaps $\\mathsf{FINDEP} =\\TOAST$.\nIt is however a result of Holroyd et al \\cite[Corollary~25]{HolroydSchrammWilson2017FinitaryColoring} that proper vertex $3$-coloring, $\\Pi^{d}_{3,\\chi}$, is not in $\\mathsf{FINDEP}$ for every $d>1$.\nThe exact relationship between these classes remain mysterious.\n\n\n\n\\begin{comment}\n\\begin{theorem}\nLet $\\Pi$ be an LCL on $\\mathbb{Z}^d$.\nSuppose that $\\Pi$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\nThen there is finitely dependent process $\\mu$ on $X_\\Pi$.\n\\end{theorem}\n\\begin{proof}\nIt follows from Theorem~\\ref{thm:classification_of_local_problems_on_grids} that there is $r,\\ell \\in \\mathbb{N}$ such that $\\ell\\le r-t$ and a rule $\\mathcal{A}$ of locality at most $\\ell$ that produces a $\\Pi$-coloring for every $(r+t)$-power proper vertex coloring of $\\mathbb{Z}^d$.\n\nSuppose that $\\mu_{(r+t)}$ is a finitely dependent $(r+t)$-power proper vertex coloring of $\\mathbb{Z}^d$ process with locality $k\\in \\mathbb{N}$.\nWe claim that composing $\\mu_{(r+t)}$ with $\\mathcal{A}$ yields a finitely dependent process with locality $(2\\ell+k)$.\nLet $A\\subseteq \\mathbb{Z}^d$ be a finite set.\nThen the output of $\\mathcal{A}$ depends on $\\mathcal{B}(A,\\ell)$.\nTherefore if $A,B\\subseteq \\mathbb{Z}^d$ satisfy $d(A,B)>2\\ell+k$, then\n$$d(\\mathcal{B}(A,\\ell),\\mathcal{B}(B,\\ell))>k$$\nand we conclude that the outputs of $\\mathcal{A}$ on $A$ and $B$ are independent.\n\nIt remains to find $\\mu_{r}$ finitely dependent $r$-power proper vertex coloring of $\\mathbb{Z}^d$ process for every $r>0$.\nThis is done by Holroyd and Liggett \\cite[Theorem~20]{FinDepCol}.\n{\\color{red} or something like that, they use a lot of colors fr $r$-power coloring...}\n\\end{proof}\n\nGiven a fiid $F$ we define $F^*\\lambda$ to be the push-forward of the product measure via $F$.\nSince $F$ is equivariant it follows that $F^*\\lambda$ is shift-invariant.\n\nThe following theorem was essentially proven by Spinka\\cite{?}. His result nicely fits to our definitions in \\cref{sec:Toast}. \n\\begin{theorem}\n\n\\end{theorem}\n\n\n\\begin{theorem}\nLet $\\Pi$ be an LCL problem on $\\mathbb{Z}^d$ and $\\mu$ be finitely dependent process on $X_\\Pi$.\nThen for every $\\epsilon>0$ there is a ffiid $F$ such that $F^*\\lambda=\\mu$ and the tail of $F$ has the $(1-\\epsilon)$-moment finite.\nMoreover, $\\Pi$ is in the class $\\TOAST$.\n\\end{theorem}\n\n\\begin{remark}\nMaybe the random toast construction works for any finitely dependent processes not just LCL.\n\nThere is the notion of strongly irreducible shifts of finite type, some comment about it...\n\nDefine $\\mathsf{FINDEP}$ to be the class of all LCL problems that admit a finitely dependent distribution $\\mu$.\nIt follows that $\\mathsf{TAIL}(\\tow(\\Omega(r)))\\subseteq \\mathsf{FINDEP} \\subseteq \\mathsf{TOAST}$ and we have $\\mathsf{FINDEP}\\not = \\TOAST$ because proper vertex $3$-coloring $\\Pi^d_{\\chi,3}$ is not in $\\mathsf{FINDEP}$ for every $d>1$, by \\cite[Corollary~25]{HolroydSchrammWilson2017FinitaryColoring}.\n\\end{remark}\n\n\n\\begin{proof}\nWe define a spontaneous randomized toast algorithm $\\Omega$ such that $\\Omega(\\mathcal{D},\\ell)$ is a $\\Pi$-coloring for every $k$-toast $\\mathcal{D}$ and every $\\ell\\in [0,1]^{\\mathbb{Z}^d}$.\nThis already implies the moreover part by Theorem~\\ref{thm:derandomOfToast}.\n\nLet $A\\subseteq \\mathbb{Z}^d$ be a finite subset of $\\mathbb{Z}^d$.\nWrite $\\mathbb{P}_A$ for the distribution of $\\Pi$-patterns of $\\mu$ on $A$, i.e., $\\mathbb{P}_A=\\mu\\upharpoonright A$.\nSuppose now that $(A,c,\\ell)$ are given, where $A\\subseteq \\mathbb{Z}^d$ is finite subset, $c$ is a partial coloring of $A$ and $\\ell\\in [0,1]^{A}$.\nConsider the conditional distribution $\\mathbb{P}_{A,c}$ of $\\mathbb{P}_A$ given the event $c$.\nIf $\\mathbb{P}_{A,c}$ is non-trivial, then define $\\Omega$ such that $\\Omega(A,c,{-})$ induces the same distribution as $\\mathbb{P}_{A,c}$ when $\\ell$ is fixed on $\\operatorname{dom}(c)$.\nOtherwise define $\\Omega$ arbitrarily.\n\nLet $\\mathcal{D}$ be a $k$-toast.\nWe show that $\\Omega(\\mathcal{D},\\ell)$ is a $\\Pi$-coloring for every $\\ell\\in [0,1]^{\\mathbb{Z}^d}$ and $\\Omega(\\mathcal{D},{-})$ produces $\\mu$.\nNote that the first claim shows that $\\Pi$ is in the class $\\RTOAST$, while the second produces $\\mu$ as ffiid as follows.\nSuppose that every $v\\in \\mathbb{Z}^d$ is assigned two independent reals $l_{v,0}$ and $l_{v,1}$.\nUse the first real to produce a $k$-toast as in Theorem~\\ref{thm:toast moment} and use the second real together with the produced toast as an input to $\\Omega$.\nIt follows from the independence of $l_{v,0}$ and $l_{v,1}$ that this produces $\\mu$.\n\nWe show that\n$$\\Omega(\\mathcal{D},{-})\\upharpoonright D=\\mu\\upharpoonright D$$\nfor every $D\\in \\mathcal{D}$ inductively on $\\mathfrak{r}_{\\mathcal{D}}$.\nIf $\\mathfrak{r}_{\\mathcal{D}}(D)=0$, then it follows from the definition.\nSuppose that $\\mathfrak{r}_{\\mathcal{D}}(D)>0$ and let $E_1,\\dots, E_t\\in \\mathcal{D}$ be an enumeration of $E\\subset D$.\nIt follows from the inductive hypothesis together with the assumption that $\\mu$ is $k$-independent and $\\mathcal{D}$ is a $k$-toast that \n$$\\Omega(\\mathcal{D},{-})\\upharpoonright \\bigcup_{i\\le t}E_i=\\mu\\upharpoonright \\bigcup_{i\\le t} E_i.$$\nNow when $\\ell$, and therefore a partial coloring $c$, is fixed on $\\bigcup_{i\\le t}E_i=\\operatorname{dom}(c)$, the definition $\\Omega(D,c,{-})$ extends $c$ in such a way that $\\mathbb{P}_{D,c}=\\mu\\upharpoonright \\operatorname{dom}(c)$.\nSince this holds for any such assignment $\\ell$ of real numbers on $\\bigcup_{i\\le t}E_i$ we deduce that\n$$\\Omega(\\mathcal{D},{-})\\upharpoonright D=\\mu\\upharpoonright D$$\nand the proof is finished.\n\\end{proof}\n\\end{comment}\n\n\n\n\n\n\n\\subsection{$\\mathsf{TAIL}$ Complexities in Distributed and Centralised Models}\n\\label{subsec:congestLCA}\n\nAlthough we are not aware of a systematic work on $\\mathsf{TAIL}$ complexities in the distributed algorithms community, similar notions like oblivious algorithms \\cite{Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring}, node-average complexity \\cite{feuilloley2017node_avg_complexity_of_col_is_const,barenboim2018avg_complexity,chatterjee2020sleeping_MIS} or energy complexity \\cite{chang2018energy_complexity_exp_separation,chang2020energy_complexity_bfs} are studied. \nThe breakthrough MIS algorithm of Ghaffari \\cite{ghaffari2016MIS} may have found many applications exactly because of the fact that its basis is an algorithm with exponential tail behavior. \n\nWe discuss two applications of algorithms with nontrivial $\\mathsf{TAIL}$ complexities. First, the node-average complexity literature studies the average coding radius that a node needs to answer. Whenever an oblivious algorithm has its first moment finite, we have an algorithm with constant average coding radius. \nWith a little more work, one can see that the result from \\cref{thm:3in toast}, that is $3$-coloring of grids has $\\mathsf{TAIL}$ complexity $r^{1-O(1\/\\sqrt{\\log r})}$ can be adapted to the setting of finite grids where it gives a $3$-coloring algorithm with average complexity $2^{O(\\sqrt{\\log n})}$. \nFor nonconstant $\\Delta$, the situation is more complicated. In \\cite{barenboim2018avg_complexity} it was shown that the Luby's $\\Delta+1$ coloring algorithm has tail $2^{\\Omega(r)}$ even for nonconstant $\\Delta$, but the question of whether MIS can have constant average coding radius in nonconstant degree graphs is open \\cite{chatterjee2020sleeping_MIS}. \n\nAnother application of the $\\mathsf{TAIL}$ bound approach comes from the popular centralised $\\mathsf{LCA}$ model \\cite{lca11,lca12}: there one has a huge graph and instead of solving some problem, e.g., coloring, on it directly, we are only required to answer queries about the color of some given nodes. It is known that, as in the $\\mathsf{LOCAL}$ model, one can solve $\\Delta+1$ coloring with $O(\\log^* n)$ queries on constant degree graphs by a variant of Linial's algorithm \\cite{even03_conflictfree}. But using the tail version of the Linial's algorithm instead, we automatically get that $O(\\log^* n)$ is only a worst-case complexity guarantee and on average we expect only $O(1)$ queries until we know the color of a given node. That is, the amortized complexity of a query is in fact constant. The same holds e.g. for the $O(\\log n)$ query algorithm for LLL from \\cite{brandt_grunau_rozhon2021LLL_in_LCA}. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Open Questions}\n\\label{sec:open_problems}\n\nHere we collect some open problems that we find interesting. \n\n\\begin{enumerate}\n \\item Is it true that $\\RLTOAST = \\LTOAST$? If yes, this implies $\\mathsf{TAIL} \\subseteq \\mathsf{BOREL}$. \n \\item Is it true that $\\RLTOAST = \\RTOAST$? If yes, this implies $\\mathsf{TAIL} = \\mathsf{TAIL}(r^{1-o(1)})$ and answers the fourth question of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n\t\\item Can the tail complexity $r \/ 2^{O(\\sqrt{\\log r})}$ in the construction of toast be improved to $r \/ \\operatorname{poly} \\log (r)$? The same can be asked for the complexity of intermediate problems from \\cref{sec:Inter}. \n\t\t\\item Is there a local coloring problem $\\Pi$ such that ${\\bf M}(\\Pi)$ is non-integer value or $0$, i.e., ${\\bf M}(\\Pi)\\in [0,d-1]\\setminus \\{1,2,\\dots, d-1\\}$?\n\t\t\\item Can you find an example of a local problem without input labels which does not belong to any class in \\cref{fig:big_picture_grids} but the solutions to this problem are not ``periodic'' (this cannot happen for $d=1$). Maybe (a version of) Penrose tilings can be helpful there?\n\t\\item What is the value ${\\bf M}(\\Pi^d_{pm})$, where $\\Pi^d_{pm}$ is the problem of finding a perfect matching of $\\mathbb{Z}^d$, where $d>2$?\n \\item Let $d>1$.\n\tFor what finite collections of $d$-dimensional rectangles $\\mathcal{R}$ is $\\Pi^d_{\\mathcal{R}}$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$? Is there a finite collection of $d$-dimensional rectangles $\\mathcal{R}$ that would satisfy $1< {\\bf M}(\\Pi^d_{\\mathcal{R}})\\le d-1$?\n\t\\item Is it true (on $\\mathbb{Z}^d$) that $\\mathsf{BOREL}=\\mathsf{MEASURE}$ or $\\mathsf{BOREL} = \\mathsf{BAIRE}$?\n\\end{enumerate}\n\n\\section*{Acknowledgement}\nWe would like to thank Ferenc Bencs, Anton Bernshteyn, Sebastian Brandt, Yi-Jun Chang, Mohsen Ghaffari, Aranka Hru\u0161kov\u00e1, Stephen Jackson, Oleg Pikhurko, Brandon Sewart, Yinon Spinka, L\u00e1szlo T\u00f3th, and Zolt\u00e1n Vidny\u00e1szky for many engaging discussions. \n\nThe first author was supported by Leverhulme Research Project Grant RPG-2018-424. \nThis project has received funding from the European Research Council (ERC) under the European\nUnions Horizon 2020 research and innovation programme (grant agreement No. 853109). \n\n\\bibliographystyle{alpha}\n\n\\section{Introduction}\n\nIn this paper, we study local problems on grid graphs from three different perspectives: (a) the perspective of the theory of distributed algorithms, (b) the perspective of the theory of random processes, (c) the perspective of the field of descriptive combinatorics. \nAlthough different communities in these three fields ask very similar questions, no systematic connections between them were known, until the insightful work of Bernshteyn \\cite{Bernshteyn2021LLL} who applied results from the field (a) to the field (c). In this work, our most important results follow by applying techniques from the field (a) to (b), but perhaps more importantly, we present all three different perspectives as a part of one common theory (see \\cref{fig:big_picture_grids}). \n\nIn this introductory section, we first quickly present the three fields and define the object of interest, local problems, a class of problems that contains basic problems like vertex or edge coloring, matchings, and others. \nThen, we briefly present our results, most of which can be seen as inclusions between different classes of local problems shown in \\cref{fig:big_picture_grids}. \n\n\\paragraph{Distributed Computing}\n\nThe $\\mathsf{LOCAL}$ model of computing \\cite{linial1987LOCAL} was motivated by understanding distributed algorithms in huge networks. \nAs an example, consider the network of all wifi routers, where two routers are connected if they are close enough to exchange messages. In this case, they should better communicate with user devices on different channels, so as to avoid interference. In the graph-theoretic language, we want to color the network. Even if the maximum degree is $\\Delta$ and we want to color it with $\\Delta+1$ colors, the problem remains nontrivial, because the decision of each node should be done after few communication rounds with its neighbors. \n\nThe $\\mathsf{LOCAL}$ model formalizes this setup: we have a large network, with each node knowing its size, $n$, and perhaps some other parameter like the maximum degree $\\Delta$. In case of randomized algorithms, each node has an access to a random string, while in case of deterministic algorithms, each node starts with a unique identifier from a range of size polynomial in $n$. \nIn one round, each node can exchange any message with its neighbors and can perform an arbitrary computation. We want to find a solution to a problem in as few communication rounds as possible. \nImportantly, there is an equivalent view of $t$-rounds $\\mathsf{LOCAL}$ algorithms: such an algorithm is simply a function that maps $t$-hop neighbourhoods to the final output. An algorithm is correct if and only if applying this function to each node solves the problem. \n\nThere is a rich theory of distributed algorithms and the local complexity of many problems is understood. One example: if the input graph is the $d$-dimensional torus (think of the grid $\\mathbb{Z}^d$ and identify nodes with the same remainder after dividing the coordinate with $n^{1\/d}$), a simple picture emerges. Any local problem (we define those formally later) is solvable with local complexity $O(1)$, $\\Theta(\\log^* n)$, or we need to see the whole graph to solve it, i.e., the local complexity is $\\Theta(n^{1\/d})$ (see \\cref{thm:classification_of_local_problems_on_grids}). For example, a proper coloring of nodes with $4$ colors has local complexity $\\Theta(\\log^* n)$, while coloring with $3$ colors has already complexity $\\Theta(n^{1\/d})$. \n\n\\paragraph{Finitary factors of iid processes}\n\\todo{zeptat se Spinky}\nConsider the following example of the so-called anti-ferromagnetic Ising model. \nImagine atoms arranged in a grid $\\mathbb{Z}^3$. Each atom can be in one of two states and there is a preference between neighboring atoms to be in a different state. We can formalize this by defining a potential function $V(x,y)$ indicating whether $x$ and $y$ are equal. \nThe Ising model models the behavior of atoms in the grid as follows: for the potential $V$ and inverse temperature $\\beta$, if we look at the atoms, we find them in a state $\\Lambda$ with probability proportional to $\\exp(-\\beta \\sum_{u \\sim v \\in E(\\mathbb{Z}^d)} V(\\Lambda(u), \\Lambda(v)))$. If $\\beta$ is close to zero, we sample essentially from the uniform distribution. If $\\beta$ is infinite, we sample one of the two $2$-colorings of the grid. \n\nTo understand better the behavior of phase transitions, one can ask the following question. For what values of $\\beta$ can we think of the distribution $\\mu$ over states from the Ising model as follows: there is some underlying random variable $X(u)$ at every node $u$, with these variables being independent and identically distributed. We can think of $\\mu$ as being generated by a procedure, where each node explores the random variables in its neighborhood until it decides on its state, which happens with probability $1$. If this view of the underlying distribution is possible, we say it is a finitary factor of an iid process (ffiid). \nIntuitively, being an ffiid justifies the intuition that there is a correlation decay between different sites of the grid. To understand ffiids, we try to analyze solutions to which local problems can be sampled from an ffiid distribution (2-coloring of the grid is not among them). \n\nAlthough this definition was raised in a different context then distributed computing, it is actually almost identical to the definition of a distributed algorithm! The differences are two. First, the ffiid procedure does not know the value of $n$ since, in fact, it makes most sense to consider it in infinite graphs (like the infinite grid $\\mathbb{Z}^d$). Second, the natural measure of the complexity is different: it is the speed of the decay of the random variable corresponding to the time until the local algorithm finds the solution. \nNext, note that the algorithm has to return a correct solution with probability $1$, while the usual randomized local algorithm can err with a small probability. We call ffiids \\emph{oblivious local algorithms} to keep the language of distributed computing (see \\cref{sec:preliminaries}).\nWe define the class $\\mathsf{TAIL}(f(r))$ as the class of problems solvable by an oblivious local algorithm if the probability of the needed radius being bigger than $r$ is bounded by $1\/f(r)$ (see \\cref{fig:big_picture_grids}). \n\n\\paragraph{Descriptive Combinatorics}\n\nA famous Banach-Tarski theorem states that one can split a three-dimensional unit volume ball into five pieces, translate and rotate each piece and end up with two unit volume balls. \nThis theorem serves as an introductory warning in the beginner measure theory courses: not every subset of $\\mathbb{R}^3$ can be measurable.\nSimilarly surprising was when in 1990 Laczkovich answered the famous Circle Squaring Problem of Tarski\\todo{cite laczkovich}: A square of unit area can be decomposed into finitely many pieces such that one can translate each of them to get a circle of unit area.\nUnlike in Banach-Tarski paradox, this result was improved in recent years to make the pieces measurable in various sense \\cite{OlegCircle, Circle, JordanCircle}.\n\n\nThis theorem and its subsequent strengthenings are the highlights of a field nowadays called descriptive combinatorics \\cite{kechris_marks2016descriptive_comb_survey,pikhurko2021descriptive_comb_survey} that has close connections to distributed computing as was shown in an insightful paper by Bernshteyn \\cite{Bernshteyn2021LLL}. \nGoing back to the Circle Squaring Problem, the proofs of its versions first choose a clever set of constantly many translations $\\tau_1, \\dots, \\tau_k$. Then, one constructs a bipartite graph between the vertices of the square and the cycle and connects two vertices $x, y$ if there is a translation $\\tau_i$ mapping the point $x$ in the square to the point $y$ in the cycle. The problem can now be formulated as a perfect matching problem and this is how combinatorics enters the picture. In fact, versions of a matching or flow problems are then solved on the grid graph $\\mathbb{Z}^d$ using a trick called \\emph{toast construction} also used in the area of the study of ffiids \\cite{HolroydSchrammWilson2017FinitaryColoring,ArankaFeriLaci,Spinka}. \n\nWe discuss the descriptive combinatorics classes in a more detail in the expository \\cref{sec:OtherClasses} but we will not need them in the main body. Let us now give an intuitive definition of the class $\\mathsf{BOREL}$. \nBy a result of Chang and Pettie \\cite{chang2016exp_separation}, we know that the maximal independent set problem is complete for the class $\\mathsf{LOCAL}(O(\\log^* n))$ in the sense that this problem can be solved in $O(\\log^* n)$ rounds and any problem in that class can be solved in constant number of rounds if we are given access to an oracle computing MIS (we make this precise in \\cref{rem:chang_pettie_decomposition}). Imagine now that the input graph is infinite and our local algorithm has access to MIS oracle and local computations but now both can be used countably many times instead of just once or, equivalently, fixed number of times as is the case with the class $\\mathsf{LOCAL}(O(\\log^* n))$. That is, we can think of an algorithm that constructs MIS of larger and larger powers of the input graph and uses this infinite sequence to solve an LCL. Those constructions are in the class $\\mathsf{BOREL}$ from \\cref{fig:big_picture_grids}. In fact, the exact relationship between this definition and full power of $\\mathsf{BOREL}$ is not clear.\nHowever, it should give an idea about the difference between descriptive combinatorics and distributed computing -- from the descriptive combinatorics perspective, reasonable classes are closed under repeating basic operations countably many times.\n\n\n\n\n\\paragraph{Locally Checkable Problems on Grids as a ``Rosetta Stone''}\n\nLocally checkable problems (LCLs) are graph problems where a correctness of a solution can be checked locally. That is, there is a set of local constraints such that if they are satisfied at every node, the solution to a problem is correct. \nAn examples of such constraints include vertex coloring, edge coloring, or perfect matching. We also allow inputs on each node, so the list coloring is also an example of LCL (given that the total number of colors in all lists is finite). One should think of this class as an analogue of the class NP from complexity theory. \n\nIn this paper, we study locally checkable problems on grid graphs in the three settings introduced above. This approach reveals number of connections. For example, in \\cref{fig:big_picture_grids}, we can see that the two classes of local complexities $\\mathsf{LOCAL}(O(1))$ and $\\mathsf{LOCAL}(O(\\log^* n))$ are equivalent to different natural classes from the perspective of factors of iid and descriptive combinatorics. On the other hand, those two worlds are richer. Many of our discussions in fact apply to graphs of bounded growth, but the grids serve as the simplest, but still quite nontrivial model in which one can try to understand all three settings together (the case of lines is understood quite well \\cite{grebik_rozhon2021LCL_on_paths} and we study aspects of the connection for trees in a parallel paper \\cite{brandt_chang_grebik_grunau_rozhon_vidnyaszky2021LCLs_on_trees_descriptive}). \n\n\\begin{landscape}\n\\begin{figure}\n \\centering\n \\includegraphics[width = 1.5\\textwidth]{fig\/big_picture_grids.png}\n \\caption{The big picture of local complexity classes on grids from the perspective of distributed algorithms, factors of iid, and descriptive combinatorics. Canonical example of each class is on the left side. }\n \\label{fig:big_picture_grids}\n\\end{figure}\n\\end{landscape}\n\n\n\\subsection{Our Contribution }\n\nOur contribution can be mostly summed up as several new arrows in \\cref{fig:big_picture_grids}. In general, this clarifies several basic questions regarding the connections of the three fields and helps to view all three fields as part of a general theory of locality. \n \n\n\nFor example, consider the following question from the paper of Brandt et al. \\cite{brandt_grids} that considered LCL problems on grids: \n\\begin{quote}\n\\emph{\nHowever, despite the fact that techniques seem to translate between finitary colorings and distributed complexity, it remains unclear how to directly translate results from one setting to the other in a black-box manner; for instance, can we derive the lower bound for 3-colouring from\nthe results of Holroyd et al., and does our complexity classification imply answers to the open questions they pose?\n}\n\\end{quote}\nWe show how to 'directly translate' results between the setting of distributed algorithms and finitary factors of iid in \\cref{subsec:preliminaries_tail,subsec:speedup_warmup}. This indeed answers two open questions of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n\nTo understand the connection between the world of factors of iids and descriptive combinatorics, we formalize the notion of toast construction used in both setups. We prove general results connecting the toast constructions with oblivious local algorithms and this enables us to understand the $\\mathsf{TAIL}$ complexity of a number of concrete problems like vertex $3$-coloring, edge $4$-coloring, perfect matchings and others. \n\nIn general, we hope that the concept of $\\mathsf{TAIL}$ complexities will find other usecases. We discuss in \\cref{subsec:congestLCA} how it relates to studied notions of oblivious algorithms, energy complexity, or its relevance to the $\\mathsf{LCA}$ model.\\todo{Oleg se ptal jestli uz ma tady vedet co je LCA a jestli to nekde neprehlid}\nIn fact, the $\\mathsf{TAIL}$ complexity approach was already considered e.g. in the celebrated MIS paper of \\cite{ghaffari2016MIS} (see the introduction of his paper) and the $\\mathsf{TAIL}$ bound on the MIS algorithm there may be the reason of its wide applicability. \n\nWe now list concrete results that we prove (or they follow from the work of others, see \\cref{fig:big_picture_grids}). \n\n\\paragraph{Classification Theorems for $\\mathsf{TAIL}$}\n\nWe prove several results that one can view as arrows in \\cref{fig:big_picture_grids}. Those can be viewed as an analogy to the line of work aiming for the classification of $\\mathsf{LOCAL}$ complexities discussed in \\cref{subsec:preliminaries_local}. \n\n\\begin{itemize}\n \\item $\\mathsf{TAIL}(\\mathrm{tower}(\\omega(r))) = \\mathsf{LOCAL}(O(1))$\\hspace*{1cm} We show that any oblivious local algorithm with tail decay faster than tower function can be in fact solved in $O(1)$ local complexity. This answers a question of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n \\item $\\mathsf{LOCAL}(O(\\log^* n)) = \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r))) = \\mathsf{CONTINUOUS}$\\hspace*{1cm} There are three natural definitions of classes of basic symmetry breaking problems in all three considered fields and they are the same \\cite{GJKS,HolroydSchrammWilson2017FinitaryColoring}. \n As a direct application one gets that from the three proofs that $3$-coloring of grids is not in $\\mathsf{LOCAL}(O(\\log^* n))$ \\cite{brandt_grids}, not in $ \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$ \\cite{HolroydSchrammWilson2017FinitaryColoring} and not in $ \\mathsf{CONTINUOUS}$ \\cite{GJKS} any one implies the others. These three classes are the same in a much more general setup (see \\cref{subsec:CONT}). \n \\item $\\mathsf{TAIL}(\\omega(r^{d-1})) = \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$\\hspace*{1cm} We prove that any problem that can be solved with an oblivious local algorithm with tail decay faster than $r^{d-1}$, can in fact be solved with a tower tail decay. This answers a question of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n \\item Similarly to the time hierarchy theorems of the $\\mathsf{LOCAL}$ model \\cite{chang2017time_hierarchy}, we study the range of possible $\\mathsf{TAIL}$ complexities and give examples of problems LCL problems with $\\mathsf{TAIL}$-complexities $\\Theta(r^{j - o(1)})$ for any $1 \\le j \\le d-1$. \n\\end{itemize}\n\n\\paragraph{$\\TOAST$ constructions}\nWe formally define $\\TOAST$ algorithms (we mentioned earlier that these constructions are used in descriptive combinatorics and the theory of random processes) and prove results about them. \nLet us first explain the toast constructions in a more detail. A $q$-toast in the infinite graph $\\mathbb{Z}^d$ is a collection $\\mathcal{D}$ of finite connected subsets of $\\mathbb{Z}^d$ that we call \\emph{pieces} such that (a) for any two nodes of the graph we can find a piece that contains both,\\todo{it is enough that every vertex is contained in a piece} (b) boundaries of any two different pieces are at least distance $q$ apart. \nThat is, imagine a collection of bigger and bigger pieces that form a laminar family (only inclusions, not intersections) that, in the limit, covers the whole infinite graph.\n\nThis structure can be built in the descriptive combinatorics setups \\cite{FirstToast,GJKS,Circle} but also with oblivious local algorithms \\cite{HolroydSchrammWilson2017FinitaryColoring,Spinka}. In fact, we show that one can construct the structure with $1\/r^{1-o(1)}$ $\\mathsf{TAIL}$ decay in \\cref{thm:toast_in_tail}. This is best possible up to the $o(1)$ loss in the exponent. \nIf one has an inductive procedure building the toast, we can try to solve an LCL problem during the construction. That is, whenever a new piece of the toast is constructed, we define how the solution of the problem looks like at this piece. This is what we call a $\\TOAST$ algorithm. \nThere is a randomized variant of the $\\TOAST$ algorithm and a lazy variant. In the lazy variant one can postpone the decision to color given piece, but it needs to be colored at some point in the future. \nNext, we prove several relations between the $\\TOAST$ classes (see \\cref{fig:big_picture_grids}). \n\n\\begin{itemize}\n \\item $\\TOAST \\subseteq \\mathsf{TAIL}(r^{1-o(1)})$\\hspace*{1cm} This result allows us to translate toast-algorithm constructions used in descriptive combinatorics to the $\\mathsf{TAIL}$-setting. This includes problems like vertex $3$-coloring (this answers a question of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}), edge $2d$-coloring, or perfect matching. \n \\item $\\RTOAST = \\TOAST$\\hspace*{1cm} This derandomization result for example implies on the first glance surprising inclusion $\\mathsf{FINDEP} \\subseteq \\mathsf{BOREL}$ -- these two classes are discussed in \\cref{sec:OtherClasses}. \n \\item $\\mathsf{TAIL} = \\RLTOAST$\\hspace*{1cm} Here, $\\mathsf{TAIL}$ is defined as the union of $\\mathsf{TAIL}(f)$ over all functions $f \\in \\omega(1)$. This result shows how the language of toast constructions can in fact characterize the whole hierarchy of constructions by oblivious local algorithms. \n \\item We show that the problem of $2d$ edge coloring of $\\mathbb{Z}^d$ is in $\\TOAST$, hence in $\\mathsf{BOREL}$ which answers the question of Gao et al \\cite{GJKS}. Independently, this was proven by \\cite{ArankaFeriLaci}. \n\\end{itemize}\n\n\\subsection{Roadmap}\n\nIn \\cref{sec:preliminaries} we recall the basic definitions regarding the $\\mathsf{LOCAL}$ model and explain the formal connection between finitary factors of iids. \nNext, in \\cref{sec:speedup_theorems} we prove the basic speedup theorems for $\\mathsf{TAIL}$. \nIn \\cref{sec:Toast} we define $\\TOAST$ algorithms and relate it to the $\\mathsf{TAIL}$ model. \nIn \\cref{sec:SpecificProblems} we discuss what our results mean for specific local problems and in \\cref{sec:Inter} we construct problems with $r^{j-o(1)}$ $\\mathsf{TAIL}$ complexities for $1 \\le j \\le d-1$. Finally, in \\cref{sec:OtherClasses} we discuss connection of our work to other classes of problems and finish with open questions in \\cref{sec:open_problems}. \n\n\n\n\n\n\\begin{comment}\nHonza - Introduction\n\nA \\emph{locally checkable problems (LCL) $\\Pi$ on a family of graphs $\\mathbf{G}$} is a procedure that checks whether a given ``candidate'' vertex coloring of $G\\in \\mathbf{G}$ satisfies a local constraints given by $\\Pi$\\footnote{\nFormally, $\\Pi$ is a triplet $\\Pi=(b,t,\\mathcal{P})$, where $b$ is a set of colors, $t\\in \\mathbb{N}$ is the locality of $\\Pi$ and $\\mathcal{P}$ is a set of allowed configurations around each vertex of locality at most $t$.}.\nMore concretely, given a graph $G=(V(G),E(G))\\in \\mathbf{G}$ and a ``candidate'' coloring $c:V(G)\\to b$, we say that $c$ is a \\emph{$\\Pi$-coloring} if for every $v\\in V(G)$ the restriction of $c$ to $t$-neighborhood of $v$ is in $\\mathcal{P}$, where $t$ is the locality of $\\Pi$ and $\\mathcal{P}$ is the set of constraints of $\\Pi$.\n\nIn this paper the family $\\mathbf{G}$ always refers to graphs that look locally like the standard Cayley graph of $\\mathbb{Z}^d$, $\\mathbb{Z}^d$, for some $d\\ge 1$.\nThis includes $\\mathbb{Z}^d$ as well as various finite $d$-dimensional tori with oriented edges that are labeled by the standard generators of $\\mathbb{Z}^d$.\nThe dimension $d$ is always assumed to be bigger than $1$, since for $d=1$ there is a full classification of LCL problems.\nWe omit mentioning ${\\bf G}$ and refer to all LCL problems on the family ${\\bf G}$ simply as \\emph{LCL problems (on $\n\\mathbb{Z}^d$)}.\nWhen we want to stress that a given LCL problem is on $\\mathbb{Z}^d$ we write $d$ in the superscript, i.e., $\\Pi^d$.\n\nWe study a complexity of LCL problems in various settings.\nIn the most abstract terms, a complexity of a given LCL problem $\\Pi$ is measured in terms of how difficult it is to \\emph{transform} a \\emph{given labeling} of a graph into a $\\Pi$-coloring.\nThe different contexts varies in what is considered to be an allowed starting labeling and what is allowed transformation.\nMost relevant for our investigation is the following probability notion.\nStart with a graph $G$ and assign to each vertex $v\\in V(G)$ a real number from $[0,1]$ uniformly and independently, i.e., \\emph{iid labeling} of $G$.\nA function $F$ that transforms these iid labels in a measurable and equivaraint way into a different coloring is called \\emph{factor of iid (fiid)}.\nHere equivariant refers to the group of all symmetries of $G$.\nIt is not hard to see that the group of all symmetries of $\\mathbb{Z}^d$ is the group $\\mathbb{Z}^d$ itself that acts by shift-translations.\nA fiid $F$ on $\\mathbb{Z}^d$ is called \\emph{finitary fiid (ffiid)} if for almost every iid labeling of $\\mathbb{Z}^d$ the color $F(x)({\\bf 0}_d)$ only depends on the labels of $x$ on some finite neighborhood of ${\\bf 0}_d$.\nThat is to say, if $x$ and $x'$ agrees on this neighborhood, then $F(x)({\\bf 0}_d)=F(x')({\\bf 0}_d)$.\nGiven ffiid $F$, we write $R_\\mathcal{A}$ for the random variable that codes minimum diameter of such neighborhood and we refer to it as the coding radius of $F$.\n\nWe say that a ffiid $F$ \\emph{solves an LCL problem $\\Pi$} if it produces a $\\Pi$-coloring for almost every iid labeling of $\\mathbb{Z}^d$.\nan LCL problem $\\Pi$ is in the class $\\mathsf{TAIL}$ if $\\Pi$ admits a ffiid solution.\nFollowing Holroyd, Schramm and Wilson \\cite{HolroydSchrammWilson2017FinitaryColoring}, the complexity of LCL problems in $\\mathsf{TAIL}$ can be measured on a finer scale by the properties of the coding radius $R_\\mathcal{A}$, where $F$ runs over all ffiid's that solve the given problem.\nSimilarly, we define when fiid $F$ solves $\\Pi$ and the class $\\mathsf{fiid}$.\n\nBefore we move to distributed algorithm we recall what is known about the notorious class of LCL problems that are the \\emph{proper vertex $k$-colorings of $\\mathbb{Z}^d$}, $\\Pi^d_{\\chi,k}$, where $k>1$, in the context of ffiid.\nIt is known that if $d>1$ and $k>3$, then $\\Pi^d_{\\chi,k}$ admits ffiid solution with a tower tail decay, see Holroyd et.al~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\nIf $d>0$ and $k=2$, then $\\Pi^d_{\\chi,2}$ is not even in $\\mathsf{fiid}$.\nThe most interesting case is when $d>1$ and $k=3$.\nIt was shown by Holroyd et al~\\cite{HolroydSchrammWilson2017FinitaryColoring} that in that case $\\Pi^d_{\\chi,3}$ is in the class $\\mathsf{TAIL}$ and every ffiid that solves $\\Pi^d_{\\chi,3}$ must have infinite the $2$-moment of its coding radius. \nMoreover, they found a ffiid that solve $\\Pi^d_{\\chi,3}$ and has finite some moments of its coding radius.\nHere we strengthen their result by showing that ${\\bf M}(\\Pi^d_{\\chi,3})=1$ and every ffiid solution of $\\Pi^d_{\\chi,3}$ has infinite the $1$-moment of its coding radius.\n\nIn the context of deterministic distributed computing the setting is the following.\nStart with finite graph $G$ on $n$-vertices (in our case $n$ is big enough and $G$ is a torus) and assign to every vertex of $G$ its unique identifier from $[n]$.\\footnote{In generality, the identifier could be assigned from some domain that has size polynomial in $n$.}\nA \\emph{local algorithm $\\mathcal{A}_n$ of locality $t_n$} is a rule that computes color of a vertex $v$ based only on its $t$-neighborhood and the unique identifiers in this neighborhood.\nWe say that $\\mathcal{A}_n$ \\emph{solves} $\\Pi$ if for every (admissible)\\footnote{As a torus $G$ needs to be wider than the locality of $\\mathcal{A}_n$} graph $G$ on $n$-vertices and any assignment of unique identifiers to $G$ the rule $\\mathcal{A}_n$ produces a $\\Pi$-coloring.\nWe say that $\\Pi$ is in the class $\\mathsf{LOCAL}(O(\\log^*(n)))$ if there is a sequence of local rules \\todo{co je to local rule?}$\\{\\mathcal{A}_n,t_n\\}_{n\\in \\mathbb{N}}$ such that $\\mathcal{A}_n$ has locality $t_n$, solves $\\Pi$ for every $n\\in\\mathbb{N}$ and $n\\mapsto t_n$ is in the class $O(\\log^*(n))$.\nThe reason why we picked $\\log^*(n)$ is that there is a complete classification of LCL problems on oriented labeled grids.\nRoughly speaking, an LCL problem $\\Pi$ is either \\emph{trivial}, i.e., there is a single color that produces $\\Pi$-coloring, \\emph{local}, i.e., there is an algorithm that needs locality $\\Theta(\\log^*(n))$, or \\emph{global}, i.e., every local algorithm that solves $\\Pi$ needs to see essentially the whole graph.\\footnote{Same classification holds in the context of randomized distributed algorithms.}\n\nWe use this classification to provide several speed-up results in the context of ffiid solutions.\nWe note that the connection between distributed computing and probability setting was previously observed by several authors.\nFor example, it follows easily from the result of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring} that local coloring problems that are not global in the distributed setting admit ffiid solution with tower tail decay.\nIn fact, Brandt et al. \\cite{brandt_grids} asked:\n\\begin{quote}\n\\emph{\nHowever, despite the fact that techniques seem to translate between finitary colourings\nand distributed complexity, it remains unclear how to directly translate results from one setting to\nthe other in a black-box manner; for instance, can we derive the lower bound for 3-colouring from\nthe results of Holroyd et al. [24], and does our complexity classification imply answers to the open\nquestions they pose?\n}\n\\end{quote}\nAs one of our main result we address their question.\n\nThe most abstract setting that we consider is Borel combinatorics.\nIn this context one studies graphs that additionally posses a completely metrizable and separable topological structure (or a standard Borel structure) on its vertex set.\nIt follows from the work of Seward and Tucker-Drob \\cite{ST} and Gao, Jackson, Krohne and Seward \\cite{GJKS} that the canonical object to investigate is the graph structure on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$, the free part of the shift action $\\mathbb{Z}^d\\curvearrowright 2^{\\mathbb{Z}^d}$, induced by the standard generating set of $\\mathbb{Z}^d$. \nThe topological and Borel structure on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ is the one inherited from the product topology on $2^{\\mathbb{Z}^d}$.\nWe say that an LCL problem $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ if there is a continuous and equivariant map from $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ into the space of $\\Pi$-colorings.\nThe class $\\mathsf{BOREL}$ is defined similarly, only the requirement on the map to be continuous is relaxed to be merely Borel measurable.\nBernshteyn \\cite{Bernshteyn2021LLL} proved that LCL problems that are not global in the distributed setting admit continuous and therefore Borel solution.\nIn another words, he showed that the class $\\mathsf{LOCAL}(O(\\log^*(n)))$ is contained in $\\mathsf{CONTINUOUS}$.\nIt is not hard to show that the other inclusion follows from the Twelve Tile Theorem of Gao et al \\cite{GJKS}, i.e., the class $\\mathsf{CONTINUOUS}$ in fact coincide with the class $\\mathsf{LOCAL}(O(\\log^*(n)))$.\nHere we investigate in greater detail what is the connection between existence of a Borel solution and a tail behavior of ffiid.\nIn fact all known Borel constructions uses the following structure that can be rephrased without mentioning topology or Borel measurability.\n\nA \\emph{$q$-toast on $\\mathbb{Z}^d$} is a collection $\\mathcal{D}$ of finite connected subsets of $\\mathbb{Z}^d$ such that\n\\begin{itemize}\n \\item for every $g,h\\in \\mathbb{Z}^d$ there is $D\\in\\mathcal{D}$ such that $g,h\\in D$,\n \\item if $D,E\\in \\mathcal{D}$ and $d(\\partial D,\\partial E)\\le q$, then $D=E$.\n\\end{itemize}\nWe say that an LCL problem $\\Pi$ is in the class $\\TOAST$ if there is $q\\in \\mathbb{N}$ and an equivariant rule $\\Omega$ that extends partial colorings such that if we apply $\\Omega$ on every $q$-toast $\\mathcal{D}$ inductively on the depth, then we produce a $\\Pi$-coloring.\nIt is not known whether the classes $\\mathsf{BOREL}$ and $\\TOAST$ coincide, however, it follows from the work of Gao et al.~\\cite{GJKS2} that $\\TOAST\\subseteq \\mathsf{BOREL}$.\nThis is proved by constructing a toast structure in a Borel way on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$, see Marks and Unger~\\cite{Circle} for application and proof of this fact.\nThe specific problems that belong to the class $\\TOAST$ include $3$-vertex coloring, perfect matching, $2d$-edge coloring, Schreier decoration (for non-oriented grids), some rectangular tilings, space filling curve etc.\nThese results were proved by many people in slightly different language, \\cite{GJKS2, ArankaFeriLaci, spencer_personal}.\nWe utilize here all the constructions and show that these all belong to $\\TOAST$.\n\nWe connect the classes $\\TOAST$ and $\\mathsf{TAIL}$ as follows.\nWe show that for every $q\\in \\mathbb{N}$ and $\\epsilon>0$ there is a ffiid $F$ that produces $q$-toast that has the $(1-\\epsilon)$-moment finite.\nA slightly stronger result implies that, in fact, ${\\bf M}(\\Pi)\\ge 1$, whenever an LCL problem $\\Pi$ is in the class $\\TOAST$.\nThis shows that ${\\bf M}(\\Pi)\\ge 1$ for all the problems mentioned above.\nWe note that in the special case when $d=2$ and the corresponding LCL problem $\\Pi$ does not admit ffiid solution with a tower tail decay, we deduce that ${\\bf M}(\\Pi)=1$.\nThis follows from our speed-up result in the context of ffiid.\n\nTraditionally there are two more classes that are investigated in Borel combinatorics, $\\mathsf{BAIRE}$ and $\\mathsf{MEASURE}$.\nTo define the class $\\mathsf{BAIRE}$ we replace the topological requirement on continuity by \\emph{Baire measurability}, i.e., we require a solution to be continuous only on a \\emph{comeager} set, see the work of Bernshteyn \\cite{Bernshteyn2021LLL}.\nThe class $\\mathsf{MEASURE}$ generalizes the setting of fiid.\nThe goal here is to find a Borel solution on a conull set for a given Borel graph that is endowed with a Borel probability measure.\nThis direction has been recently investigated by Bernshteyn \\cite{Bernshteyn2021LLL} for general graphs, where he proved a measurable version of Lovasz's Local Lemma.\nSee \\cref{fig:big_picture_grids} for all the relations that are known to us.\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{fig\/big_picture_grids.png}\n \\caption{Big picture of lcl for $\\mathbb{Z}^d$}\n \n\\end{figure}\n\n\n\n\n\n\n\n\\subsection{The Results}\n\nIn this subsection we formulate our main results.\nThe formulation uses mostly the language of ffiid because the tail behavior of the coding radius allows to compare problems on a quantitative and, as it turns out, rich scale.\nOne should always re-interpret the result with the help of Figure~\\ref{fig:big_picture_grids} to different contexts.\nIronically enough, the techniques that are most valuable when investigating ffiid solutions come either from local algorithms or Borel combinatorics, where such a rich and quantitative distinction is not possible.\nRecall that in the distributed algorithms there is a trichotomy of LCL problems, i.e., trivial, local or global, and in Borel combinatorics we distinguish mainly between trivial, continuous, Borel, and global problems.\n\n\nWe start with results that use ideas from distributed algorithms.\nFirst, we prove that LCL problems that are not global in distributed algorithms are precisely problems that admit ffiid solution with a tower tail decay.\n\n\\begin{theorem*}\nLet $d\\in \\mathbb{N}$ and $\\Pi$ be an LCL problem on $\\mathbb{Z}^d$.\nThen $\\Pi$ is not global in the local world, i.e., there is a deterministic algorithm in the local model that runs in $O(\\log^*(n))$ rounds, if and only if there is an ffiid solution $F$ of $\\Pi$ such that\n$$\\P(R_\\mathcal{A}>r)\\le \\frac{1}{tower(Cr)}$$\nfor some $C>0$ and large enough $r\\in \\mathbb{N}$.\n\\end{theorem*}\n\nThe translation between ffiid and distributed algorithms allows to deduce the following result that answers problem (ii) of Holroyd at el.~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\n\n\\begin{theorem*}\nLet $d\\in \\mathbb{N}$ and $\\Pi$ be an LCL problem on $\\mathbb{Z}^d$.\nIf $\\Pi$ admits an ffiid solution $F$ that has faster tail decay than any tower function, then $\\Pi$ is trivial, i.e., there is a color such that a constant coloring with this color is a correct solution.\n\\end{theorem*}\n\nAs a last application of ideas from the distributed algorithms, we show the following speed-up theorem for ffiid.\n\n\\begin{theorem*}\nLet $d,t\\in \\mathbb{N}$.\nThere is $C_{d,t}>0$ such that if an LCL problem $\\Pi$ admits an ffiid solution $F$ such that\n$$\\P(R_\\mathcal{A}>r)\\le C_{d,t}r^{d-1},$$\nthen there is a deterministic algorithm in the local model that runs in $O(\\log^*(n))$ rounds and solves $\\Pi$.\n\\end{theorem*}\n\nTo understand better LCL problems that are global in the distributed algorithms or admit no continuous solution in the context of Borel combinatorics we introduce the following numerical invariant.\nThe inspiration comes from the paper of Holroyd et al.~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\n\n\\begin{definition*}\nLet $\\Pi$ be an LCL on $\\mathbb{Z}^d$.\nWe define ${\\bf M}(\\Pi)\\in [0,+\\infty]$ to be the supremum over all $\\alpha\\in [0,+\\infty)$ such that there is an ffiid solution $F$ of $\\Pi$ such that $R_F$ has finite the $\\alpha$-moment.\n\\end{definition*}\n\nCombination of previous results immediately implies the following strengthening of the upper bound on the necessary infinite moment of ffiid solution of the proper vertex $3$-coloring problem, $\\Pi^d_{\\chi,3}$, from Holroyd et al.~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\nMoreover, this answers their problem (iii).\n\n\\begin{theorem*}\nLet $d\\in \\mathbb{N}$ and $\\Pi$ be an LCL problem on $\\mathbb{Z}^d$.\nIf ${\\bf M}(\\Pi)>d-1$, then $\\Pi$ admits an ffiid solution with tower tail decay.\n\\end{theorem*}\n\nAs a first application of ideas from Borel combinatorics, i.e., the toast structure, we compute the optimal tail decay for $\\Pi^d_{\\chi,3}$, thus answering a problem (i) of Holroyd et al.~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\n\n\\begin{theorem*}\nLet $d>1$ and $\\Pi^d_{\\chi,3}$ be the proper vertex $3$-coloring problem.\nThen ${\\bf M}(\\Pi^d_{\\chi,3})=1$.\n\\end{theorem*}\n\nA high level explanation of a toast construction in the context of ffiid says that if $r\\in \\mathbb{N}$ is given, then the fraction of vertices not covered by first $r$-layers of the toast is at most $\\frac{1}{r}$.\nSimilar construction that uses only $1$-dimensional skeleton of some rectangular tiling of $\\mathbb{Z}^d$ improve this upper bound to $\\frac{1}{r^{d-1}}$.\nWe show that this type of structure is enough to find a perfect matching in a bigger graph then $\\mathbb{Z}^d$, a graph where we moreover add diagonal edges in every $2$-dimensional hyperplane.\nThis yields an existence of intermediate problems from the point of view of ${\\bf M}$.\n\n\\begin{theorem*}\nLet $2\\le \\ell\\le d\\in \\mathbb{N}$.\nThen there is an LCL problem $\\Pi$ on $\\mathbb{Z}^d$ that satisfies ${\\bf M}(\\Pi)=\\ell-1$.\n\\end{theorem*}\n\nAs a last result we summarize what is known about specific problems.\nSince we work mainly with oriented grids we state everything in this context, however, some of the results can be deduced for non-oriented grids as well, see Section~\\ref{sec:SpecificProblems}.\nIn particular, we answer Question~6.5 of Gao et al.~\\cite{GJKS} about Borel edge chroamtic number of $\\operatorname{Free}(2^{\\mathbb{Z}^d})$.\n\n\\begin{theorem*}\nLet $d\\ge 2$ and $\\Pi$ be any of the following LCL problems:\n\\begin{enumerate}\n \\item proper vertex $3$-coloring, \\cite{GJKS}, \\cite{spencer_personal},\n \\item proper edge $2d$-coloring,\n \\item rectangular tiling with rectangles from some set $\\mathcal{R}$ that satisfies \\cref{eq:GCD}, \\cite{Unger},\n \\item perfect matching, \\cite{GJKS}.\n\\end{enumerate}\nThen $\\Pi$ is in the class $\\TOAST$.\nConsequently, $\\Pi$ is in the class $\\mathsf{BOREL}$ and satisfies $1\\le {\\bf M}(\\Pi)$.\n\\end{theorem*}\n\n\n\nThe paper is organized as follows...\n\n\n\\end{comment}\n\n\n\n\n\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\n\nLet $d\\in \\mathbb{N}$ be fixed.\nThe class of graphs that we consider in this work contains mostly the infinite grid $\\mathbb{Z}^d$, that is $\\mathbb{Z}^d$ with the Cayley graph structure given by the standard set of generators \n$$S=\\{(1,0,\\dots, 0),(0,1,\\dots, 0), \\dots (0,\\dots,0,1)\\}. $$\nWe also used the finite variant of the graph: the torus of side length $\\Theta(n^{1\/d})$ with edges oriented and labeled by $S$.\nUsually, when we talk about a graph $G$ we mean one of the graphs above and it follows from the context whether we mean finite or infinite graph. \n\nGiven a graph $G$, we denote as $V(G)$ the set of vertices, as $E(G)$ the set of edges and as $d_G$ the \\emph{graph distance} on $G$, i.e., $d_G(v,w)$ is the minimal number of edges that connects $v$ and $w$.\nWe write $\\mathcal{B}_G(v,r)$ for the \\emph{ball of radius $r\\in \\mathbb{N}$} around $v\\in V(G)$. When $G$ is the infinite grid $\\mathbb{Z}^d$, then we leave the subscripts blank.\nA \\emph{rooted graph} is a pair $(G,v)$ where $G$ is a graph and $v\\in V(G)$.\nWe denote as $[G,v]$ the \\emph{isomorphism type} of the rooted graph $(G,v)$, where $(G,v)$ and $(G',v')$ are isomorphic if there is a bijection $\\varphi:V(G)\\to V(G')$ that sends $v$ to $v'$ and preserves edge orientation and labeling.\n\nLet $D$ be a set, $G$ a graph and $c:V(G)\\to D$ be a labeling of vertices.\nThen we write $G_c=(G,c)$ for the pair graph + labeling.\nSimilarly we denote as $(G_c,v)$ the rooted graph $(G,v)$ with the labeling $c$, $[G_c,v]$ its isomorphism class and $\\mathcal{B}_{G_c}(v,r)$ the labeled ball of radius $r\\in \\mathbb{N}$ around $v$, where $v\\in V(G)$.\nOften we do not mention the labels and tacitly assume that they are understood from the context.\nAlso, the label set might depend on the size of of the vertex set, e.g., unique labels from $[|V(G)|]=\\{1,\\dots,|V(G)|\\}$ etc.\n\n\\begin{comment}\n\nWe always consider the group $\\mathbb{Z}^d$ with the standard set of generators $S$, i.e.,\n$$S=\\{(1,0,\\dots, 0),(0,1,\\dots, 0), \\dots (0,\\dots,0,1)\\}.$$\nThe \\emph{Cayley graph} $\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right):=\\operatorname{Cay}\\left(\\mathbb{Z}^d,S\\right)$ of $\\mathbb{Z}^d$ is then defined on the vertex set $\\mathbb{Z}^d$ with oriented and $S$-labeled edges \n$$\\left\\{(g,s+g):s\\in S, g\\in \\mathbb{Z}^d\\right\\}=E\\left(\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)\\right).$$\nWe denote the identity in $\\mathbb{Z}^d$ as ${\\bf 0}_d$. \nWhen we talk about $\\mathbb{Z}^d$ next, we mean formally the appropriate Cayley graph with labelled edges, that is, we think of the graph together with its consistent orientation. \n\n\nA \\emph{graph} $G$ is a pair $(V(G),E(G))$, where $V(G)$ is a vertex set and $E(G)$ consists of oriented and $S$-labeled edges.\nGiven a graph $G$, we denote as $d_G$ the \\emph{graph distance} on $G$, i.e., $d_G(v,w)$ is the minimal number of edges that connects $v$ and $w$.\nWe write $\\mathcal{B}_G(v,r)$ for the \\emph{ball of radius $r\\in \\mathbb{N}$} around $v\\in V(G)$. Usually, $G$ is just the grid $\\mathbb{Z}^d$, then we leave the subscript blank. \nA \\emph{rooted graph} is a pair $(G,v)$ where $G$ is a graph and $v\\in V(G)$.\nWe denote as $[G,v]$ the isomorphism type of the rooted graph $(G,v)$, where $(G,v)$ and $(G',v')$ are isomorphic if there is a bijection $\\varphi:V(G)\\to V(G')$ that sends $v$ to $v'$ and preserves edge orientation and labeling.\n\n\\todo{vsechny strany toru maji delku n na 1\/d; zjevne vsechny problemy maji lokalitu nejvyse n na 1\/d}\n\n\\end{comment}\n\n\n\\begin{comment}\nWe write $\\operatorname{\\bf FG}^{S}$ for the set of all graphs that look in their $2$-neighborhoods as $\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)$, i.e., a graph $G=(V(G),E(G))$ is in $\\operatorname{\\bf FG}^{S}$ if $[\\mathcal{B}_G(v,2)]=\\left[\\mathcal{B}_{\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)}({\\bf 0}_d,2)\\right]$ for every $v\\in V(G)$.\nNote that $\\mathbb{Z}^d\\in \\operatorname{\\bf FG}^{S}$ and there are finite graphs in $\\operatorname{\\bf FG}^{S}$, as an example take any torus that is big enough.\nWrite $\\operatorname{\\bf FG}_{\\bullet}^{S}$ for the rooted graphs of the form\n$$(\\mathcal{B}_G(v,r),v)$$\nwhere $G\\in \\operatorname{\\bf FG}^{S}$, $v\\in V(G)$ and $r\\in \\mathbb{N}$.\nWe say that a rooted graph $(G,v)\\in \\operatorname{\\bf FG}_{\\bullet}^{S}$ has \\emph{diameter at most $m\\in \\mathbb{N}$} if $(\\mathcal{B}_G(v,m),v)=(G,v)$.\nNote that, in particular, we have $\\mathcal{B}_{\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)}({\\bf 0}_d,r)\\in \\operatorname{\\bf FG}_{\\bullet}^{S}$ for every $r\\in \\mathbb{N}$.\n\nLet $D$ be a set, $G$ a graph and $c:V(G)\\to D$ be a labeling of vertices.\nThen we write $G_c=(G,c)$ for the pair graph + labeling.\nSimilarly we denote as $(G_c,v)$ the rooted graph $(G,v)$ with the labeling $c$, $[G_c,v]$ its isomorphism class and $\\mathcal{B}_{G_c}(v,r)$ the labeled ball of radius $r\\in \\mathbb{N}$ around $v$, where $v\\in V(G)$.\nOften the label set $D$ depends on the size of of the vertex set, e.g., unique labels from $[|V(G)|]=\\{1,\\dots,|V(G)|\\}$ etc.\n\\end{comment}\n\n\n\n\\subsection{LCLs on $\\mathbb{Z}^d$}\n\nOur goal is to understand locally checkable problems (LCL) on $\\mathbb{Z}^d$. \n\n\\begin{definition}[Locally checkable problems \\cite{naorstockmeyer}]\nA \\emph{locally checkable problems (LCL)} on $\\mathbb{Z}^d$ is a quadruple $\\Pi=(\\Sigma_{in}, \\Sigma_{out},t,\\mathcal{P})$ where $\\Sigma_{in}$ and $\\Sigma_{out}$ are finite sets, $t\\in \\mathbb{N}$ and $\\mathcal{P}$ is an assignment\n$$[H_c,w]\\mapsto \\mathcal{P}([H_c,w])\\in \\{0,1\\},$$\nwhere $(H,w)$ is a rooted grid of radius at most $t$ together with an input and output coloring $c: V(H)\\to \\Sigma_{in}\\times\\Sigma_{out}$. \n \nGiven a graph $G$ together with a coloring $c:V(G)\\to \\Sigma_{in}\\times\\Sigma_{out}$, we say that $c$ is a \\emph{$\\Pi$-coloring} if\n$$\\mathcal{P}([\\mathcal{B}_{G_c}(v,t),v])=1$$\nfor every $v\\in V(G)$.\n\\end{definition}\n\n\n\n\\begin{comment}\n\nClasses of functions\n\nBefore we proceed to definition of local algorithms and their complexity we recall here the relevant notions concerning functions from $\\mathbb{N}$ to $\\mathbb{N}$. \nImportantly, we often use asymptotic notation as usual in Computer Science, e.g., we define the class of functions $O(f(n))$ as the union over all functions $g:\\mathbb{N} \\rightarrow \\mathbb{N}$ such that there exists $C > 0$ such that for all $n$ we have $g(n) \\le C f(n)$. Other symbols $\\Omega, \\Theta, o, \\omega$ are defined analogously and their usage is analogous (see e.g. \\cite{?}). \n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{LOCAL model}\n\\label{subsec:preliminaries_local}\n\nHere we formally introduce the $\\mathsf{LOCAL}$ model for both deterministic and randomized algorithms and state the classification of LCLs on $\\mathbb{Z}^d$ from the $\\mathsf{LOCAL}$ point of view.\nInformally, a local algorithm $\\mathcal{A}$ of radius $r\\in \\mathbb{N}$ is a rule that takes a labeled graph $G_c$ and then re-labels each vertex $v\\in V(G)$ depending only on the isomorphism type of its neighborhood $[\\mathcal{B}_{G_c}(v,r),v]$ with a new label from some finite set $b$.\n\n\\paragraph{Local algorithms}\n\nHere, we formally define the complexity classes $\\mathsf{LOCAL}(\\mathcal{F})$ of deterministic local algorithms, where $\\mathcal{F}$ is a class of functions from $\\mathbb{N}$ to $\\mathbb{N}$. \nThe somewhat unintuitive part of the definition is that the algorithm takes as input \\emph{unique identifiers}. Usually, they are taken from some range of values that is polynomial in $n$, the number of vertices of the input graph. However, in most scenarios this is equivalent to the assumption that they are from range $\\{1,\\dots n\\}$, i.e., every identifier is used exactly once. We also use this definition here, for simplicity.\nIn all forthcoming definition we have some fixed LCL $\\Pi$ in mind.\n\n\\begin{definition}[A deterministic local algorithm for $\\mathbb{Z}^d$]\n\\label{def:local_algorithm\nLet $\\Sigma_{out}$ be a finite set.\nA \\emph{local (deterministic) algorithm $\\mathcal{A}$} is a sequence of pairs $\\{\\mathcal{A}_n,\\ell_n\\}_{n\\in\\mathbb{N}}$, where $\\ell_n\\in \\mathbb{N}$ and $\\mathcal{A}_n$ is an assignment, such that\n$$[\\mathcal{B}_{G_\\ell}(v,\\ell_n),v]\\mapsto \\mathcal{A}_n([\\mathcal{B}_{G_\\ell}(v,\\ell_n),v])\\in \\Sigma_{out}$$\nfor every $\\Sigma_{in}$-labeled graph $G$ (torus) on $n$ vertices, $v\\in V(G)$ and a bijection $\\ell:V(G)\\to [n]$.\nIn another words, $\\mathcal{A}_n$ is a function that is defined on isomorphism classes of rooted graphs of diameter at most $\\ell_n$ with injectively labeled vertices from the set $\\{1,\\dots, n\\}$ and with range $b$. \n\nGiven such a $G$ and a bijection $\\ell:V(G)\\to |V(G)|$, we define\n$$\\mathcal{A}(G_\\ell)(v):=\\mathcal{A}_{|V(G)|}([\\mathcal{B}_{G_\\ell}(v,\\ell_n),v])$$\nfor every $v\\in V(G)$.\n\\end{definition}\n\nNext we define the deterministic complexity of a given local coloring problem $\\Pi$.\n\n\\begin{definition}[Complexity of deterministic local algorithm for $\\mathbb{Z}^d$]\nWe say that a local algorithm $\\mathcal{A}=\\{\\mathcal{A}_n,\\ell_n\\}_{n\\in \\mathbb{N}}$ \\emph{solves} an LCL $\\Pi=(\\Sigma_{in},\\Sigma_{out},t,\\mathcal{P})$ if $\\mathcal{A}(G_l)$ is a $\\Pi$-coloring of $G$ for every finite $\\Sigma_{in}$-labeled graph $G$ and any bijection $\\ell:V(G)\\to |V(G)|$.\n\nWe say that $\\Pi$ is in the class $\\mathsf{LOCAL}(\\mathcal{F})$, where $\\mathcal{F}$ is a class of functions, if $n\\mapsto \\ell_n$ is in $\\mathcal{F}$.\n\\end{definition}\n\nA randomized local algorithm and its complexity is defined analogously but instead of labeling with unique identifiers each node has an iid random string and the algorithm succeeds with probability $1 - 1\/n$.\n\nCrucial for our investigation are so-called \\emph{oblivious local algorithms}.\nRoughly speaking an algorithm is \\emph{oblivious} if it does not have knowledge of $n$ (see \\cite{Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring} for details). \nSince we do not consider deterministic oblivious algorithms, by an oblivious local algorithm we mean an oblivious randomized local algorithm.\n\n\\begin{definition}[Oblivious local algorithm]\n\\label{def:uniform_local}\nA \\emph{oblivious local algorithm} $\\mathcal{A}$ is a pair $(\\mathcal{C},f)$, where $\\mathcal{C}$ is a (measurable) collection of rooted graphs labeled with $\\Sigma_{in}$ and real numbers from $[0,1]$, and $f: \\mathcal{C} \\rightarrow \\Sigma_{out}$ is a measurable assignment.\n\nWe define $\\mathcal{A}(G)$ to be the labeling computed by $\\mathcal{A}$ on $G$, where for a given node $u$ we have $\\mathcal{A}(G)(u)=g([B_G(u, t), u])$ where $t$ is the smallest integer such that $[B_G(u, t),u] \\in \\mathcal{C}$.\n\\end{definition}\n\nNote that since there is no $n$ in the definition of oblivious local algorithm, the algorithm is correct if it always finishes with probability $1$, while randomized algorithms can have small probability of error dependent on $n$. Hence, one should think about such algorithms as an analogue of Las Vegas algorithms, while the classical randomized local algorithms correspond to Monte Carlo algorithms. \nAn advantage of not knowing $n$ is that it is possible to run $\\mathcal{A}$ on, e.g., the infinite grid $\\mathbb{Z}^d$ instead of just a torus. Usually, these two models give the same results but for toast algorithms that we consider later, there is a difference: While we define the algorithms for an infinite grid, for example the $3$-coloring algorithm makes sense also for a finite torus, while the perfect matching algorithm does not make sense in the finite world for the simple reason that the input graph can have an odd number of vertices. \n\n\n\n\n\n\n\n\n\\begin{comment}\n\\paragraph{Randomized local algorithms}\\todo{merge with previous paragraph}\n\nThe main difference between deterministic and randomized model is that the input of the local rule in the randomized case are graphs with vertices labeled uniformly independently at random by elements of the interval $[0,1]$ and we require that the output coloring is correct everywhere with high probability.\nThe complexity of local algorithm is then defined as the locality it needs.\n\n\\begin{definition}[Randomized local algorithm for $\\mathbb{Z}^d$]\nLet $b$ be a set.\nWe say that $\\mathbb{A}=\\{\\mathbb{A}_n,\\ell_n\\}_{n\\in\\mathbb{N}}$ is a randomized local algorithm for the class $\\operatorname{\\bf FG}^{S}$ if $\\ell_n\\in \\mathbb{N}$ and \n$$[\\mathcal{B}_{G_l}(v,\\ell_n),v]\\mapsto \\mathbb{A}_n([\\mathcal{B}_{G_l}(v,\\ell_n),v])\\in b$$\nis a measurable assignment, where $G\\in \\operatorname{\\bf FG}^{S}$, $|V(G)|=n$, $v\\in V(G)$ and $\\ell:V(G)\\to [0,1]$.\nIn another words, $\\mathbb{A}_n$ is a measurable function that is defined on isomorphism classes of rooted graphs of diameter at most $\\ell_n$ with vertices labeled by $[0,1]$ and with range $b$. \n\nGiven $G\\in \\operatorname{\\bf FG}^{S}$ such that $n=|V(G)|$ and a map $\\ell:V(G)\\to [0,1]$, we define\n$$\\mathbb{A}(G_l)(v):=\\mathbb{A}_n([\\mathcal{B}_{G_l}(v,\\ell_n),v])$$\nfor every $v\\in V(G)$.\n\\end{definition}\n\nNext we define when a randomized algorithm solves an LCL $\\Pi$ and the randomized complexity classes.\n\n\\begin{definition}[Complexity of randomized local algorithm for $\\mathbb{Z}^d$]\nLet $\\Pi=(b,t,\\mathcal{P})$ be an LCL.\nWe say that a randomized algorithm $\\mathbb{A}=\\{\\mathbb{A}_n,\\ell_n\\}_{n\\in \\mathbb{N}}$ solves $\\Pi$ if \n$$\\lambda^{\\oplus n}\\left(\\left\\{\\ell\\in [0,1]^{V(G)}:\\mathbb{A}(G_l) \\ \\operatorname{is \\ a} \\ \\Pi\\operatorname{-coloring} \\right\\}\\right)>1-\\frac{1}{n}$$\nholds for every $n\\in \\mathbb{N}$ and $G\\in \\operatorname{\\bf FG}^{S}$ such that $|V(G)|=n$ and\n$$[\\mathcal{B}_{G}(v,\\ell_n+t),v]=\\left[\\mathcal{B}({\\bf 0}_d,\\ell_n+t),{\\bf 0}_d\\right]$$\nfor every $v\\in V(G)$.\n\nWe say that $\\Pi$ is in the class $\\mathsf{R-LOCAL}(\\mathcal{F})$, where $\\mathcal{F}$ is a class of functions, if $n\\mapsto \\ell_n$ is in $\\mathcal{F}$.\n\\end{definition}\n\\end{comment}\n\n\\begin{comment}\n\\label{sec:clocal_definition}\n\nThe last class of local algorithm that we introduce serves on one hand as a main technical tool for deterministic speed up tricks and on the other hand allows a smooth translation of local algorithm into the measurable setting.\nIntuitively, an algorithm is in the class $\\mathsf{C-LOCAL}$ if it only needs a fixed finite locality (not depending on the size of a graph) given some coloring of the vertices with fixed number of colors (not depending on the size of a graph).\n\nLet $s\\in \\mathbb{N}$ and $G$ be a graph.\nDenote as $G^s$ the $s$-power graph of $G$, i.e., $V(G^s)=V(G)$ and $(x,y)\\in E(G^s)$ if and only if $0 r),$$\nwhat we mean is the following. \nFirst, for a given rooted neighborhood $(G,v)$ with a fixed $\\Sigma_{in}$-labeling we ask what is the probability that, after labeling $G$ with real numbers uniformly at random, that $(G,v)$ is in $\\mathcal{C}$. \nSecond, we take the supremum over all such values over all $\\Sigma_{in}$-labels.\n\n\\begin{definition}[$\\mathsf{TAIL}(f)$]\n\\label{def:tail}\nWe say that an LCL $\\Pi$ is in the class $\\mathsf{TAIL}(f)$ for some function $f:\\mathbb{N}\\to \\mathbb{N}$ if there is an oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi$ such that\n$$\\P(R_\\mathcal{A} > r)\\le \\frac{1}{f(r)}$$\nfor every large enough $r\\in \\mathbb{N}$. \n\nWe say that $\\Pi \\in \\mathsf{TAIL}$ if there exists $f \\in \\omega(1)$ such that $\\Pi \\in \\mathsf{TAIL}(f)$. \nWe say that $\\Pi$ is solved by a \\emph{block factor} if there is an oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi$ such that for some $r_0$ we have $\\P(R_\\mathcal{A} > r_0) = 0$.\n\\end{definition}\n\nHere things are made simple because we think of the algorithm as being run on an infinite graph so nothing depends on $n$ in this complexity measure (while it is a crucial parameter in the classical local complexity definition). \n\nWe also use the asymptotic notation when talking about $\\mathsf{TAIL}$ complexities. For example, an important class $\\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$ is defined as the class of LCLs $\\Pi$ such that there is a constant $C\\in \\mathbb{N}$ and oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi$ that satisfies\n$$\\P(R_\\mathcal{A}> r)\\le \\frac{1}{\\mathrm{tower}(Cr)}$$\nwhere $\\mathrm{tower}(x)=\\exp\\circ\\dots\\circ \\exp(1)$ is $\\lfloor x\\rfloor$-many times iterated exponential.\n\nA coarser way to measure the complexity of a problem $\\Pi$ is to ask what moments of $R_\\mathcal{A}$ could be finite, when $\\mathcal{A}$ is an oblivious local algorithm.\nFor example, the first moment finite corresponds to the fact that average time until an algorithm finishes is defined (for finite graphs this is independent on the size of the graph). \nIn particular, we investigate the question what is the least infinite moment for any oblivious local algorithm to solve a given problem $\\Pi$, as this language was used in \\cite{HolroydSchrammWilson2017FinitaryColoring}. \nRecall that $R_\\mathcal{A}$ has a \\emph{finite $\\alpha$-moment}, where $\\alpha\\in [0,+\\infty)$, if\n$$\\mathbb{E}\\left(\\left(R_\\mathcal{A}\\right)^\\alpha\\right)=\\sum_{r \\in \\mathbb{N}} r^\\alpha \\cdot \\P(R_\\mathcal{A} = r) <+\\infty.$$\n\n\\begin{definition}\nLet $\\Pi$ be an LCL.\nWe define ${\\bf M}(\\Pi)\\in [0,+\\infty]$ to be the supremum over all $\\alpha\\in [0,+\\infty)$ such that there is an oblivious $\\mathcal{A}$ that solves $\\Pi$ and has the $\\alpha$-moment of $R_\\mathcal{A}$ finite.\n\\end{definition}\n\nThe $\\alpha$ moment being finite roughly corresponds to $\\Omega(r^{\\alpha})$ $\\mathsf{TAIL}$ complexity. In fact we now recall the following observation.\n\n\\begin{fact}\n\\label{fact:moment_implies_tail}\nSuppose that the coding radius of an oblivious local algorithm $\\mathcal{A}$ has the $\\alpha$-moment finite.\nThen for every $C>0$ we have\n$$\\P(R_\\mathcal{A} > r) \\le C\\frac{1}{r^\\alpha}$$\nfor large enough $r\\in \\mathbb{N}$.\n\\end{fact}\n\\begin{proof}\nWe prove the contrapositive. Suppose the tail of the coding radius $R$ satisfies $\\P(R > r_i) > C \\cdot \\frac{1}{r^\\alpha_i}$ for some $C > 0$ and an infinite sequence $(r_i)_{i\\in \\mathbb{N}}$. \nAs the sequence $\\P(R> r_i)$ goes to zero, we may pass to a subsequence such that for each $i\\in \\mathbb{N}$ we have $\\P(R > r_{i+1}) \\le \\frac{1}{2}\\P(R > r_i)$. We call the new subsequence still $(r_i)_{i\\in \\mathbb{N}}$. \nWe have \n\\begin{align*}\n\\mathbb{E}(R^\\alpha) &\\ge \\sum_{i \\in \\mathbb{N}} r_i^\\alpha \\cdot (\\P(R> r_i) - \\P(R > r_{i+1}))\\\\\n&\\ge \\sum_{i \\in \\mathbb{N}} r_i^\\alpha \\cdot \n\\frac{1}{2}\\P(R > r_i)\n\\ge \\sum_{i \\in \\mathbb{N}} r_i^\\alpha \\cdot \n\\frac{C}{2} \\cdot \\frac{1}{r_i^\\alpha}\n= +\\infty\n\\end{align*}\nwhich finishes the proof. \n\\end{proof}\n\nFinally, an important fact is that a small $\\mathsf{TAIL}$ complexity of an oblivious randomized local algorithm implies that it has small randomized complexity (on finite graphs). This is because if we run an oblivious algorithm with tail decay $1\/f(r)$ on an $n$-node graph and mark its run as a failure if the coding radius exceeds $f^{-1}(n^2)$, the failure at some node happens only with probability $1\/n$. This was formally proven in \\cite{brandt_grebik_grunau_rozhon2021classification_of_lcls_trees_and_grids} and here we only state the result. \n\n\\begin{lemma}\n\\label{lem:tail->local}\nLet $\\mathcal{A}$ be an oblivious local randomized algorithm with $\\mathsf{TAIL}$ complexity $f(r)$. Then the randomized complexity of $\\mathcal{A}$ is $O(f^{-1}(n^2))$. \n\\end{lemma}\n\nIn fact, one can prove that for $o(\\log n)$ local complexities the two notions are the same. We refer to the reader to the discussion in \\cite{brandt_grebik_grunau_rozhon2021classification_of_lcls_trees_and_grids} for more details. \n\n\\begin{theorem}[\\cite{brandt_grebik_grunau_rozhon2021classification_of_lcls_trees_and_grids}]\n\\label{thm:tail=uniform}\nLet $\\mathcal{A}$ be an oblivious local randomized algorithm with randomized complexity $f(n) = o(\\log n)$. Then this algorithm in fact never returns an incorrect solution and its tail complexity is $\\Omega(f^{-1}(r))$. \n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\\subsection{Subshifts of finite type}\n\\label{subsec:subshifts}\n\nHere we connect our definition of LCLs and oblivious local algorithms with the well-studied notions of subshifts of finite type and finitary factors of iid labelings.\n\nTo define subshifts of finite type we restrict ourselves to LCLs without input.\nThat is $\\Sigma_{in}$ is empty or contains only one element.\nMoreover, for convenience we set $b:=\\Sigma_{out}$.\nThen an LCL $\\Pi$ is of the form $\\Pi=(b,t,\\mathcal{P})$.\nWe define $X_\\Pi\\subseteq b^{\\mathbb{Z}^d}$ to be the space of all $\\Pi$-colorings.\nThat is we consider all possible labelings of $\\mathbb{Z}^d$ by elements from $b$ and then include in $X_{\\Pi}$ only those that satisfies the finite constraints $\\mathcal{P}$.\nViewing $\\mathbb{Z}^d$ as a group, there is a natural shift action of $\\mathbb{Z}^d$ on (the Cayley graph) $\\mathbb{Z}^d$.\nNamely, $g\\in \\mathbb{Z}^d$ shifts $h$ to $g+h$.\nThis action naturally ascend to an action $\\cdot $ on $b^{\\mathbb{Z}^d}$ as\n$$(g\\cdot x)(h)=x(g+h)$$\nwhere $x\\in b^{\\mathbb{Z}^d}$.\nIt is not hard to see (since $\\mathcal{P}$ is defined on isomorphism types and not rooted in any particular vertex) that $X_\\Pi$ is invariant under the shift map.\nThen we say that $X_\\Pi$ together with the shift structure is a \\emph{subshift of finite type}.\n\nA \\emph{finitary factor of iid process (ffiid) $F$} is a function that takes as an input $\\mathbb{Z}^d$ labeled with real numbers from $[0,1]$, independently according to Lebesgue measure, and outputs a different labeling of $\\mathbb{Z}^d$, say with elements of some set $b$. \nThis labeling satisfies the following: (a) $F$ is equivariant, that is for every $g\\in \\mathbb{Z}^d$ shifting the real labels with $g$ and applying $F$ is the same as applying $F$ and then shifting with $g$, (b) the value of $F(x)({\\bf 0})$ depends almost surely on the real labels on some random but finite neighborhood of ${\\bf 0}$.\nSimilarly as for oblivious local algorithms one defines the random variable $R_F$ that encodes minimal such neighborhood.\nWe say that ffiid $F$ solves $\\Pi$ if $F(x)\\in X_{\\Pi}$ with probability $1$.\n\nThe difference between ffiid and oblivious local algorithms is just formal.\nNamely, given an oblivious local algorithm $\\mathcal{A}$, we can turn it into an ffiid $F_\\mathcal{A}$ simply by running it simultaneously in every vertex of $\\mathbb{Z}^d$.\nWith probability $1$ this produces a labeling of $\\mathbb{Z}^d$ and it is easy to see that both conditions from the definition of ffiid are satisfied.\nOn the other hand, if $F$ is ffiid, then we collect the finite neighborhoods that determine the value at ${\\bf 0}$ together with the decision about the label.\nThis defines a pair $\\mathcal{A}_F=(\\mathcal{C},f)$ that satisfies the definition of an oblivious local algorithm.\nIt follows that the coding radius $R_\\mathcal{A}$, resp $R_F$, is the same after the translations.\nWe chose to work with the language of distributed algorithms as we borrow the techniques mainly from that area. \n\n\\begin{remark}\nIf in (b), in the definition of ffiid, we only demand that the value at ${\\bf 0}$ depends measurably on the labeling, i.e., $F$ is allowed to see the labels at the whole grid, then we say that $F$ is a \\emph{factor of iid labels (fiid)}.\n\nGiven an LCL $\\Pi$, we say that $\\Pi$ is in the class $\\mathsf{fiid}$, if there is a fiid $F$ that produces element of $X_\\Pi$ almost surely.\nIt is easy to see that $\\mathsf{TAIL}\\subseteq \\mathsf{fiid}$.\nWe remark that in the setting without inputs we have $\\mathsf{TAIL}=\\mathsf{fiid}$ on $\\mathbb{Z}$ and it is unknown, however, still plausible, whether this holds for $d>1$.\nFor general LCLs with inputs (where the definition of $\\mathsf{fiid}$ is extended appropriately) we have $\\mathsf{TAIL}\\not=\\mathsf{fiid}$ even for $d=1$, see \\cite{grebik_rozhon2021LCL_on_paths}.\n\\end{remark}\n\n\\begin{comment}\nGiven an LCL $\\Pi=(b,t,\\mathcal{P})$, we define $X_\\Pi\\subseteq b^{\\mathbb{Z}^d}$ to be the space of all $\\Pi$-colorings.\nWe say that an LCL $\\Pi$ is \\emph{satisfiable} if $\\emptyset\\not=X_\\Pi$, i.e., if there is a $\\Pi$-coloring of $\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)$.\nWe say that an LCL $\\Pi$ is \\emph{trivial} if there is $\\alpha\\in b$ such that the constant function $\\alpha$ is a $\\Pi$-coloring.\n\nThere is a canonical (right-)shift action $\\cdot$ of $\\mathbb{Z}^d$ on $b^{\\mathbb{Z}^d}$ (and in general on any $X^{\\mathbb{Z}^d}$, where $X$ is a set) that is defined as\n$$(g\\cdot x)(h)=x(h + g)$$\nfor every $g,h\\in \\mathbb{Z}^d$ and $x\\in b^{\\mathbb{Z}^d}$.\nIt is easy to see that $X_\\Pi$ is invariant under this action, i.e., $x\\in X_\\Pi$ implies $g\\cdot x\\in X_\\Pi$ for every $g\\in \\mathbb{Z}^d$.\nMoreover, since $b$ is a finite set, the product topology turns $b^\\Gamma$ into a compact metric space and it is easy to see $X_\\Pi$ is closed subspace.\nThe set $X_\\Pi$ together with the shift action is called a \\emph{sub-shift of finite type}.\n\nSuppose that $\\mathbb{Z}^d$ has an action $\\star_Y$ and $\\star_Z$ on spaces $Y$ and $Z$, respectively. \nWe say that a function $f:Y\\to Z$ is \\emph{equivariant} if $f(g\\star_Y x)=g\\star_Z f(x)$ for every $g\\in \\mathbb{Z}^d$ and $x\\in Y$.\n\nGiven an LCL $\\Pi$, our aim is to study complexity of finitary factors of iid that produce elements in $X_\\Pi$ almost surely.\nWe say that $\\Pi$ admits a \\emph{fiid solution} if there is a measurable equivariant map $F:[0,1]^{\\mathbb{Z}^d}\\to X_\\Pi$ that is defined on a $\\mu$-conull set, where $\\mu$ is the product measure on $[0,1]^{\\mathbb{Z}^d}$.\n\nWrite $\\varphi_F:[0,1]^{\\mathbb{Z}^d}\\to b$ for the composition of $F$ with the projection to ${\\bf 0}_d$-coordinate, i.e., \n$$x\\mapsto \\varphi_F(x)=F(x)({\\bf 0}_d).$$\nWe say that $F$ is a \\emph{finitary fiid (ffiid)} if for $\\mu$-almost every $x\\in [0,1]^{\\mathbb{Z}^d}$ the value of $\\varphi_F$ depends only on values of $x$ on some finite neighborhood of ${\\bf 0}_d$.\nThat is to say, there is a random variable $R_F$ that takes values in $\\mathbb{N}$, is defined $\\mu$-almost everywhere and satisfies for $\\mu$-almost every $x\\in [0,1]^{\\mathbb{Z}^d}$ the following: if $y\\in [0,1]^{\\mathbb{Z}^d}$ is such that\n$$x\\upharpoonright \\mathcal{B}_{\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)}\\left({\\bf 0}_d,R_\\varphi(x)\\right)=y\\upharpoonright \\mathcal{B}\\left({\\bf 0}_d,R_\\varphi(x)\\right),$$\nthen $\\varphi_F(x)=\\varphi_F(y)$.\n\\todo{nejak to zakoncit tim ze to nebudeme potrebovat}\n\\end{comment}\n\n\n\n\n\\begin{comment}\n\nWarm-up: Equivalence of $\\mathsf{LOCAL}(\\log^* n)$, $\\mathsf{TAIL}(\\mathrm{tower})$, $\\mathsf{CONTINUOUS}$\n\nRecall that $\\mathsf{LOCAL}(O(\\log^* n))$ denotes the class of satisfiable LCLs that satisfy (1) or (2) in Theorem~\\ref{thm:classification_of_local_problems_on_grids}.\nIn this section we show that every problem from $\\mathsf{LOCAL}(O(\\log^* n))$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\nIn the proof we use the fact that $\\mathsf{LOCAL}=\\mathsf{C-LOCAL}(O(1))$, see~Theorem~\\ref{thm:classification_of_local_problems_on_grids}.\n\nWe let $\\Delta_{s}:=|\\mathcal{B}({\\bf 0}_d,s)|$, where $s\\in \\mathbb{N}$.\nDenote as $\\mathcal{M}_s$ the subspace of $(\\Delta_s+1)^{\\mathbb{Z}^d}$ that consists of all proper vertex $(\\Delta_s+1)$-colorings of the $s$-power graph of $\\mathbb{Z}^d$.\nIt is easy to see that $\\mathcal{M}_s$ is nonempty, $\\mathbb{Z}^d$-invariant and compact.\nIn fact it is a subshift of finite type.\nThe following result holds for any graph of bounded degree, as was proved in \\cite{HolroydSchrammWilson2017FinitaryColoring}.\n\n\\begin{theorem}\\cite[Theorem~6]{HolroydSchrammWilson2017FinitaryColoring}\\label{th:ColorTower}\nLet $s\\in\\mathbb{N}$.\nThen there is an ffiid $F:[0,1]^{\\mathbb{Z}^d}\\to \\mathcal{M}_s$ and a constant $C_s>0$ such that\n$$\\mu(R_F>r)<\\frac{1}{tower(C_sr)}.$$\nIn particular, the LCL of coloring the $s$-power graph of $\\mathbb{Z}^d$ with $(\\Delta_s+1)$-many colors is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\n\\end{theorem}\n\n\\begin{proposition}\\label{pr:LOCALandCOLORING}\nLet $\\Pi=(b,t,\\mathcal{P})$ be a satisfiable LCL problem that is in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\nThen there is a continuous equivariant function $\\varphi:\\mathcal{M}_s\\to \\mathcal{C}_{\\Pi}$ for some $s\\in \\mathbb{N}$.\n\\end{proposition}\n\\begin{proof}\nSuppose that $\\Pi$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\nBy Theorem~\\ref{thm:classification_of_local_problems_on_grids}, we have that $\\Pi$ is in the class $\\mathsf{C-LOCAL}(O(1))$.\nFix $r,\\ell\\in \\mathbb{N}$ and $\\mathcal{A}$ that witnesses this fact.\nLet $x\\in \\mathcal{M}_{r+t}$.\nThen define\n$$\\varphi(x)(h):=\\mathcal{A}([\\mathcal{B}_{\\mathbb{Z}^d_x}(h,\\ell),h]).$$\nIt is clear that $\\varphi$ is well defined, because $\\ellr)<\\frac{\\Delta_{\\ell}}{tower(C_s(r-\\ell))}\\le \\frac{1}{tower(Dr)}$$\nfor $r\\in \\mathbb{N}$ large enough and some constant $D\\in (0,+\\infty)$.\n\\end{proof}\n\n\n\n\n\ngives that\n\n\n\n\n\n\n\n\n\n$\\mathsf{TAIL}(\\tow(\\Omega(r)))$ $\\Rightarrow$ $\\mathsf{LOCAL}(O(\\log^* n))$\n\n\nFirst we show how to get randomized local algorithm from ffiid in a qualitative way.\nWe note that similar results holds for any bounded degree graph.\n\n\\begin{theorem}\\label{th:TailToRandom}\nLet $\\Pi$ be an LCL problem that is in the class $\\mathsf{TAIL}(f)$ for some $f:\\mathbb{N}\\to \\mathbb{N}$.\nSet $g(n)=\\min\\{m\\in \\mathbb{N}:f(m)\\ge n^2\\}$ and suppose that $g\\in o(n^{\\frac{1}{d}})$.\nThen $\\Pi$ is in the class $\\mathsf{R-LOCAL}(O(g(n)))$.\n\\end{theorem}\n\\begin{proof}\nFix an ffiid $F:[0,1]^{\\mathbb{Z}^d}\\to X_\\Pi$ such that\n$$\\mu(R_F>r)\\le \\frac{1}{f(r)}$$\nfor large enough $r\\in \\mathbb{N}$.\nDenote as $\\varphi_F$ the finitary rule and as $\\operatorname{dom}(\\varphi_F)$ its domain.\nThat is $\\alpha\\in \\operatorname{dom}(\\varphi_F)$ if $\\alpha\\in [0,1]^{V(\\mathcal{B}({\\bf 0}_d,s))}$ for some $s\\in \\mathbb{N}$ and satisfies\n$$F(x)=F(y)$$\nfor every extensions $\\alpha\\sqsubseteq x,y\\in [0,1]^{\\mathbb{Z}^d}$.\nIn that case we set $\\varphi_F(\\alpha)=F(x)$ for any such extension $\\alpha\\sqsubseteq x$.\n\nDefine $\\ell_n:=g(n)$.\nLet $(G,v)\\in \\operatorname{\\bf FG}_{\\bullet}^{S}$ be of a radius at most $\\ell_n$.\nConsider any map $\\ell:V(G)\\to [0,1]$.\nDefine\n$$\\mathbb{A}_n([G_l,v])=\\varphi_F(\\ell)$$\nin the case when $\\ell\\in \\operatorname{dom}(\\varphi_F)$.\nOtherwise let $\\mathbb{A}_n([G_l,v])$ to be any element in $b$.\nIt is easy to verify that this defines a randomized local algorithm.\n\nLet $G\\in \\operatorname{\\bf FG}^{S}$ be such that $|V(G)|=n$ and\n$$[\\mathcal{B}_{G}(v,\\ell_n+t),v]=\\left[\\mathcal{B}({\\bf 0}_d,\\ell_n+t),{\\bf 0}_d\\right]$$\nholds for every $v\\in V(G)$.\nNote that by the assumption on $g$ such graphs do exist, e.g., large enough torus.\nFor every $v\\in V(G)$ define the event\n$$A_v=\\left\\{\\ell\\in [0,1]^{V(G)}:\\ell\\upharpoonright \\mathcal{B}_G(v,\\ell_n)\\not \\in \\operatorname{dom}(\\varphi_F)\\right\\}.$$\nNote that if $\\ell\\not \\in \\bigcup_{v\\in V(G)} A_v$, then $\\mathbb{A}(G_l)$ is a $\\Pi$-coloring.\nTo see this pick $v\\in V(G)$ and any extension $x\\in [0,1]^{\\mathbb{Z}^d}$ of $\\ell\\upharpoonright \\mathcal{B}_G(v,\\ell_n+t)$ that is in the domain of $F$, where we interpret $v$ as the origin ${\\bf 0}_d$.\nThen $F(x)\\in X_\\Pi$ and\n$$F(x)\\upharpoonright \\mathcal{B}({\\bf 0}_d,t)\\simeq\\mathbb{A}(G_l)\\upharpoonright \\mathcal{B}_G(v,t).$$\nConsequently, we have $\\mathcal{P}([\\mathcal{B}_{G_{\\mathbb{A}(G_l)}}(v,t),v])=1$.\n\nLet $v\\in V(G)$.\nWe have $\\ell\\in A_v$ if and only if $R_F(x)>\\ell_n$ for every $x\\in [0,1]^\\Gamma$ that extends $\\ell$, i.e., every $x\\in [0,1]^{\\mathbb{Z}^d}$ that satisfies\n$$[B_{G_l}(v,\\ell_n),v]=[B_{\\mathbb{Z}^d_x}({\\bf 0}_d,\\ell_n),{\\bf 0}_d].$$ \nThis implies that\n$$\\lambda^{\\oplus n}(A_v)=\\mu\\left(\\left\\{x\\in [0,1]^{\\mathbb{Z}^d}:R_F(x)>\\ell_n\\right\\}\\right)\\le \\frac{1}{f(\\ell_n)}\\le \\frac{1}{n^2}.$$\nA union bound then shows that $\\lambda^{\\oplus n}\\left(\\bigcup_{v\\in V(G)}A_v\\right)\\le \\frac{1}{n}$ and the proof is finished.\n\\end{proof}\n\n\n\n\\begin{corollary}\\label{cor:tower=local}\n$\\mathsf{TAIL}(\\tow(\\Omega(r)))=\\mathsf{LOCAL}(O(\\log^* n))$.\n\\end{corollary}\n\\begin{proof}\nIt follows from Corollary~\\ref{cor:locallocal} itself has the following implications: \n\n\\begin{corollary}\n\\label{cor:block_factor}\n$\\Pi$ can be solved by a block factor (recall \\cref{def:tail}) if and only if $\\Pi \\in \\mathsf{LOCAL}(O(1))$. \n\\end{corollary}\n\\begin{proof}\nThis follows from \\cref{thm:classification_of_local_problems_on_grids,rem:chang_pettie_decomposition}. \n\\end{proof}\n\n\\begin{corollary}\n\\label{cor:BelowTOWERtrivial}\nIf $\\Pi \\in \\mathsf{TAIL}(\\mathrm{tower}(\\omega(r)))$, then $\\Pi \\in \\mathsf{LOCAL}(O(1))$ and if $\\Sigma_{in} = \\emptyset$ then $\\Pi$ is trivial. \n\\end{corollary}\n\\begin{proof}\n\\cref{lem:tail->local} gives that $\\Pi \\in\\mathsf{LOCAL}(o(\\log^* n))$ and \\cref{thm:classification_of_local_problems_on_grids} then implies the rest. \n\\end{proof}\n\nIn particular, Corollary~\\ref{cor:BelowTOWERtrivial} answers in negative Problem~(ii) in \\cite{HolroydSchrammWilson2017FinitaryColoring} that asked about the existence of a nontrivial local problem (without input labels) with super-tower function tail. \n\nThe second corollary is the following:\n\n\\begin{corollary}\n\\label{cor:local=tower}\n$\\Pi \\in \\mathsf{TAIL}(\\tow(\\Omega(r)))$ if and only if $\\Pi \\in \\mathsf{LOCAL}(O(\\log^* n))$. \n\\end{corollary}\n\\begin{proof}\nIf $\\Pi \\in \\mathsf{TAIL}(\\tow(\\Omega(r)))$, \\cref{lem:tail->local} yields $\\Pi \\in \\mathsf{RLOCAL}(O(\\log^* n))$. The other direction (special case of \\cref{thm:tail=uniform}) follows from \\cref{rem:chang_pettie_decomposition} that essentially states that it suffices to solve the independent set problem in the power graph. But this follows from a version of Linial's algorithm from \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n\\end{proof}\n\nFinally, we get the following. \n\\begin{corollary}\n\\label{cor:Below_d_local}\nIf $\\Pi \\in \\mathsf{TAIL}(\\omega(r^{2d}))$, then $\\Pi \\in \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$. \n\\end{corollary}\n\\begin{proof}\n\\cref{lem:tail->local} gives that $\\Pi \\in \\mathsf{LOCAL}(o(n^{1\/d}))$ and \\cref{thm:classification_of_local_problems_on_grids} then implies that $\\Pi \\in \\mathsf{LOCAL}(O(\\log^* n))$ which is equivalent to $\\Pi \\in \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$ by \\cref{cor:local=tower}. \n\\end{proof}\n\nCorollary~\\ref{cor:Below_d_local} answers in negative Problem~(iii) in \\cite{HolroydSchrammWilson2017FinitaryColoring} that asked about the existence of a problem with all its moments finite but with sub-tower function decay. Recall that \\cref{fact:moment_implies_tail} implies that if the $2d$-th moment is finite, then the tail is $\\omega(r^{2d})$ and, hence, the problem can in fact be solved with a tower tail decay. \nHowever, the $2d$ is not the right dependency and in the next section we prove a tight version of this result.\n\\subsection{Tight Grid Speedup for $\\mathsf{TAIL}$}\n\\label{subsec:tail_d-1_speedup}\n\nWhile \\cref{cor:Below_d_local} already answers the Problem of \\cite{HolroydSchrammWilson2017FinitaryColoring}, we now improve the speedup from the tail of $\\omega(r^{d-1})$. This result is tight (up to $o(1)$ in the exponent, see the discussion in \\cref{subsec:toast_construction} and \\cref{cor:toast_lower_bound}) and it is the first nontrivial difference between the $\\mathsf{TAIL}$ and $\\mathsf{LOCAL}$ model that we encountered so far. \n\n\n\n\n\\begin{theorem}\n\\label{thm:grid_speedup}\nLet $d,t\\in \\mathbb{N}$.\nFor $d \\ge 2$ there is $C_{d,t}>0$ such that if an LCL $\\Pi$ is in $\\mathsf{TAIL}(C_{d,t}r^{d-1})$, then $\\Pi \\in \\mathsf{TAIL}(\\tow(\\Omega(r)))$.\n\nSimilarly, for $d=1$ we have that a local coloring problem $\\Pi$ admits a solution by an oblivious random algorithm (it is in $\\mathsf{TAIL}(f(r))$ for any function $f = \\omega(1)$) if and only if $\\Pi$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\n\\end{theorem}\nNote that \\cref{fact:moment_implies_tail} hence shows that ${\\bf M}(\\Pi) > d-1$ already implies that $\\Pi$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$. \n\nThe proof is a simple adaptation of Lov\u00e1sz local lemma speedup results such as \\cite{chang2017time_hierarchy,fischer2017sublogarithmic}. \nThe proof goes as follows. Split the grid into boxes of side length roughly $r_0$ for some constant $r_0$. This can be done with $O(\\log^* (n))$ local complexity (or, equivalently, with tower tail decay). \n\nNext, consider a box $B$ of side length $r_0$ and volume $r_0^d$. In classical speedup argument we would like to argue that we can deterministically set the randomness in nodes of $B$ and its immediate neighborhood so that we can simulate the original oblivious local algorithm $\\mathcal{A}$ with that randomness and it finishes in at most $r_0$ steps. \nIn fact, ``all nodes of $B$ finish after at most $r_0$ steps'' is an event that happens with constant probability if the tail decay of $\\mathcal{A}$ is $r^d$, since then each node fails with probability $1\/r_0^d$ and there are $r_0^d$ nodes in $B$. \n\nHowever, we observe that it is enough to consider only the boundary of $B$ with $O(r_0^{d-1})$ nodes and argue as above only for the boundary nodes, hence the speedup from $r^{d-1}$ tail decay. \nThe reason for this is that although the coding radius of $\\mathcal{A}$ inside $B$ may be very large, just the fact that there \\emph{exists} a way of filling in the solution to the inside of $B$, given the solution on its boundary, is enough for our purposes. The new algorithm simply simulates $\\mathcal{A}$ only on the boundaries and then fills in the rest. \n\n\\begin{proof}\n\n\nWe will prove the result for $d \\ge 2$, the case $d=1$ is completely analogous and, in fact, simpler. \nSet $C=C_{d,t}:=d\\cdot t\\cdot(2\\Delta)^{2\\Delta}$, where $\\Delta=20^d$.\nSuppose that there is an oblivious local algorithm $\\mathcal{A}$ that explores the neighborhood of a given vertex $u$ and outputs a color of $u$ after seeing a radius $R$ such that \\begin{equation}\\label{eq:tail}\n\\P(R > r)\\le \\frac{1}{C r^{d-1}}\n\\end{equation}\nfor $r\\in \\mathbb{N}$ large enough.\nLet $r_0\\in \\mathbb{N}$ be large enough such that \\eqref{eq:tail} is satisfied.\nWrite $\\Box_{r_0}$ for the local coloring problem of tiling the input graph (which is either a large enough torus or an infinite grid $\\mathbb{Z}^d$) into boxes with length sizes in the set $\\{r_0,r_0+1\\}$ such that, additionally, each box $B$ is colored with a color from $[\\Delta+1]$ which is unique among all boxes $B'$ with distance at most $3(r_0+t)$. \n\nThe problem of tiling with boxes of side lengths in $\\{r_0, r_0+1\\}$ can be solved in $O(\\log^* n)$ local complexity \\cite[Theorem~3.1]{GaoJackson} but we only sketch the reason here: Find a maximal independent set of a large enough power of the input graph $\\mathbb{Z}^d$. Then, each node $u$ in the MIS collects a Voronoi cell of all non-MIS nodes such that $u$ is their closest MIS node. As the shape of the Voronoi cell of $u$ depends only on other MIS vertices in a constant distance from $u$, and their number can be bounded as a function of $d$ (see the ``volume'' argument given next), one can discretize each Voronoi cell into a number of large enough rectangular boxes. Finally, every rectangular box with all side lengths of length at least $r^2$ can be decomposed into rectangular boxes with all side lengths either $r$ or $r+1$. \n\nNote that in additional $O(\\log^* n)$ local steps, the boxes can then be colored with $(\\Delta+1)$-many colors in such a way that boxes with distance at most $3(r_0+t)$ must have different colors.\nTo see this, we need to argue that there are at most $\\Delta$ many boxes at distance at most $3(r_0+t)$ from a given box $B$.\nThis holds because if $B,B'$ are two boxes and $d(B,B')\\le 3(r_0+t)$, then $B'$ is contained in the cube centered around the center of $B$ of side length $10(r_0+t)$.\nThen a volume argument shows that there are at most $\\Delta\\ge \\frac{(10(r_0+t))^d}{r_0^d}$ many such boxes $B'$ (when $B$ is fixed).\n\nWe show now that there is a local algorithm $\\mathcal{A'}$ that solves $\\Pi$ when given the input graph $G$ together with a solution of $\\Box_{r_0}$ as an input.\nThis clearly implies that $\\Pi$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\n\n\n\nLet $\\{B_i\\}_{i\\in I}$ be the input solution of $\\Box_{r_0}$, together with the coloring $B_i\\mapsto c(B_i)\\in [\\Delta+1]$.\nWrite $V(B)$ for the vertices of $B$ and $\\partial B$ for the vertices of $B$ that have distance at most $t$ from the boundary of $B$.\nWe have $|\\partial B|< d\\cdot t \\cdot (r_0+1)^{d-1}$.\nDefine $N(B)$ to be the set of rectangles of distance at most $(r_0+t)$ from $B$.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width= .9\\textwidth]{fig\/tail_LLL.eps}\n \\caption{The picture shows one block $B$ with a red boundary $\\partial B$ of length $t$. Boxes in $N(B)$, that is with distance at most $(r_0+t)$ from $B$ are dark grey. }\n \\label{fig:tail_LLL}\n\\end{figure}\n\nWe are now going to describe how $\\mathcal{A}'$ iterates over the boxes in the order of their colors and for each node $u$ in the box $B$ it carefully fixes a random string $\\mathfrak{r}(u)$. \nThis is done in such a way that after fixing the randomness in all nodes, we can apply the original algorithm $\\mathcal{A}$ to each vertex in $\\partial B$ for each $B$ with the randomness given by $\\mathfrak{r}$ and the local algorithm at $u \\in \\partial B$ will not need to see more than its $r$-hop neighborhood. \n\nGiven a partial function $\\mathfrak{r_\\ell}:V(G)\\to [0,1]$ corresponding to nodes such that their randomness was already fixed, every box $B$ can compute a probability $\\mathbb{P}(\\fE_{\\ell}(B))$, where $\\fE_{\\ell}(B)$ is the (bad) event that there is a vertex in $\\partial B$ that needs to look further than $r_0$ to apply the local algorithm $\\mathcal{A}$.\nNote that for computing $\\mathbb{P}(\\fE_{\\ell}(B))$ it is enough to know the $r_0$-neighborhood of $B$.\n\nThe algorithm $\\mathcal{A}$ has $\\Delta+1$ rounds.\nIn the $0$-th round, we set $\\mathfrak{r}_0$ to be the empty function.\nNote that the probability that a vertex needs to look further than $r_0$ in the algorithm $\\mathcal{A}$ is smaller than $\\frac{1}{C r_0^{d-1}}$ by \\eqref{eq:tail}.\nThe union bound gives $\\mathbb{P}(\\fE_0(B))<\\frac{1}{(2\\Delta)^{2\\Delta}}$ for every $B$.\n\nIn each step $\\ell\\in [\\Delta+1]$ we define $\\mathfrak{r}_B:V(B)\\to [0,1]$ for every $B$ such that $c(B)=\\ell$ and put $\\mathfrak{r}_l$ to be the union of all $\\mathfrak{r}_D$ that have been defined so far, i.e., for $D$ such that $c(D)\\le \\ell$.\nWe make sure that the following is satisfied for every $B$:\n\\begin{enumerate}\n\t\\item [(a)] the assignment $\\mathfrak{r}_B$ depends only on $2(r_0+t)$-neighborhood of $B$ (together with previously defined $\\mathfrak{r}_{\\ell-1}$ on that neighborhood) whenever $c(B)=\\ell$,\n\t\\item [(b)] $\\mathbb{P}(\\fE_l(B))<\\frac{1}{(2\\Delta)^{2\\Delta-\\ell}}$,\n\t\\item [(c)] restriction of $\\mathfrak{r}_l$ to $N(B)$ can be extend to the whole $\\mathbb{Z}^d$ such that $\\mathcal{A}$ is defined at every vertex, where we view $N(B)$ embedded into $\\mathbb{Z}^d$.\n\\end{enumerate}\nWe show that this gives the required algorithm.\nIt is easy to see that by (a) this defines a local construction that runs in constantly many rounds and produces a function $\\mathfrak{r}:=\\mathfrak{r}_{\\Delta+1}:V(G)\\to [0,1]$.\nBy (b) every vertex in $\\partial B$ for some $B$ looks into its $r_0$ neighborhood and use $\\mathcal{A}$ to choose its color according to $\\mathfrak{r}$.\nFinally, by (c) it is possible to assign colors in $B\\setminus \\partial B$ for every $B$ to satisfy the condition of $\\Pi$ within each $B$.\nPicking minimal such solution for each $B\\setminus \\partial B$ finishes the description of the algorithm.\n\nWe show how to proceed from $0\\le \\ell<\\Delta+1$ to $\\ell+1$.\nSuppose that $\\mathfrak{r}_l$ satisfies the conditions above.\nPick $B$ such that $c(B)=\\ell+1$.\nNote that $\\mathfrak{r}_B$ that we want to find can influence rectangles $D\\in N(B)$ and these are influenced by the function $\\mathfrak{r}_l$ at distance at most $r_0$ from them.\nAlso given any $D$ there is at most one $B\\in N(D)$ such that $c(B)=\\ell+1$. \nWe make the decision $\\mathfrak{r}_B$ based on $2r_0$ neighborhood around $B$ and the restriction of $\\mathfrak{r}_l$ there, this guarantees that condition (a) is satisfied.\n\nFix $D\\in N(B)$ and pick $\\mathfrak{r}_B$ uniformly at random.\nWrite $\\mathfrak{S}_{B,D}$ for the random variable that computes the probability of $\\fE_{\\mathfrak{r}_B\\cup \\mathfrak{r}_l}(D)$.\nWe have $\\mathbb{E}(\\mathfrak{S}_{B,D})=\\mathbb{P}(\\fE_l(D))<\\frac{1}{(2\\Delta)^{2\\Delta-\\ell}}$.\nBy Markov inequality we have\n$$\\mathbb{P}\\left(\\mathfrak{S}_{B,D}\\ge 2\\Delta \\frac{1}{(2\\Delta)^{2\\Delta-\\ell}}\\right)<\\frac{1}{2\\Delta}.$$\nSince $|N(B)|\\le \\Delta$ we have that with non-zero probability $\\mathfrak{r}_B$ satisfies $\\mathfrak{S}_{B,D}<\\frac{1}{(2\\Delta)^{2\\Delta-\\ell-1}}$ for every $D\\in N(B)$.\nSince $\\mathcal{A}$ is defined almost everywhere we find $\\mathfrak{r}_B$ that satisfies (b) and (c) above.\nThis finishes the proof. \n\\end{proof}\n\nAs we already mentioned, the proof is in fact an instance of the famous Lov\u00e1sz Local Lemma problem. This problem is extensively studied in the distributed computing literature \\cite{chang2016exp_separation, fischer2017sublogarithmic, brandt_maus_uitto2019tightLLL, brandt_grunau_rozhon2020tightLLL} as well as descriptive combinatorics \\cite{KunLLL,OLegLLL,MarksLLL,Bernshteyn2021LLL}.\nOur usage corresponds to a simple LLL algorithm for the so-called exponential criteria setup (see \\cite{fischer2017sublogarithmic} and \\cite{Bernshteyn2021local=cont}). \n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{$\\TOAST$: General Results}\n\\label{sec:Toast}\n\nIn this section we introduce the toast construction. This construction is very standard in descriptive combinatorics \\cite{FirstToast,GJKS,Circle} and the theory of random processes (there it is often called \\emph{the hierarchy}) \\cite{HolroydSchrammWilson2017FinitaryColoring,ArankaFeriLaci,Spinka}.\n\nWhile the construction itself is well-known, in this paper we study it as a class of problems by itself. An analogy to our work is perhaps the paper of \\cite{ghaffari2018derandomizing} that defines the class $\\mathsf{S-LOCAL}$ and shows how to derandomize it.\n\n\n\nInformally, the basic toast construction works as follows. There is an adversary that has access to the (infinite) input grid $\\mathbb{Z}^d$. The adversary is sequentially choosing some connecting pieces of the grid and gives them to our algorithm $\\mathcal{A}$. The algorithm $\\mathcal{A}$ needs to label all vertices in the piece $D$ once it is given $D$. \nNote that $\\mathcal{A}$ does not know where exactly $D$ lies in $\\mathbb{Z}^d$ and its decision cannot depend on the precise position of $D$ in the input graph. In general, the piece $D$ can be a superset of some smaller pieces that were already colored by $\\mathcal{A}$. In this case, $\\mathcal{A}$ needs to extend their coloring to the whole $D$. This game is played for countable number of steps until the whole grid is colored. The algorithm solves $\\Pi$ if it never happens during the procedure that it cannot extend the labeling to $D$ in a valid manner. \n\n\n\n\n\nThe construction sketched above is what we call a spontaneous deterministic toast or just $\\TOAST$ algorithm. \nIn the next subsection we give a formal definition of $\\TOAST$ algorithms and we follow by proving relations between $\\TOAST$ algorithms and $\\mathsf{TAIL}$ complexities. \n\n\n\\subsection{Formal Definition of $\\TOAST$ classes}\n\\label{subsec:toast_definition}\nWe now give a formal definition of our four $\\TOAST$ classes. First we define the hereditary structure on which the $\\TOAST$ algorithm is interpreted. This is also called a toast (we use lowercase letters) and is parameterized by the minimum distance of boundaries of different pieces of the toast. \nThis structure also corresponds to the adversary that asks our algorithm to extend a partial solution to $\\Pi$ to bigger and bigger pieces used in the informal introduction. \n\n\\begin{definition}[$q$-toast on $\\mathbb{Z}^d$]\nFor a $q >0$, a \\emph{$q$-toast on $\\mathbb{Z}^d$} is a collection $\\mathcal{D}$ of finite connected subsets of $\\mathbb{Z}^d$ that we call \\emph{pieces} such that\n\\begin{itemize}\n \\item for every $g,h\\in \\mathbb{Z}^d$ there is a piece $D\\in\\mathcal{D}$ such that $g,h\\in D$,\n \\item if $D,E\\in \\mathcal{D}$ and $d(\\partial D,\\partial E)\\le q$, then $D=E$, where $\\partial D$ is a boundary of a set $D$.\n\\end{itemize}\n\\end{definition}\n\nGiven a $q$-toast $\\mathcal{D}$, the \\emph{rank} of its piece $D \\in \\mathcal{D}$ is the level in the toast hierarchy that $D$ belongs to in $\\mathcal{D}$. Formally, we define a \\emph{rank function $\\mathfrak{r}_\\mathcal{D}$ for the toast $\\mathcal{D}$} as $\\mathfrak{r}_\\mathcal{D}:\\mathcal{D}\\to \\mathbb{N}$, inductively as\n$$\\mathfrak{r}_\\mathcal{D}(D)=\\max\\{\\mathfrak{r}_{\\mathcal{D}}(E)+1:E\\subseteq D, \\ E\\in \\mathcal{D}\\}$$ \nfor every $D\\in \\mathcal{D}$.\n\nNext, $\\TOAST$ algorithms are described in the following language of extending functions. \n\n\\begin{definition}[$\\TOAST$ algorithm]\n\\label{def:extending_function}\nWe say that a function $\\mathcal{A}$ is a \\emph{$\\TOAST$ algorithm} for a local problem $\\Pi = (\\Sigma_{in},\\Sigma_{out}, t, \\mathcal{P})$ if \n\\begin{enumerate}\n \\item [(a)] the domain of $\\mathcal{A}$ consists of pairs $(A, c)$, where $A$ is a finite connected $\\Sigma_{in}$-labeled subset of $\\mathbb{Z}^d$ and $c$ is a partial map from $A$ to $\\Sigma_{out}$,\n \\item [(b)] $\\mathcal{A}(A,c)$ is a partial coloring from $A$ to $\\Sigma_{out}$ that extends $c$, i.e., $\\mathcal{A}(A,c)\\upharpoonright \\operatorname{dom}(c)=c$, and that depends only on the isomorphism type of $(A,c)$ (not on the position in $\\mathbb{Z}^d$). \n \n\\end{enumerate}\nA \\emph{randomized $\\TOAST$ algorithm} is moreover given as an additional input a labeling of $A$ with real numbers from $[0,1]$ (corresponding to a stream of random bits). \nFinally, if $\\mathcal{A}$ has the property that $\\operatorname{dom}(\\mathcal{A}(A,c)) =A$ for every pair $(A,c)$, then we say that $\\mathcal{A}$ is a \\emph{spontaneous (randomized) $\\TOAST$ algorithm}.\n\\end{definition}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{fig\/3-col.eps}\n \\caption{A $q$-toast and an example of an extending function for $3$-coloring. Observe that our extending function does not give a correct algorithm for $3$-coloring since the piece on the right is contained in an even bigger piece and the $3$-coloring constructed on the right cannot be extended to the node marked with an arrow}\n \\label{fig:toast_example}\n\\end{figure}\n\n\nIt is easy to see that a $\\TOAST$ algorithm $\\mathcal{A}$ can be used to color ($\\Sigma_{in}$-labeled) $\\mathbb{Z}^d$ inductively on the rank $\\mathfrak{r}_\\mathcal{D}$ of any $q$-toast $\\mathcal{D}$ for every $q\\in \\mathbb{N}$. \nNamely, define $\\mathcal{A}(\\mathcal{D})$ to be the partial coloring of $\\mathbb{Z}^d$ that is the union of $\\mathcal{A}(D)$, where $D\\in \\mathcal{D}$ and $\\mathcal{A}(D)$ is defined inductively on $\\mathfrak{r}_\\mathcal{D}$.\nFirst put $\\mathcal{A}(D):=\\mathcal{A}(D,\\emptyset)$ whenever $\\mathfrak{r}_\\mathcal{D}(D)=0$.\nHaving defined $\\mathcal{A}(D)$ for every $D\\in \\mathcal{D}$ such that $\\mathfrak{r}_\\mathcal{D}(D)=k$, we define\n$$\\mathcal{A}(D'):=\\mathcal{A}\\left(\\bigcup\\{E\\in \\mathcal{D}:E\\subseteq D'\\},\\bigcup\\{\\mathcal{A}(E):E\\subseteq D', \\ E\\in \\mathcal{D}\\}\\right).$$\nfor every $D'\\in \\mathcal{D}$ such that $\\mathfrak{r}_\\mathcal{D}(D')=k+1$. That is, we let $\\mathcal{A}$ to extend the partial solution to $\\Pi$ to bigger and bigger pieces of the toast $\\mathcal{D}$. \nObserve that if $\\mathcal{A}$ is a spontaneous $\\TOAST$ function, then $\\operatorname{dom}(\\mathcal{A}(\\mathcal{D}))=\\mathbb{Z}^d$ for every $q$-toast $\\mathcal{D}$ and every $q\\in \\mathbb{N}$.\n\n\\paragraph{$\\TOAST$ Algorithms}\nWe are now ready to define the four types of toast constructions that we are going to use (cf. \\cref{fig:big_picture_grids}). \n\n\\begin{definition}[$\\LTOAST$]\nLet $\\Pi$ be an LCL.\nWe say that $\\Pi$ is in the class $\\LTOAST$ if there is $q>0$ and an $\\TOAST$ algorithm $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is a $\\Pi$-coloring for every $q$-toast $\\mathcal{D}$.\n\nMoreover, we say that $\\Pi$ is in the class $\\TOAST$ if there is $q\\in \\mathbb{N}$ and a spontaneous $\\TOAST$ algorithm $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is a $\\Pi$-coloring for every $q$-toast $\\mathcal{D}$.\n\\end{definition}\n\nBefore we define a randomized version of these notions, we make several comments.\nFirst note that $\\TOAST\\subseteq \\LTOAST$.\nBy (c), if we construct a ($q$-)toast using an oblivious local algorithm, then we get instantly that every problem from the class $\\TOAST$ is in the class $\\mathsf{TAIL}$, see Section~\\ref{sec:Toast}.\nSimilar reasoning applies to the class $\\mathsf{BOREL}$, see Section~\\ref{sec:OtherClasses}.\nWe remark that we are not aware of any LCL that is not in the class $\\TOAST$.\n\n\\begin{comment}\n\\begin{definition}[Randomized toast algorithm]\n\\label{def:extending_function}\nLet $b$ be a set.\nWe say that a function $\\mathcal{A}$ is a randomized $\\TOAST$ algorithm if \n\\begin{enumerate}\n \\item [(a)] the domain of $\\mathcal{A}$ consists of triplets $(A, c, \\ell)$, where $A$ is a finite connected subset of $\\mathbb{Z}^d$, $c$ is a partial coloring of $A$ and $\\ell:A\\to [0,1]$. \n \\item [(b)] $\\mathcal{A}(A,c,\\ell)$ is a partial coloring of $A$ that extends $c$, i.e., $\\mathcal{A}(A,c,\\ell)\\upharpoonright \\operatorname{dom}(c)=c$. \n \\item [(c)] the function $\\mathcal{A}$ is invariant with respect to the shift action of $\\mathbb{Z}^d$ on $[0,1]^{\\mathbb{Z}^d)}$. \n\\end{enumerate}\nMoreover, if $\\mathcal{A}$ has the property that $\\operatorname{dom}(\\mathcal{A}(A,c,\\ell)) =A$ for every triplet $(A,c,\\ell)$, then we say that $\\mathcal{A}$ is a \\emph{spontaneous randomized toast algorithm}.\n\\end{definition}\n\nSimilarly as above, we define $\\mathcal{A}(\\mathcal{D},\\ell)$ for every $q$-toast $\\mathcal{D}$, $q\\in \\mathbb{N}$ and $\\ell\\in [0,1]^{\\mathbb{Z}^d}$.\nObserve that if $\\mathcal{A}$ is a spontaneous randomized toast algorithm, then $\\operatorname{dom}(\\mathcal{A}(\\mathcal{D},\\ell))=\\mathbb{Z}^d$.\n\\end{comment}\n\n\\begin{definition}[$\\RLTOAST$]\nLet $\\Pi$ be an LCL.\nWe say that $\\Pi$ is in the class $\\RLTOAST$ if there is $q>0$ and a randomized $\\TOAST$ algorithm $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is a $\\Pi$-coloring for every $q$-toast $\\mathcal{D}$ with probability $1$.\nThat is, after $\\Sigma_{in}$-labeling is fixed, with probability $1$ an assignment of real numbers from $[0,1]$ to $\\mathbb{Z}^d$ successfully produces a $\\Pi$-coloring against all possible toasts $\\mathcal{D}$ when $\\mathcal{A}$ is applied.\n\nMoreover, we say that $\\Pi$ is in the class $\\RTOAST$ if there is $q\\in \\mathbb{N}$ and a spontaneous randomized $\\TOAST$ algorithm $\\mathcal{A}$ as above.\n\\end{definition}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{$\\mathsf{TAIL}$ Construction of $\\TOAST$}\n\\label{subsec:toast_construction_in_tail_sketch}\n\n\nThe class $\\TOAST$ (or $\\RTOAST$) is especially useful since in \\cref{sec:SpecificProblems} we discuss how many local problems that are not in class $O(\\log^* n)$ but on the other hand are not ``clearly hard'' like vertex $2$-coloring are in the class $\\TOAST$. Importantly, in \\cref{subsec:toast_construction} we show how one can construct the toast decomposition with an oblivious local algorithm in $\\mathsf{TAIL}(r^{1 - O(1\/\\sqrt{\\log r})})$, which yields the following theorem.\n\n\\begin{restatable}{theorem}{toastconstruction}\n\\label{thm:toast_in_tail}\nLet $\\Pi$ be an LCL that is in the class $\\TOAST$.\nThen \\[\n\\Pi \\in \\mathsf{TAIL}\\left(\\frac{r}{2^{O(\\sqrt{\\log r})}}\\right) = \\mathsf{TAIL}\\left(r^{1-o(1)}\\right)\n\\]\nand, hence, ${\\bf M}(\\Pi)\\ge 1$.\n\\end{restatable}\n\nWe observe in \\cref{cor:toast_lower_bound} that faster then linear tail decay is not possible. \n\n\\paragraph{Proof Sketch}\n\nHere we provide an intuition for the proof that is otherwise technical. We first check that maximal independent sets of power graphs of the grid $G = \\mathbb{Z}^d$ can be constructed effectively, e.g. because of the MIS algorithm of Ghaffari \\cite{ghaffari2016MIS}. The slowdown by this construction can be neglected in this high-level overview. \n\nWe then take a sequence $r_k = 2^{k^2}$ and construct sparser and sparser MIS in graphs $G^{r_k}, k = 1, 2, \\dots$. After constructing MIS of $G^{r_k}$, we construct Voronoi cells induced by this MIS. These Voronoi cells $V_1, V_2, \\dots$ will be, roughly, the pieces of rank $k$, but we need to ensure that all the boundaries are at least $q$ hops apart. \n\nTo do so, we first shrink each Voronoi cells by $t_k = O(\\sum_{j < k} r_j) \\ll r_k$ hops (grey boundary area in \\cref{fig:toast_construction}) and get new, smaller, cells $C_1, C_2, \\dots$. The final sequence of pieces $H_1, H_2, \\dots$ is then defined inductively, namely to define $H_i$ we start with $C_1$, then add all Voronoi cells from $k-1$-th iteration that $C_1$ intersects, then add all Voronoi cells from $k_2$-th iteration that the new, bigger, cluster intersects, \\dots\nThe definition of $t_k$ ensures that the final pieces constructed after all $k$ iterations are sufficiently far from each other, and also compatible with the smaller rank pieces. In fact, we always have $H_i \\subseteq V_i$. \n\nIt remains to argue that after $k$ steps of this process, only a small fraction of nodes remains uncovered. This is done roughly as follows. The vertices from $V_i \\setminus C_i$ that are not covered by $C_i$s are roughly $t_k \/ r_k \\approx A r_{k-1} \/ r_k$ fraction of vertices where $A$ is some (large) constant. By induction, $1 - A r_{k-2} \/ r_{k-1}$ fraction of those vertices were covered in step $k-1$ and so on. The fraction of remaining vertices is then $\\frac{A r_{k-1}}{r_{k}} \\cdot \\frac{A r_{k-2}}{r_{k-1}} \\dots \\frac{A }{r_1} = \\frac{A^k}{r_k}$. This fraction is asymptotically almost equal to $1\/r_k$, which is the desired tail behavior. However, the additional term $A^k$ is the reason why the final tail decay for $r$ such that $r_k \\le r \\le r_{k+1} = 2^{(k+1)^2}$ is bounded by $2^{O(k)}\/2^{k^2} \\le \\left( \\frac{1}{r} \\right)^{\\frac{k^2 - O(k)}{(k+1)^2}} = \\left( \\frac{1}{r} \\right)^{1 - O(1\/k)} = \\left( \\frac{1}{r} \\right)^{1 - O(1\/\\sqrt{\\log r})} $, as needed. \n\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width = .95\\textwidth]{fig\/toast_construction.eps}\n \\caption{A step during the toast construction. Voronoi cell $V$ is first shrinked to $C$ and we get the final piece $H$ by taking the union of smaller-level Voronoi cells intersecting $C$ (in this picture, there is only one lower level but the construction is recursive in general). }\n \\label{fig:toast_construction}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\\subsection{$\\RTOAST = \\TOAST$}\n\\label{subsec:rtoast=toast}\n\n\n\nIn this section we show that randomness does not help spontaneous toast algorithms.\n\n\\begin{restatable}{theorem}{toastrtoast}\\label{thm:derandomOfToast}\nLet $\\Pi$ be an LCL.\nThen $\\Pi$ is in the class $\\RTOAST$ if and only if $\\Pi$ is in the class $\\TOAST$.\n\\end{restatable}\n\nThe proof goes as follows. We start with a random toast algorithm $\\mathcal{A}$ and make it deterministic as follows. When the deterministic toast algorithm gets a piece $D$ of rank $1$, it considers all possible outputs that the algorithm $\\mathcal{A}$ could produce given the input random string and it uses any one of them that $\\mathcal{A}$ would produce with nonzero probability. \nBy induction, whenever $\\mathcal{A}'$ later gets a piece $D$ containing some pieces of smaller rank with a partial solution that $\\mathcal{A}'$ already produced, the original algorithm $\\mathcal{A}$ could have produced the same partial solution with nonzero probability, hence $\\mathcal{A}$ can extend the coloring in a valid manner and $\\mathcal{A}'$ deterministically choosing any such valid extension that $\\mathcal{A}$ would use with nonzero probability. \nThis shows that $\\mathcal{A}'$ also solves $\\Pi$. \nA formal proof follows. \n\n\\begin{proof}\n\nIt is easy to see that $\\TOAST\\subseteq \\RTOAST$.\nSuppose that $\\Pi$ is in the class $\\RTOAST$ and fix a witnessing $q\\in \\mathbb{N}$ together with a spontaneous randomized $\\TOAST$ algorithm $\\mathcal{A}$.\nWe may assume that $t$, the locality of $\\Pi$, satisfies $t\\le q$.\n\nLet $(A,c)$ be such that $A\\subseteq \\mathbb{Z}^d$ is finite and connected and $\\operatorname{dom}(c)\\subseteq A$.\nWe say that $(A,c)$ comes from a \\emph{partial random $q$-toast} if there is a collection $\\mathcal{D}$ of subsets of $A$ together with an event $\\mathcal{F}$ on $\\operatorname{dom}(c)$ such that\n\\begin{itemize}\n \\item if $D,E\\in \\mathcal{D}$ and $d(\\partial D,\\partial E)\\le q$, then $D=E$,\n \\item $A\\in \\mathcal{D}$,\n \\item $\\bigcup_{E\\in \\mathcal{D}\\setminus \\{A\\}}E=\\operatorname{dom}(c)$,\n \\item $\\mathcal{F}$ has non-zero probability and if we apply inductively $\\mathcal{A}$ on $\\operatorname{dom}(c)$ we obtain $c$.\n\\end{itemize}\nIf $(A,c)$ comes from a partial random $q$-toast, then we let $\\mathcal{E}(A,c)$ to be any event $\\mathcal{F}$ as above.\nOtherwise we put $\\mathcal{E}(A,c)$ to be any event on $\\operatorname{dom}(c)$ of non-zero probability.\nIt is easy to see that $\\mathcal{E}$ does not depend on the position of $A$ in $\\mathbb{Z}^d$.\nThis is because $\\mathcal{A}$ is such.\n\nWe now define a spontaneous toast algorithm $\\mathcal{A}'$.\nGiven $(A,c)$, we let $\\mathcal{F}(A,c)$ to be any event on $A$ that has non-zero probability, satisfies\n$$\\mathcal{F}(A,c)\\upharpoonright \\operatorname{dom}(c)\\subseteq \\mathcal{E}(A,c)$$\nand $\\mathcal{A}(A,c)$ gives the same outcome on $\\mathcal{F}(A,c)$.\nLet $\\mathcal{A}'(A,c)$ be equal to this outcome on $\\mathcal{F}(A,c)$.\nIt is clear that $\\mathcal{A}'$ is a spontaneous $\\TOAST$ algorithm.\nObserve that if $(A,c)$ comes from a partial random $q$-toast, then $(A,\\mathcal{A}'(A,c))$ comes from a partial random $q$-toast.\nNamely, use the same $q$-toast witness $\\mathcal{D}$ as for $(A,c)$ together with the event $\\mathcal{F}(A,c)$.\n\nIt remains to show that $\\mathcal{A}'(\\mathcal{D})$ is a $\\Pi$-coloring for every $q$-toast $\\mathcal{D}$.\nInduction on $\\mathfrak{r}_{\\mathcal{D}}$ shows that $(D,\\mathcal{A}'(D))$ comes from a partial random $q$-toast for every $D\\in \\mathcal{D}$.\nThis is easy when $\\mathfrak{r}_{\\mathcal{D}}(D)=0$ and follows from the observation above in general.\nLet $v\\in \\mathbb{Z}^d$ and pick a minimal $D\\in \\mathcal{D}$ such that there is $D'\\in \\mathcal{D}$ that satisfies $v\\in D'$ and $D'\\subseteq D$.\nSince $t\\le q$ we have that $\\mathcal{B}(v,t)\\subseteq D$.\nSince $(D,\\mathcal{A}'(D))$ comes from a partial random $q$-toast it follows that there is a non-zero event $\\mathcal{F}$ such that \n$$\\mathcal{A}'(D)\\upharpoonright \\mathcal{B}(v,t)=\\mathcal{A}(\\mathcal{D})\\upharpoonright \\mathcal{B}(v,t)$$\non $\\mathcal{F}$.\nThis shows that the constraint of $\\Pi$ at $v$ is satisfied and the proof is finished.\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{$\\mathsf{TAIL} = \\RLTOAST$}\n\\label{subsec:tail=randomlazytoast}\n\nIn this section we prove that the class $\\RLTOAST$ in fact covers the whole class $\\mathsf{TAIL} = \\bigcup_{f = \\omega(1)} \\mathsf{TAIL}(f(r))$. \n\n\\begin{restatable}{theorem}{tailrltoast}\n\\label{thm:tail=rltoast}\nLet $\\Pi$ be an LCL.\nThen $\\Pi$ is in the class $\\RLTOAST$ if and only if $\\Pi$ is in the class $\\mathsf{TAIL}$.\n\\end{restatable}\n\nWe first give an informal proof. Suppose that there is a random lazy $\\TOAST$ algorithm $\\mathcal{A}$ for $\\Pi$. We construct an oblivious algorithm with nontrivial tail behavior $\\mathcal{A}'$ for $\\Pi$ by constructing a toast via \\cref{thm:toast_in_tail} and during the construction, each node moreover simulates the algorithm $\\mathcal{A}$. The algorithm $\\mathcal{A}$ finishes after some number of steps with probability $1$, hence $\\mathcal{A}'$ finishes, too, with probability $1$ for each node. \n\nOn the other hand, suppose that there is oblivious local algorithm $\\mathcal{A}'$ for $\\Pi$. \nThen we can construct a lazy randomized toast algorithm $\\mathcal{A}$ that simply simulates $\\mathcal{A}'$ for every node. This is possible since as the pieces around a given vertex $u$ that has gets are growing bigger and bigger, the radius around $u$ that $\\mathcal{A}$ sees also grows, until it is large enough and $\\mathcal{A}$ sees the whole coding radius around $u$ and applies $\\mathcal{A}'$ then. \n\nA formal proof follows. \n\n\n\\begin{proof}\nSuppose that $\\Pi$ is in the class $\\RLTOAST$, i.e, there is $q\\in\\mathbb{N}$ together with a randomized toast algorithm $\\mathcal{A}$.\nIt follows from Theorem~\\ref{thm:toast moment} that there is an oblivious local algorithm $F$ that produces $q$-toast with probability $1$.\n\nDefine an oblivious local algorithm $\\mathcal{A}'$ as follows.\nLet $\\mathcal{B}(v,r)$ be a $\\Sigma_{in}$-labeled neighborhood of $v$.\nAssign real numbers from $[0,1]$ to $\\mathcal{B}(v,r)$ and run $F$ and $\\mathcal{A}$ on $\\mathcal{B}(v,r)$.\nLet $a_{v,r}$ be the supremum over all $\\Sigma_{in}$-labelings of $\\mathcal{B}(v,r)$ of the probability that $\\mathcal{A}'$ fails.\nIf $a_{v,r}\\to 0$ as $r \\to \\infty$, then $\\mathcal{A}'$ is an oblivious local algorithm.\n\nSuppose not, i.e., $a_{v,r}>a>0$.\nSince $\\Sigma_{in}$ is finite, we find by a compactness argument a labeling of $\\mathbb{Z}^d$ with $\\Sigma_{in}$ such that the event $\\mathcal{F}_r$ that $\\mathcal{A}'$ fails on $\\mathcal{B}(v,r)$ with the restricted labeling has probability bigger than $a$.\nNote that since the labeling is fixed, we have $\\mathcal{F}_r\\supseteq \\mathcal{F}_{r+1}$.\nSet $\\mathcal{F}=\\bigcap_{r\\in \\mathbb{N}} \\mathcal{F}_r$.\nThen by $\\sigma$-additivity we have that the probability of the event $\\mathcal{F}$ is bigger than $a>0$.\nHowever, $F$ produces toast almost surely on $\\mathcal{F}$ and $\\mathcal{A}$ produces $\\Pi$-coloring with probability $1$ for any given toast.\nThis contradicts the fact that probability of $\\mathcal{F}$ is not $0$.\n\n\n\n\\begin{comment}\nIn another words, with probability $1$ any node $v$ can reconstruct bigger and bigger piece of the toast $\\mathcal{D}$ when exploring its neighborhood.\nIn particular, $v$ will eventually became element of some piece of the toast and $\\mathcal{A}$ has to label $v$ with $\\Sigma_{out}$ after finitely many steps.\nThis defines an oblivious local algorithm $\\mathcal{B}$.\n\n\n\nPick a labeling $\\ell\\in [0,1]^{\\mathbb{Z}^d}$ such that $F(\\ell)$ is a $q$-toast and $\\mathcal{A}(\\mathcal{D},\\ell)$ is a $\\Pi$-coloring for $q$-toast $\\mathcal{D}$.\nNote that the picked labeling satisfies these conditions with probability $1$.\n\nDefine $G(\\ell)=\\mathcal{A}(F(\\ell),\\ell)$.\nIt remains to observe that $G$ is an oblivious local algorithm.\nNote that $G(\\ell)({\\bf 0}_d)=\\mathcal{A}(F(\\ell),\\ell)({\\bf 0}_d)$ depends on some $D\\in \\mathcal{D}$ such that ${\\bf 0}_d\\in D$, $\\{E\\subseteq D:E\\in \\mathcal{D}\\}$ and $\\ell\\upharpoonright D$.\nSince $F$ is an oblivious local algorithm, we see that $G$ is too.\nConsequently, $\\Pi$ is in the class $\\mathsf{TAIL}$.\n\\end{comment}\n\nThe idea to prove the converse is to ignore the $q$-toast part and wait for the coding radius.\nSuppose that $\\Pi$ is in the class $\\mathsf{TAIL}$.\nBy the definition there is an oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi$ and an unbounded function $f$ such that\n$$\\P(R_\\mathcal{A}>r)\\le \\frac{1}{f(r)}.$$\nGiven an input labeling of $\\mathbb{Z}^d$ by $\\Sigma_{in}$, we see by the definition of $R_\\mathcal{A}$ that every node $v$ must finish after finitely many steps with probability $1$.\nTherefore it is enough to wait until this finite neighborhood of $v$ becomes a subset of some piece of a given toast $\\mathcal{D}$.\nThis happens with probability $1$ independently of the given toast $\\mathcal{D}$.\n\\end{proof}\n\n\\begin{comment}\nLet $(A,c,\\ell)$ be a triplet as in definition of randomized toast algorithm.\nTake $v\\in A\\setminus \\operatorname{dom}(c)$, and suppose that $v={\\bf 0}_d$.\nOutput $\\mathcal{A}(A,c,\\ell)({\\bf 0}_d)$ if and only if there is $t\\in \\mathbb{N}$ such that $\\mathcal{B}({\\bf 0}_d,t)\\subseteq A$ and $\\ell\\upharpoonright \\mathcal{B}({\\bf 0}_d,t)$ corresponds to some coding neighborhood of $F$.\nThat is to say, any extension $\\ell\\upharpoonright \\mathcal{B}({\\bf 0}_d,t)\\sqsubseteq x\\in [0,1]^{\\mathbb{Z}^d}$ outputs the same color at $F(x)({\\bf 0}_d)$.\nIn that case output $\\mathcal{A}(A,c,\\ell)({\\bf 0}_d)=F(x)({\\bf 0}_d)$.\n\nWe show that $\\mathcal{A}$ produces a $\\Pi$-coloring for $\\mu$-almost every $\\ell\\in [0,1]^{\\mathbb{Z}^d}$ and $1$-toast $\\mathcal{D}$.\nLet $\\ell\\in [0,1]^{\\mathbb{Z}^d}$ be such that $F$ computes a $\\Pi$-coloring an need a finite coding radius for every $v\\in \\mathbb{Z}^d$.\nNote that almost every labeling $\\ell$ satisfies this condition.\nTake a $1$-toast $\\mathcal{D}$.\nIt is easy to see that $\\mathcal{A}(\\mathcal{D},\\ell)$ does not depend on $\\mathcal{D}$ and that $\\mathcal{A}(\\mathcal{D},\\ell)=F(\\ell)$, i.e., $\\mathcal{A}(\\mathcal{D},\\ell)$ is a $\\Pi$-coloring.\nThis finishes the proof.\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\\section{$\\TOAST$: Specific Problems}\\label{sec:SpecificProblems}\n\\todo{explain that some of this is work of others and we show it to substantiate definition of toast}\n\nIn this section we focus on specific problems that are not in the class $\\mathsf{LOCAL}(O(\\log^* n))$ but they admit a solution by a (spontaneous) toast algorithm. Recall, that proving that $\\Pi \\in \\TOAST$ implies that there is an oblivious local algorithm with $1\/r^{1-o(1)}$ tail decay via \\cref{thm:toast_in_tail} and we discuss in \\cref{subsec:descriptive_combinatorics_discussion} that by a result of \\cite{GJKS2} it also implies $\\Pi\\in \\mathsf{BOREL}$. \n\nThe problems we discuss are the following: (a) vertex coloring, (b) edge coloring, (c) tiling the grid with boxes from a prescribed set (this includes, e.g., a perfect matching problem). \n\n\n\n\n\n\nAll these problems were recently studied in the literature from different points of view, see \\cite{ArankaFeriLaci, HolroydSchrammWilson2017FinitaryColoring, GJKS, spencer_personal}.\nIn all cases the main technical tool, sometimes in a slight disguise, is the toast construction.\nOur aim here is to put these problems into a unified context.\nWe note that sometimes the results in the aforementioned literature are stronger or work in a bigger generality (for example, all results discussed here can also be proven to hold on unoriented grids as discussed in \\cite{ArankaFeriLaci}, or the constructed coloring could be aperiodic \\cite{spencer_personal}, etc)\nIn what follows we only give a ``proof by picture'', sketch the description of the corresponding spontaneous toast algorithm and refer the reader to other papers for more details\/stronger results.\n\nThroughout this section we assume that $d>1$.\nWe refer the reader to \\cref{sec:OtherClasses} for definition of classes $\\mathsf{CONTINUOUS}$, $\\mathsf{BOREL}$, etc.\n\n\n\nThe main contribution if this section is the following:\n\\begin{enumerate}\n \\item The fact that the problem of vertex $3$-coloring is in $\\TOAST$ together with the speed-up \\cref{thm:grid_speedup} answers a question (i) from \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n \\item We observe that the problem of edge $2d$-coloring the grid $\\mathbb{Z}^d$ is in $\\TOAST$, thereby answering a question of \\cite{GJKS} coming from the area of descriptive combinatorics. Independently, this was proven in \\cite{ArankaFeriLaci}. \n\\end{enumerate}\n\nIn general, our approach gives that if an LCL $\\Pi$ is in the class $\\TOAST\\setminus \\mathsf{LOCAL}(O(\\log^*(n)))$, then $1\\le {\\bf M}(\\Pi)\\le d-1$ by \\cref{thm:toast_in_tail} and \\cref{thm:grid_speedup}.\nThe examples in Section~\\ref{sec:Inter} show that there are LCLs with ${\\bf M}(\\Pi)=\\ell$ for every natural number $1\\le \\ell\\le d-1$, but for most natural LCLs we do not know the exact value of ${\\bf M}(\\Pi)$, unless $d=2$.\n\n\n\\subsection{Vertex Coloring}\n\\label{subsec:3coloring}\n\nLet $d>1$.\nRecall that $\\Pi^d_{\\chi,3}$ is the \\emph{proper vertex $3$-coloring problem}.\nThis problem was studied extensively in all contexts.\nIt is known that $\\Pi^d_{\\chi,3}\\not\\in \\mathsf{LOCAL}(O(\\log^* n))$ \\cite{brandt_grids}, $\\Pi^d_{\\chi,3}\\not\\in \\mathsf{CONTINUOUS}$ \\cite{GJKS}, $\\Pi^d_{\\chi,3}\\in \\mathsf{BOREL}$ \\cite{GJKS2} and $0<{\\bf M}(\\Pi^d_{\\chi,3})\\le 2$ \\cite{HolroydSchrammWilson2017FinitaryColoring}.\nA version of the following theorem is proven in \\cite{GJKS2}, here we give a simple ``proof by picture''. \n\n\\begin{theorem}[\\cite{GJKS2}]\\label{thm:3in toast}\nLet $d>1$.\nThe LCL $\\Pi^d_{\\chi,3}$ is in the class $\\TOAST$.\n\\end{theorem}\n\n\\begin{proof}[Proof Sketch]\nThe toast algorithm tries to color all nodes in a given piece with two colors, blue and red. However, it can happen that two partial colorings of small pieces that have different parity in a bigger piece have to be connected (see \\cref{fig:3-coloring}). \nIn this case, we take the small pieces colored with incorrect parity and ``cover'' their border with the third color: green. From the inside of a small piece, the green color corresponds to one of the main two colors, while from the outside it corresponds to the other color. This enables us to switch the color parity. \n\\end{proof}\n\nAs a consequence of results from \\cref{sec:Toast} we can now fully determine the moment behavior of the problem which answers a question from \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n\n\\begin{corollary}\nLet $d>1$.\nEvery oblivious local algorithm that solves $\\Pi^d_{\\chi,3}$ has the $1$-moment infinite, but there is an oblivious local algorithm solving $\\Pi^d_{\\chi,3}$ with all $(1-\\varepsilon)$-moments finite. \nIn particular, ${\\bf M}(\\Pi^d_{\\chi,3})=1$. \n\\end{corollary}\n\\begin{proof}\nBy \\cref{thm:3in toast}, $\\Pi^d_{\\chi,3}$ is in the class $\\TOAST$.\nIt follows from \\cref{thm:toast_in_tail} that ${\\bf M}(\\Pi^d_{\\chi,3})\\ge 1$.\nBy the speedup \\cref{thm:grid_speedup} and the fact that $3$-coloring was proven not to be in class $\\mathsf{LOCAL}(O(\\log^* n)) = \\mathsf{TAIL}(\\tow(\\Omega(r))) = \\mathsf{CONTINUOUS}$ \\cite{brandt_grids,HolroydSchrammWilson2017FinitaryColoring,GJKS}, we have that every oblivious local algorithm $\\mathcal{A}$ solving $\\Pi^2_{\\chi,3}$ must have the $1$-moment infinite.\nIn particular, ${\\bf M}(\\Pi^2_{\\chi,3})=1$.\nIt remains to observe that every vertex $3$-coloring of $\\mathbb{Z}^d$ yields a vertex $3$-coloring of $\\mathbb{Z}^2$ by forgetting the last $(d-2)$ dimensions.\nConsequently, every oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi^d_{\\chi,3}$ defines an oblivious local algorithm $\\mathcal{A}'$ that solves $\\Pi^2_{\\chi,3}$ with the same tail behavior.\nThis finishes the proof.\n\\end{proof}\n\nThe $3$-coloring discussion above also implies that the toast itself cannot be constructed faster then with linear tail, otherwise $3$-coloring could be constructed with such a decay, too. \n\\begin{corollary}\n\\label{cor:toast_lower_bound}\nThere is a constant $C > 0$ such that a $q$-toast cannot be constructed in $\\mathsf{TAIL}(C r)$ on any $\\mathbb{Z}^d$.\nHere, by constructing a $q$-toast we mean that the coding radius of each node $u$ is such that $u$ can describe the minimal toast piece $D \\in \\mathcal{D}$ such that $u \\in D$. \n\\end{corollary}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width = .95\\textwidth]{fig\/3-col-real.eps}\n \\caption{The $\\TOAST$ algorithm is given small pieces of a toast that are already vertex-colored. First step is to match the parity of colors with respect to the bigger piece. Then color the bigger piece.}\n \\label{fig:3-coloring}\n\\end{figure}\n\n\n\\subsection{Edge Coloring}\n\\label{subsec:edgecoloring}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width = .95\\textwidth]{fig\/4-edge-col.png}\n \\caption{The $\\TOAST$ algorithm is given small pieces of a toast that are already edge-colored. First step is to match the parity of colors with respect to the bigger piece. This is done by adding a three layered gadget in the problematic directions. After that the coloring is extended to the bigger piece.}\n \\label{fig:4-edge-coloring}\n\\end{figure}\\todo{dopsat}\nLet $d>1$.\nWe denote as $\\Pi^d_{\\chi',2d}$ the \\emph{proper edge $2d$-coloring problem}.\nObserve that a solution to $\\Pi^d_{\\chi',2d}$ yields a solution to the perfect matching problem $\\Pi^d_{pm}$ (because $\\mathbb{Z}^d$ is $2d$-regular).\nThis shows that $\\Pi^d_{\\chi',2d}$ is not in the class $\\mathsf{LOCAL}(O(\\log^* n))$, because the perfect matching problem clearly cannot be solved with a local algorithm: if it was, we could find a perfect matching in every large enough finite torus, which is clearly impossible since it can have an odd number of vertices \\cite{brandt_grids}. \n\n\n\n\\begin{theorem}[See also \\cite{ArankaFeriLaci}]\\label{thm:edge toast}\nLet $d>1$.\nThen $\\Pi^d_{\\chi',2d}$ is in the class $\\TOAST$.\n\\end{theorem}\n\\begin{proof}[Proof Sketch]\nThe spontaneous toast algorithm is illustrated for $d=2$ in \\cref{fig:4-edge-coloring}.\nThe ``interface'' used by the toast algorithm between smaller and bigger pieces is: smaller pieces pretend they are colored such that in the vertical direction violet and green edges alternate, while in the horizontal direction blue and red edges alternate. If all small pieces in a bigger one have the same parity, this interface is naturally extended to the bigger piece. Hence, we need to work out how to fix pieces with bad horizontal\/vertical parity, similarly to \\cref{thm:3in toast}. \n\n\\cref{fig:4-edge-coloring} shows how to fix the vertical parity, adding the highlighted gadget in the middle picture above and below given pieces keeps its interface the same but switches the parity. The same can be done in the horizontal direction. \n\nTo show the result for $d>2$ it is enough to realize that switching parity in one dimension can be done with a help of one extra additional dimension, while fixing the parity in the remaining dimensions.\nConsequently, this allows to iterate the previous argument at most $d$-many times to fix all the wrong parities.\n\\end{proof}\n\n\n\\begin{corollary}\nLet $d>1$.\nThen $1 \\le {\\bf M}(\\Pi^d_{\\chi',2d}) \\le d-1$. \nIn particular, ${\\bf M}(\\Pi^2_{\\chi',4})=1$.\n\\end{corollary}\n\nWe leave as an open problem to determine the exact value of ${\\bf M}(\\Pi^d_{\\chi',2d})$, where $d>2$.\n\nThe following corollary answers a question of \\cite{GJKS}. See \\cref{sec:OtherClasses} for the definition of class $\\mathsf{BOREL}$. \n\\begin{corollary}\nLet $d\\ge 1$.\nThen $\\Pi^d_{\\chi',2d}$ is in the class $\\mathsf{BOREL}$.\nIn particular, if $\\mathbb{Z}^d$ is a free Borel action and $\\mathcal{G}$ is the Borel graph generated by the standard generators of $\\mathbb{Z}^d$, then $\\chi'_\\mathcal{B}(\\mathcal{G})=2d$, i.e., the Borel edge chromatic number is equal to $2d$.\n\\end{corollary}\n\n\n\\begin{remark}[Decomposing grids into finite cycles]\nBencs, Hru\u0161kov\u00e1, T\u00f3th \\cite{ArankaFeriLaci} showed recently that there is a $\\TOAST$ algorithm that decomposes an unoriented grid into $d$-many families each consisting of finite pairwise disjoint cycles.\nHaving this structure immediately gives the existence of edge coloring with $2d$-colors as well as Schreier decoration.\nSince every $\\TOAST$ algorithms imply Borel solution on graphs that look locally like unoriented grids \\cite{AsymptoticDim}, this result implies that all these problems are in the class $\\mathsf{BOREL}$.\nWe refer the reader to \\cite{ArankaFeriLaci} for more details.\n\\end{remark}\n\n\n\n\n\n\n\n\\begin{comment}\n\\paragraph{Decomposing grids into finite cycles} \\todo{Jsem pro zkratit to na jeden odstavec}\n\nWe sketch how to use toast structure to decompose $\\mathbb{Z}^d$ into $d$-many families each consisting of finite pairwise disjoint cycles, formally, a collection $\\{\\mathcal{C}_i\\}_{i\\in [d]}$ such that $\\mathcal{C}_i$ is family of pairwise vertex disjoint cycles for every $i\\in [d]$ and $\\bigcup_{i\\in [d]} \\mathcal{C}_i=\\mathbb{Z}^d$.\nNote that finding such a decomposition is not an LCL because we do not have bound on the length of the cycles.\nOn the other hand, if we allow infinite paths, then such problem is LCL problem and we denote it as $\\Pi^d_{\\operatorname{cyc},d}$.\nMoreover, it is easy to see that since we work with oriented grids $\\Pi^d_{\\operatorname{cyc},d}$ is in the class $\\mathsf{LOCAL}(O(1))$.\n\nWe show that there is a spontaneous extension function $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is always a $\\Pi^d_{\\operatorname{cyc},d}$-coloring with the additional property that all the cycles in $\\mathcal{A}(\\mathcal{D})$ are finite for every $37$-toast $\\mathcal{D}$.\nMoreover, it is easy to see that our toastable algorithm also works for non-oriented grids, where the LCL problem $\\Pi^d_{\\operatorname{cyc},d}$ is not trivial.\nThe combinatorial construction is inspired by result of Bencs, Hruskova and Toth \\cite{ArankaFeriLaci} who originally proved similar result for $d=2$.\nAfter reading their paper we realized that the argument goes through for any dimension $d>1$ and we refer the reader to their paper for a detailed proof of this fact.\nHere we only sketch the main idea.\n\n\\begin{theorem}\nLet $d>1$.\nThen there is a spontaneous extension function $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is always a $\\Pi^d_{\\operatorname{cyc},d}$-coloring for every $37$-toast $\\mathcal{D}$ with the additional property that all the cycles in $\\mathcal{A}(\\mathcal{D})$ are finite.\n\\end{theorem}\n\n\\begin{remark}\nA straightforward use of the toastable algorithm together with the fact that Borel graphs with finite Borel asymptotic dimension admit Borel toast structure (see~\\cite{AsymptoticDim}) gives a decomposition of any Borel graph that looks locally like a $d$-dimensional grid (oriented or non-oriented) into $d$-many Borel families of edges each consisting of pairwise disjoint cycles.\nSince each such cycle is necessarily of even length, we get immediately that such graphs admit Borel Schreier decoration, Borel balanced orientation and their Borel chromatic index is $2d$.\n\\end{remark}\n\n\\begin{proof}[Proof Sketch]\nLet $d>1$.\nIt is easy to see that decomposing $\\mathbb{Z}^d$ into $d$ families of cycles (possibly infinite) is the same as coloring the edges with $d$-colors, such that every vertex $v\\in \\mathbb{Z}^d$ has exactly two adjacent edges of each color.\nLet $\\{\\alpha_1,\\dots,\\alpha_d\\}$ be the set of the colors.\n\nAn \\emph{(edge) coloring} of a connected finite set $A\\subseteq \\mathbb{Z}^d$ is a coloring of all edges that are adjacent to $A$.\nWe say that an edge $e$ is a \\emph{boundary edge}, if only one vertex of $e$ is contained in $A$.\nA coloring $c$ is \\emph{suitable}, if after choosing a reference point and orientation of $A$ we have that parallel boundary edges in the $i$-th direction are assigned the color $\\alpha_i$.\n\nThere is a constant $q\\in \\mathbb{N}$ such that for every $i\\not=j\\in [d]$, finite connected set $A\\subseteq \\mathbb{Z}^d$ and a suitable edge coloring $c$, there is a suitable edge coloring $c'$ of $A':=\\mathcal{B}(A,q)$ that extends $c$ such that \n\\begin{itemize}\n \\item every $c'$-cycle of color $\\alpha_i$ that contains a vertex from $A$ is finite for every $i\\in [d]$,\n \\item the $i$-th and $j$-th directions of $A$ and $A'$ are swapped, e.g., parallel $A'$-boundary edges in the $i$-th direction from the point of view of $A$ are $c'$-colored with $\\alpha_j$,\n \\item other directions remain the same.\n\\end{itemize}\nIt is easy to see that iterating this extension possibly $2d$-many times produces a spontaneous toast algorithm $\\mathcal{A}$ with all the desired properties.\n\nIt remains to sketch how to find such extension $c'$.\nFirst, since $c$ is suitable and we want to swap $i$-th and $j$-th direction, we may assume that $d=2$.\nSecond, we may assume that $A$ is a union of rectangles, such that each side has even number of vertices adjacent to a perpendicular boundary edge.\nThis could be done as follows, pick a reference point and consider decomposition of $\\operatorname{Cay}(\\mathbb{Z}^2)$ into a square of a fixed even number of vertices on a side.\nTake the union of all such squares that intersect $A$.\nThird, see Figure~\\ref{fig:SwapingColors} how to swap colors and close all cycles coming out of $A$.\nThis finishes the sketch.\n\\end{proof}\n\\end{comment}\n\n\n\n\n\\subsection{Tiling with Rectangles}\n\\label{subsec:rectangle_tiling}\n\n\nLet $d>1$ and $\\mathcal{R}$ be a finite collection of $d$-dimensional rectangles.\nWe denote as $\\Pi^d_{\\mathcal{R}}$ the problem of \\emph{tiling $\\mathbb{Z}^d$ with elements of $\\mathcal{R}$}.\nIt is not fully understood when the LCL $\\Pi^d_{\\mathcal{R}}$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$ (or equivalently $\\mathsf{CONTINUOUS}$).\nCurrently the only known condition implying that $\\Pi^d_{\\mathcal{R}}$ is \\emph{not} in $\\mathsf{LOCAL}(O(\\log^* n))$ is when GCD of the volumes of elements of $\\mathcal{R}$ is not $1$.\nMinimal unknown example is when $\\mathcal{R}=\\{2\\times 2, 2\\times 3,3\\times 3\\}$. \nFor the other complexity classes the situation is different.\nGiven a collection $\\mathcal{R}$ we define a condition \\eqref{eq:GCD} as\n\\begin{equation}\n\\label{eq:GCD}\\tag{$\\dagger$}\n \\text{GCD of side lengths in the direction of} \\ i \\text{-th generator is} \\ 1 \\ \\text{for every} \\ i\\in [d].\n\\end{equation}\nIt follows from the same argument as for vertex $2$-coloring, that if $\\mathcal{R}$ does not satisfy \\eqref{eq:GCD}, then $\\Pi^d_{\\mathcal{R}}$ is a global problem, i.e., it is not in any class from \\cref{fig:big_picture_grids}.\nOn the other hand, the condition \\eqref{eq:GCD} is sufficient for a spontaneous toast algorithm. For example, this implies $\\Pi^2_{\\{2\\times 3,3\\times 2\\}}\\in \\mathsf{BOREL} \\setminus \\mathsf{CONTINUOUS}$.\nA version of the following theorem was announced in \\cite{spencer_personal}, here we give a simple ``proof by picture''.\n\n\\begin{theorem}[\\cite{spencer_personal, stephen}]\n\\label{thm:tilings}\nLet $d>1$ and $\\mathcal{R}$ be a finite collection of $d$-dimensional rectangles that satisfy \\cref{eq:GCD}.\nThen $\\Pi^d_{\\mathcal{R}}$ is in the class $\\TOAST$.\nConsequently, ${\\bf M}(\\Pi^d_{\\mathcal{R}})\\ge 1$.\n\\end{theorem}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=.9\\textwidth]{fig\/rectangle_filling.eps}\n \\caption{The $\\TOAST$ algorithm is given small pieces of a toast that are already tiled. First step is to match the parity of the tiling with respect to the bigger piece. Then tile the bigger piece.}\n \\label{fig:Tiling}\n\\end{figure}\n\n\\begin{proof}[Proof Sketch]\nThe two-dimensional version of the construction is shown in \\cref{fig:Tiling}. The interface of pieces this time is that they cover themselves with boxes of side length $L$ which is the least common multiple of all the lengths of basic boxes. As in the previous constructions, we just need to work out how to ``shift'' each piece so that all small pieces agree on the same tiling with boxes of side length $L$ which can then be extended to the whole piece. This can be done as in \\cref{fig:Tiling}. \n\\end{proof}\n\n\n\\begin{corollary}\nLet $d=2$ and $\\mathcal{R}$ be a finite collection of $2$-dimensional rectangles.\nThen $\\Pi^2_{\\mathcal{R}}$ is either in the class $\\mathsf{LOCAL}(O(\\log^* n))$, satisfies ${\\bf M}(\\Pi^2_{\\mathcal{R}})=1$, or is global (that is, in no class in \\cref{fig:big_picture_grids}).\n\\end{corollary}\n\nFor any $d$, \\cref{thm:tilings} implies that if a tiling problem is ``hard'' (in no class in \\cref{fig:big_picture_grids}), it is because of a simple reason: a projection to one dimension gives a periodic problem. It would be interesting to understand whether there are problems that are hard but their solutions do not have any periodicity in them (see \\cref{sec:open_problems}). \n\nWe do not know if there is $\\mathcal{R}$ that would satisfy $1< {\\bf M}(\\Pi^d_{\\mathcal{R}})\\le d-1$.\nLet $d\\ge \\ell>0$ and denote as $\\mathcal{R}_\\ell$ the restriction of rectangles to first $\\ell$-dimensions.\nAs in the case of proper $3$-vertex coloring problem, we have that ${\\bf M}(\\Pi^d_{\\mathcal{R}})\\le {\\bf M}(\\Pi^\\ell_{\\mathcal{R}_{\\ell}})$.\nIn particular, if, e.g., $\\Pi^d_{\\mathcal{R}_2}$ is not in the class $\\mathsf{LOCAL}(O(\\log^* n))$, then ${\\bf M}(\\Pi^d_{\\mathcal{R}})\\le 1$.\n\n\\begin{remark}\nIn the context of non-oriented grids we need to assume that $\\mathcal{R}$ is invariant under swapping coordinates. \nSimilar argument, then shows that $\\Pi^d_\\mathcal{R}\\in \\TOAST$ if and only if $\\mathcal{R}$ satisfies \\eqref{eq:GCD}.\nCombined with the fact that there is a Borel toast on such Borel graphs (see~\\cite{AsymptoticDim}), we deduce that \\eqref{eq:GCD} for $\\mathcal{R}$ is equivalent with existence of Borel tiling with rectangles from $\\mathcal{R}$.\n\\end{remark}\n\n\n\\todo{2x3 a 3x2 se daji poskladat z elek, coz odpovida nejake partition kruznice}\n\n\n\n\n\n\\paragraph{Perfect matching}\n\nLet $d>1$.\nA special case of a tiling problem is the \\emph{perfect matching problem} $\\Pi^d_{pm}$.\nSince there are arbitrarily large tori of odd size we conclude that $\\Pi^d_{pm}\\not \\in \\mathsf{LOCAL}(O(\\log^* n))$.\nConsequently, $\\Pi^d_{pm}\\not \\in \\mathsf{TAIL}(\\tow(\\Omega(r)))$ by \\cref{thm:tail=uniform}, and it follows from the Twelve Tile Theorem of Gao et al \\cite[Theorem 5.5]{GJKS} that $\\Pi^d_{pm}\\not \\in \\mathsf{CONTINUOUS}$.\nSimilarly as with the vertex $3$-coloring problem, the proof of $\\Pi^d_{pm}\\in \\mathsf{BOREL}$, where $d>1$, from \\cite{GJKS2} uses the toast construction and, in our notation, yields the following result. \n\n\\begin{theorem}[\\cite{GJKS2}]\\label{thm:perfect_matching_intoast}\nLet $d>1$.\nThe LCL $\\Pi^d_{pm}$ is in the class $\\TOAST$.\n\\end{theorem}\n\n\\begin{corollary}\nLet $d>1$.\nThen ${\\bf M}(\\Pi^d_{pm})\\ge 1$ and every ffiid $F$ that solves $\\Pi^d_{\\chi,3}$ has the $(d-1)$-moment infinite.\nIn particular, ${\\bf M}(\\Pi^2_{pm})=1$.\n\\end{corollary}\n\nWe leave as an open problem to determine the exact value of ${\\bf M}(\\Pi^d_{pm})$, where $d>2$.\nIn fact, this problem is similar to intermediate problems that we study in \\cref{sec:Inter}. \n\n\n\n\n\n\n\n\n\n\n\n\\renewcommand{\\pm}[2]{\\operatorname{PM}^{#1, #2}_\\boxtimes}\n\\renewcommand{\\pm}[1]{\\operatorname{PM}^{#1}_\\boxtimes}\n\\newcommand{\\mathbb{Z}^d_\\boxtimes}{\\mathbb{Z}^d_\\boxtimes}\n\n\\section{Intermediate Problems}\\label{sec:Inter}\n\nIn this section we construct local problems with $\\mathsf{TAIL}$ complexities $\\Theta(r^{j-o(1)})$ for $1 \\le j \\le d-1$. \nWe first define the problem $\\pm{d}$: this is just a perfect matching problem, but in the graph $\\mathbb{Z}^d_\\boxtimes$ where two different nodes $u = (u_1, \\dots, u_d)$ and $v = (v_1, \\dots, v_d)$ are connected with an edge if $\\sum_{1 \\le i \\le d} |u_i - v_i| \\le 2$. That is, we also add some ``diagonal edges'' to the grid. We do not know whether perfect matching on the original graph (Cayley graph of $\\mathbb{Z}^d$) is also an intermediate problem (that is, not $\\mathsf{LOCAL}$ but solvable faster than $\\TOAST$). \n\nWe will show that the tail complexity of $\\pm{d}$ is $\\Theta(r^{d - 1 - o(1)})$. \nAfter this is shown, the hierarchy of intermediate problems is constructed as follows. \nConsider the problem $\\pm{d,\\ell}$. This is a problem on $\\mathbb{Z}^d$ where one is asked to solve $\\pm{d}$ in every hyperplane $\\mathbb{Z}^\\ell$ given by the first $\\ell$ coordinates in the input graph. \nThe $\\mathsf{TAIL}$ complexity of $\\pm{d,\\ell}$ is clearly the same as the complexity of $\\pm{\\ell}$, that is, $\\Theta(r^{\\ell-1-o(1)})$. \nHence, we can focus just on the problem $\\pm{d}$ from now on. \n\n\\begin{claim}\\label{cl:PM is global}\nLet $2\\le d\\in \\mathbb{N}$.\nThen the local coloring problem $\\pm{d}$ is not in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\nVia \\cref{thm:grid_speedup} this is equivalent to $\\pm{d} \\not\\in \\mathsf{TAIL}(Cr^{d-1})$ for large enough $C$. \n\\end{claim}\n\nThis follows from the fact that if there is a local algorithm, then in particular there exists a solution on every large enough torus. But this is not the case if the torus has an odd number of vertices. \n\nSecond, we need to show how to solve the problem $\\pm{d}$ faster than with a $\\TOAST$ construction algorithm. \n\n\\begin{restatable}{theorem}{intermediate}\n\\label{thm:intermediate}\nThe problem $\\pm{d}$ can be solved with an oblivious local algorithm of tail complexity $r^{d-1-O(\\frac{1}{\\sqrt{\\log r}})}$. \n\\end{restatable}\n\nHere we give only a proof sketch, the formal proof is deferred to \\cref{app:intermediate_problems}. \n\n\\begin{proof}[Proof Sketch]\nWe only discuss the case $d=3$ (and the actual proof differs a bit from this exposition due to technical reasons). As usual, start by tiling the grid with boxes of side length roughly $r_0$. \nThe \\emph{skeleton} $\\mathcal{C}_0$ is the set of nodes we get as the union of the $12$ edges of each box. \nIts \\emph{corner points} $X_0 \\subseteq \\mathcal{C}_0$ is the union of the $8$ corners of each box. \n\n\nWe start by matching vertices inside each box that are not in the skeleton $\\mathcal{C}_0$. It is easy to see that we can match all inside vertices, maybe except of one node per box that we call a \\emph{pivot}. We can ensure that any pivot is a neighbor of a vertex from $\\mathcal{C}_0$. \n\nWe then proceed similarly to the toast construction of \\cref{thm:toast_in_tail} and build a sequence of sparser and sparser independent sets that induce sparser and sparser subskeletons of the original skeleton.\nMore concretely, we choose a sequence $r_k = 2^{O(k^2)}$ and in the $k$-th step we construct a new set of corner points $X_k \\subseteq X_{k-1}$ such that any two corner points are at least $r_k$ hops apart but any point of $\\mathbb{Z}^d$ has a corner point at distance at most $2r_k$ (in the metric induced by $\\mathbb{Z}^d$). \nThis set of corner points $X_k$ induces a subskeleton $\\mathcal{C}_{k} \\subseteq \\mathcal{C}_{k-1}$ by connecting nearby points of $X_k$ with shortest paths between them in $\\mathcal{C}_{k-1}$. \n\nAfter constructing $\\mathcal{C}_k$, the algorithm solves the matching problem on all nodes of $\\mathcal{C}_{k-1} \\setminus \\mathcal{C}_k$. This is done by orienting those nodes towards the closest $\\mathcal{C}_k$ node in the metric induced by the subgraph $\\mathcal{C}_k$. \nThen, the matching is found on connected components that have radius bounded by $r_k$. There is perhaps only one point in each component, a pivot, that neighbors with $\\mathcal{C}_{k}$ and remains unmatched. \n\nAfter the $k$-th step of this algorithm, only nodes in $\\mathcal{C}_k$ (and few pivots) remain unmatched. Hence, it remains to argue that only roughly a $1\/r_k^{d-1}$ fraction of nodes survives to $\\mathcal{C}_k$. This should be very intuitive -- think about the skeleton $\\mathcal{C}_k$ as roughly looking as the skeleton generated by boxes of radius $r_k$. This is in fact imprecise, because in each step the distance between corner points of $X_k$ in $\\mathcal{C}_k$ may be stretched by a constant factor with respect to their distance in the original skeleton $\\mathcal{C}_{k-1}$. This is the reason why the final volume of $\\mathcal{C}_k$ is bounded only by $2^{O(k)} \/ r_k^{d-1}$ and by a similar computation as in \\cref{thm:toast_in_tail} we compute that the tail decay of the construction is $2^{O(\\sqrt{\\log r})}\/{r^{d-1}}$. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Other Classes of Problems}\\label{sec:OtherClasses}\n\nIn this mostly expository section we discuss other classes of problems from \\cref{fig:big_picture_grids} and how they relate to our results from previous sections. \n\n\n\n\\subsection{$\\mathsf{CONTINUOUS}$}\\label{subsec:CONT}\n\nHere we explain the fact that an LCL $\\Pi$ on $\\mathbb{Z}^d$ is in the class $\\mathsf{LOCAL}(O(\\log^*(n)))$ if and only if it is in the class $\\mathsf{CONTINUOUS}$.\nFor $\\mathbb{Z}^d$, the model $\\mathsf{CONTINUOUS}$ was extensively studied by Gao, Jackson, Krohne, Tucker-Drob, Seward and others, see \\cite{GaoJackson,GJKS,ST,ColorinSeward}.\nIn fact, the connection with local algorithms is implicit in \\cite{GJKS} under the name Twelve Tiles Theorem \\cite[Theorem~5.5]{GJKS}.\nRecently, independently Bernshteyn \\cite{Bernshteyn2021local=cont} and Seward \\cite{Seward_personal} showed that the same holds in bigger generality for all Cayley graphs of finitely generated groups.\nOur aim here is to formally introduce the model $\\mathsf{CONTINUOUS}$ and give an informal overview of the argument for $\\mathbb{Z}^d$ that follows the lines of Gao et al \\cite{GJKS}.\nThe reasons not to simply put reference to the aforementioned papers are two.\nFirst, we work with slightly more general LCLs that have inputs.\nSecond, our goal is to introduce all classes in a way accessible for a broader audience. \n\nThe setup of the $\\mathsf{CONTINUOUS}$ model is somewhat similar to the deterministic $\\mathsf{LOCAL}$ model: there is no randomness in the definition and there is a certain symmetry breaking input on the nodes. Specifically, each vertex is labelled with one label from the set $\\{0, 1\\}$; the space of all such labelings is denoted as $2^{\\mathbb{Z}^d}$. \nHowever, for the same reason as why we insist on the uniqueness of the identifiers in the $\\mathsf{LOCAL}$ model, we insist that the input $\\{0,1\\}$ labeling breaks all symmetries. Namely, we require that there is no $g$ such that shifting all elements by $g$ preserves the labeling. We write $\\operatorname{Free}(2^{\\mathbb{Z}^d}) \\subseteq 2^{\\mathbb{Z}^d}$ for this subspace. \nA \\emph{continuous algorithm} is a pair $\\mathcal{A}=(\\mathcal{C},f)$, where $\\mathcal{C}$ is a collection of $\\{0,1\\} \\times \\Sigma_{in}$-labeled neighborhoods and $f$ is a function that assigns to each element of $\\mathcal{C}$ an output label. It needs to satisfy the following. Suppose we fix a node $v$ and the input $\\{0,1\\}$-labeling $\\Lambda$ of $\\mathbb{Z}^d$ satisfies $\\Lambda \\in \\operatorname{Free}(2^{\\mathbb{Z}^d})$. If we let $v$ explore its neighborhood with an oblivious local algorithm, then we encounter an element from $\\mathcal{C}$ after finitely many steps. \nWe write $\\mathcal{A}(\\Lambda)(v)$, or simply $\\mathcal{A}(v)$, when $\\Lambda$ is understood, for the corresponding output.\n\n\\begin{comment}\n\\vspace*{5cm}\nTo break symmetries on $\\mathbb{Z}^d$ in the setting of random processes one considers iid labels on vertices.\nThe event that some symmetries remain has measure $0$ and we simply ignore such labelings.\nIn the continuous settings we label vertices with $2:=\\{0,1\\}$.\nThe space of all such labelings is denoted as $2^{\\mathbb{Z}^d}$ and we write $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ for the space of labelings that break all symmetries.\nA \\emph{continuous algorithm} is a pair $\\mathcal{A}=(\\mathcal{C},g)$, where $\\mathcal{C}$ is a collection of ${0,1},\\Sigma_{in}$-labeled neighborhoods and $g$ is a function that assigns to each element of $\\mathcal{C}$ an output label, such that if we let a node $v$ explore its labeled neighborhood $x$, then it encounters element from $\\mathcal{C}$ after finitely many steps whenever $x\\in \\Sigma_{in}^{\\mathbb{Z}^d}\\times \\operatorname{Free}(2^{\\mathbb{Z}^d})$, i.e., $x$ breaks all symmetries.\nWe write $\\mathcal{A}(x)(v)$ for the corresponding output.\n\\end{comment}\n\n\\begin{remark}\nThis is equivalent with the definition given in e.g. \\cite{GJKS} in the same way as the correspondence between oblivious local algorithms and ffiid, see \\cref{subsec:subshifts}.\nThe fact that the collection $\\mathcal{C}$ is dense, i.e., every node finishes exploring its neighborhoods in finite time, means that $\\mathcal{A}({\\bf 0})$ is \\emph{continuous} with respect to the product topology and defined in the whole space $\\operatorname{Free}(2^{\\mathbb{Z}^d})$.\nMoreover, the value computed at $v$ does not depend on the position of $v\\in \\mathbb{Z}^d$ but only on the isomorphism type of the labels, this is the same as saying that $\\mathcal{A}(\\_)$ is \\emph{equivariant}.\n\\end{remark}\n\nWe say that an LCL $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ if there is a continuous algorithm $\\mathcal{A}$ that outputs a solution for every $\\Lambda \\in \\operatorname{Free}(2^{\\mathbb{Z}^d})$.\nNote that, in general, we cannot hope for the diameters of the neighborhoods in $\\mathcal{C}$ to be uniformly bounded.\nIn fact, if that happens then $\\Pi \\in \\mathsf{LOCAL}(O(1))$. \nThis is because there are elements in $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ with arbitrarily large regions with constant $0$ label.\nApplying $\\mathcal{A}$ on that regions gives an output label that depends only on $\\Sigma_{in}$ after bounded number of steps, hence we get a local algorithm with constant local complexity.\n\nThe main idea to connect this context with distributed computing is to search for a labeling in $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ where any $\\mathcal{A}$ necessarily has a uniformly bounded diameter.\nSuch a labeling does exist!\nIn \\cite{GJKS} these labelings are called \\emph{hyperaperiodic}.\nTo show that hyperaperiodic labelings exist on $\\mathbb{Z}^d$ is not that difficult, unlike in the general case for all countable groups \\cite{ColorinSeward}. \nIn fact there is enough flexibility to encode in a given hyperaperiodic labeling some structure in such a way that the new labeling remains hyperaperiodic.\n\nWe now give a definition of a hyperaperiodic labeling. \nLet $\\Lambda \\in 2^{\\mathbb{Z}^d}$ be a labeling of $\\mathbb{Z}^d$.\nWe say that $\\Lambda$ is \\emph{hyperaperiodic} if for every $g\\in \\mathbb{Z}^d$ there is a finite set $S\\subseteq \\mathbb{Z}^d$ such that \nfor every $t\\in \\mathbb{Z}^d$ we have \n$$\\Lambda(t+S) \\not= \\Lambda(g+t+S)$$\nThis notation means that the $\\Lambda$-labeling of $S$ differs from the $\\Lambda$-labeling of $S$ shifted by $g$ everywhere. \nIn another words, given some translation $g\\in \\mathbb{Z}^d$, there is a uniform witness $S$ for the fact that $\\Lambda$ is nowhere $g$-periodic.\nIt follows from a standard compactness argument that any continuous algorithm $\\mathcal{A}$ used on a hyperaperiodic labeling finishes after number of steps that is uniformly bounded by some constant that depends only on $\\mathcal{A}$. \n\n\n\n\\begin{comment}\n\\vspace*{5cm}\n\nLet $\\Lambda \\in 2^{\\mathbb{Z}^d}$ be a labeling of $\\mathbb{Z}^d$.\nWe say that $\\Lambda$ is \\emph{hyperaperiodic} if for every $g\\in \\mathbb{Z}$ there is a finite set $S\\subseteq \\mathbb{Z}^d$ such that \nfor every $s\\in \\mathbb{Z}^d$ we have \n$$x(s+t)\\not=x(g+s+t)$$\nfor some $t\\in S$.\nIn another words, given a shift $g\\in \\mathbb{Z}^d$ there is a uniform witness $S$ for the fact that $x$ is nowhere $g$-periodic.\nIt follows from some standard compactness argument that every hyperaperiodic element needs uniformly bounded neighborhoods for each continuous algorithm $\\mathcal{A}$, the bound of course depends on the algorithm. \n\\end{comment}\n\n\\begin{theorem}[\\cite{GJKS,Bernshteyn2021local=cont}]\nLet $d>0$.\nThen an LCL $\\Pi$ on $\\mathbb{Z}^d$ is in the class $\\mathsf{CONTINUOUS}$ if and only if it is in the class $\\mathsf{LOCAL}(O(\\log^*(n)))$.\n\\end{theorem}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=.9\\textwidth]{fig\/continuous.eps}\n \\caption{On the left side we have examples of a labeled local tiling configurations compared with a possible continuity window $w$. These are then sparsely embedded into $\\mathbb{Z}^2$.}\n \\label{fig:my_label}\n\\end{figure}\n\n\\begin{proof}[Proof Sketch]\nOne direction was observed by Bernshteyn \\cite{Bernshteyn2021local=cont} and follows from the fact that there is continuous algorithm that produces a maximal $r$-independent set for every $r\\in \\mathbb{N}$ (recall \\cref{thm:classification_of_local_problems_on_grids}).\nThis is implicit, even though in a different language, already in Gao and Jackson \\cite{GaoJackson}.\n\nNext, suppose that $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ and write $\\mathcal{A}$ for the continuous algorithm for $\\Pi$. \nWe will carefully construct a certain hyperaperiodic element $\\Lambda$ and let $\\mathcal{A}$ solve $\\Pi$ on it. \nWe know that there is a certain number $w$ such that $\\mathcal{A}$ solves $\\Pi$ on $\\Lambda$ with local complexity $w$. \nWe note that even though we do not know the value $w$ in advance, it depends on $\\Lambda$ that is due to be constructed, the flexibility of the construction guarantees to ``encode'' in $\\Lambda$ all possible scenarios.\nSee the discussion at the end of the argument. \n\nThe overall approach is as follows. \nOur local algorithm $\\mathcal{A}'$ first tiles the plane with tiles of length much bigger than $w$ with $O(\\log^* n)$ local complexity. We cannot tile the plane in a periodic manner in $O(\\log^* n)$ steps, but it can be done with a constant number of different tiles such that the side length of each tile is either $r$ or $r+1$ for some $r \\gg w$ (this construction is from \\cite[Theorem~3.1]{GaoJackson}, see also the proof of \\cref{thm:grid_speedup}). \nFor each tile $T$, $\\mathcal{A}'$ labels it with certain $\\{0,1\\}$ labeling. \nThis labeling depends only on the shape of the tile.\nWe make sure that not only $T$ with its labeling, but also $T$ together with all possible local configurations of tiles around it and corresponding labelings appears in $\\Lambda$.\nIn particular, as $r\\gg t,w$ ($t$ is the radius necessary to check the solution $\\Pi$), we have that the $(w+t)$-hop neighborhood of any node is labelled in a way that also appears somewhere in the labeling $\\Lambda$.\nAfter labeling each tile, $\\mathcal{A}'$ simulates $\\mathcal{A}$. We know that $\\mathcal{A}$ finishes after $w$ steps on $\\Lambda$ and is correct there, hence $\\mathcal{A}'$ is correct and has local complexity $O(\\log^* n)$. What remains is to describe is how to get the labeling of each tile. \n\nWe use the fact that there is a lot of flexibility when constructing a hyperaperiodic labeling. \nWe define the hyperaperiodic labeling $\\Lambda$ of the input graph $\\mathbb{Z}^d$ in such a way that, after fixing $r$, each one of constantly many local configurations of $\\{r,r+1\\}$-tiles are embedded in $\\Lambda$, each one in fact in many copies that exhaust possible $\\Sigma_{in}$-labelings.\n\nThis is done as follows.\nFix any hyperaperiodic element $\\Lambda'$, these exist by \\cite[Lemma~2.8]{GJKS}.\nGiven a tile $T$, that depends on $r$, $\\{0,1\\}$-label it by any $T$-pattern that appears in $\\Lambda'$.\nConsider the collection of all possible local configurations of such labeled tiles that are moreover labeled by $\\Sigma_{in}$.\nThis collection is finite. Since we do not know $w$ in advance, do this for any value of $r$. This yields a countable collection of possible configurations. \nEmbed this collection sparsely into $\\Lambda'$, \\cref{fig:my_label}. Extend the $\\Sigma_{in}$-labeling arbitrarily. \nThis defines $\\Lambda$.\nIt is routine to check that $\\Lambda$ is hyperaperiodic, see \\cite[Lemma~5.9]{GJKS}.\n\n\n\nIt remains to recall that since $\\Lambda$ is hyperaperiodic, there is $w\\in \\mathbb{N}$ such that $\\mathcal{A}$ has uniformly bounded locality by $w$ on $\\Lambda$.\nPick $r\\gg w,t$.\nThis works as required since any $w+t$-hop neighborhood $\\mathcal{A}'$ can encounter is present in $\\Lambda$.\n\\end{proof}\n\n\\begin{comment}\n\\vspace*{4cm}\nNext, suppose that $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ and write $\\mathcal{A}$ for the continuous algorithm.\nA high-level strategy for $d=2$ is as follows.\nThere are 12 type of tiles that are parametrized by natural numbers $n,p,q\\in \\mathbb{N}$, i.e., for fixed $n,p,q$ there are $12$ tiles.\nIf $n,p,q$ are fixed, then there is a local deterministic algorithm that tiles given, large enough, torus with these tiles in such a way that the boundary parts matches.\\todo{picture here ad reference to the nice pictures in \\cite{GJKS}}\nIt is not clear at this point how large parameters we should choose.\nThe trick is to encode all such possible tiles for all $n,p,q\\in \\mathbb{N}$ in one hyperaperiodic element $x$.\nThe encoding of the tiles in $x$ guarantees that whenever the regions \\todo{picture} are the same for the fixed parameters $n,p,q$, then $\\mathcal{A}$ outputs the same label.\nSince $x$ is hyperaperiodic, we have that $\\mathcal{A}$ stops after $T$-many steps to output $\\mathcal{A}(x)(v)$ for some $T\\in \\mathbb{N}$.\nWe choose $n,p,q$ according to $T$ in such a way that for any vertex at any region there is a tile such that the $T$-neighborhood of this vertex is fully contained in the tile.\nConsequently, this defines local rule once a given, large enough, torus is tiled into these tiles of 12 types.\n\nThis description exactly follows the proof of (1) $\\rightarrow$ (2) in \\cite[Theorem~5.5]{GJKS}, where (1) is the same as saying that there is a continuous algorithm and (2) fixes the coloring of the tiles for some parameters $n,p,q$.\nHowever, we note that we are skipping some technical details about additional parameter $w$ etc.\nThe additional step to produce a local algorithm by tiling a large enough torus into these 12 tiles is standard.\nIn fact, this is proved in \\cite[Theorem~5.10]{GJKS}, again in different language, to show (2) $\\rightarrow$ (1) in \\cite[Theorem~5.5]{GJKS}.\n\nThe main difference that we encounter are the input labels.\nWe explain how to handle them and hint, simultaneously, what we mean by encoding tiles in hyperaperiodic element, again ignoring a technical parameter $w$, see \\cite[Figures~13,14,etc]{GJKS}.\nFix a hyperaperiodic element $y$.\nFor $n,p,q\\in \\mathbb{N}$ and a tile $G$, see \\eqref{fig:???}, assign to each region labels from $\\{0,1\\}$ in such a way that (a) same regions get the same pattern, (b) this pattern is part of $y$.\nConsider moreover any possible input labeling of such $\\{0,1\\}$-labeled tiles.\nNow place these labeled tiles, for every $n,p,q\\in \\mathbb{N}$, sparsely in $y$ (Figure ???) and extend the input labeling on them to the whole $\\mathbb{Z}^2$, this defines $x$.\nIt is not hard to see that $x$ remains hyperaperiodic.\n\nIn higher dimension, i.e., when $d>2$, the same argument works as well with the only difference that there are more than 12 types of tiles.\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\\subsection{Descriptive Combinatorics Classes}\n\\label{subsec:descriptive_combinatorics_discussion}\n\nIn this section we define the classes of problems that are studied in descriptive combinatorics.\nThere are two equivalent ways how the notion of a \\emph{Borel graph} can be defined.\nThe less abstract one is as follows.\nConsider the unit interval $[0,1]$ with the $\\sigma$-algebra of Borel sets, that is the smallest algebra of subsets that contains sub-intervals and is closed under countable unions and complements.\nLet $A,B\\subseteq [0,1]$ be Borel sets and consider a bijection $\\varphi:A \\to B$ between them.\nWe say that $\\varphi$ is \\emph{Borel measurable} if $\\varphi^{-1}(D\\cap B)$ is a Borel set for every Borel set $D$.\nNow, let $\\mathcal{G}$ be a graph with vertex set $[0,1]$ and assume that the degree of every vertex is bounded by some $\\Delta<\\infty$.\nWe say that $\\mathcal{G}$ is a \\emph{Borel graph} if the edge relation is induced by Borel bijections, that is there are sets $A_1,\\dots, A_\\ell$, $B_1,\\dots, B_\\ell$ and Borel bijections $\\varphi_1,\\dots, \\varphi_\\ell$ such that $(x,y)$ is an edge in $\\mathcal{G}$ if and only if there is $1\\le i\\le \\ell$ such that either $\\varphi_i(x)=y$ or $\\varphi_i(y)=x$.\n\nThe more abstract way how to define Borel graphs uses the notion of a \\emph{standard Borel space}.\nFor example $[0,1]$ with the $\\sigma$-algebra of Borel sets, when we forget that it comes from the standard topology, is a standard Borel space.\nFormally, a standard Borel space is a pair $(X,\\mathcal{B})$, where $X$ is a set and $\\mathcal{B}$ is a $\\sigma$-algebra that comes from some separable and complete metric structure on $X$.\nA uniformly bounded degree graph $\\mathcal{G}$ on $(X,\\mathcal{B})$ is Borel if it satisfies the same condition as in the case of the particular space $[0,1]$.\n\nIt is easy to see that with this definition all the basic constructions from graph theory are \\emph{relatively} Borel, e.g. if $A$ is a Borel set of vertices, then $N_\\mathcal{G}(A)$ the neighbors of elements in $A$ form a Borel set.\nAlso note that since we assume that the degree is uniformly bounded, every connected component in $\\mathcal{G}$ is at most countable.\nTypical questions that are studied in this context are Borel versions of standard notions, e.g., \\emph{Borel} chromatic number, \\emph{Borel} chromatic index, \\emph{Borel} perfect matching etc.\nWe refer the reader to \\cite{pikhurko2021descriptive_comb_survey,kechris_marks2016descriptive_comb_survey} for more details.\n\nThe most relevant Borel graphs for us are the ones that are induced by a free Borel action of $\\mathbb{Z}^d$.\nThat is, there are $d$-many aperiodic Borel bijections $\\varphi_i:X\\to X$ that pairwise commute, where \\emph{aperiodic} means that $\\varphi_i^k(x)\\not=x$ for every $k>0$, $i\\in [d]$ and $x\\in X$.\nThen the Borel (labeled) graph $\\mathcal{G}$ that is induced by this action consists of edges of the form $(x,\\varphi_i(x))$, where $x\\in X$ and $i\\in [d]$.\nThe most important result in this context is that $\\mathcal{G}$ admits a \\emph{Borel toast structure}.\nThis follows from work of Gao et al \\cite{GJKS2}, see Marks and Unger \\cite{Circle} for proof.\nWe note that it is not understood in general what Borel graphs admit a Borel toast structure.\nThis is connected with the notoriously difficult question about Borel hyperfiniteness, see \\cite{AsymptoticDim}.\n\nGiven an LCL $\\Pi$, we say that $\\Pi$ is in the class $\\mathsf{BOREL}$ if there is a Borel measurable $\\Pi$-coloring of every Borel graph $\\mathcal{G}$ as above, i.e., induced by a free Borel action of $\\mathbb{Z}^d$.\nEquivalently, it is enough to decide whether $\\Pi$ can be solved on the canonical Borel graph $\\mathcal{G}$ on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$.\nMoreover, if $\\Pi$ has some inputs $\\Sigma_{in}$, then we consider any Borel labeling $X\\to \\Sigma_{in}$ and then ask if there is a Borel solution that satisfies all the constraints on $\\Sigma_{in}\\times \\Sigma_{out}$.\n\n\\begin{theorem}[Basically \\cite{GJKS}]\nLet $\\Pi$ be an LCL that is in the class $\\LTOAST$.\nThen $\\Pi\\in\\mathsf{BOREL}$.\n\\end{theorem}\n\nIn fact, all the LCL problems without input that we are aware of and that are in the class $\\mathsf{BOREL}$ are in fact in the class $\\TOAST$.\nIt is an open problem to decide whether it is always the case.\n\nAnother interesting perspective how to see Borel constructions is as a countably infinite analogue of $\\mathsf{LOCAL}(O(\\log^*(n)))$.\nNamely, suppose that you are allowed to inductively (a) construct Borel MIS with some parameter (b) do local $O(1)$-construction that depends on the previously build structure.\nIt is clear that objects that are build in this way are Borel, however, it is not clear if this uses the whole potential of Borel sets to find Borel solutions of LCLs.\n\nAs a last remark we note that traditionally in descriptive combinatorics, people consider relaxations of Borel constructions.\nThis is usually done via ignoring how a solution behave on some negligible Borel set.\nProminent examples are \\emph{measure $0$ sets} for some Borel probability measure on $(X,\\mathcal{B})$ or \\emph{meager sets} for some compatible complete metric on $(X,\\mathcal{B})$.\nWe denote the classes of problems that admit such a solution for \\emph{every} Borel probability measure or compatible complete metric as $\\mathsf{MEASURE}$ and $\\mathsf{BAIRE}$, respectively.\nWithout going into any details we mention that on general graphs these classes are much richer than $\\mathsf{BOREL}$, see e.g. \\cite{DetMarks,BrooksMeas}.\nHowever, it is still plausible that on $\\mathbb{Z}^d$ we have $\\mathsf{BOREL}=\\mathsf{BAIRE}=\\mathsf{MEASURE}$.\nIt is well-know that this is true in $\\mathbb{Z}$ \\cite{grebik_rozhon2021LCL_on_paths}.\n\n\n\\begin{comment}\nIn both cases we are allowed to neglect some small sets, where we either do not care about the solution or are allowed to use the axiom of choice.\nSuppose that $\\mathcal{G}$ is a Borel graph on a standard Borel space $(X,\\mathcal{B})$ and $\\Pi$ is an LCL.\nSuppose, in addition, that there is a Borel probability measure $\\mu$ on $(X,\\mathcal{B})$, that is a $\\sigma$-additive measure on $\\mathcal{B}$, see \\cite[Section~?]{Kec}.\nWe say that $\\Pi$ admits \n\n\n\n\n\n\n\n\n\n\nand show the arrows that are left to complete Figure~\\ref{fig:big_picture_grids}.\nRecall that a \\emph{Borel graph} $\\mathcal{G}$ is a triplet $\\mathcal{G}=(X,\\mathcal{B},E)$, where $(X,\\mathcal{B})$ is a standard Borel space, $(X,E)$ is a graph and $E\\subseteq X\\times X$ is an element of the product $\\sigma$-algebra.\nFor the purposes of this paper we always assume that $\\mathcal{G}$ is induced by a free Borel action of $\\mathbb{Z}^d$ for some $d>0$, i.e., there is a Borel action $\\mathbb{Z}^d\\curvearrowright X$ with trivial stabilizers such that $\\mathcal{G}$ is induced by the standard generators of $\\mathbb{Z}^d$.\n\nLet $\\Pi$ be an LCL.\nThen $\\Pi$ belongs to the class $\\mathsf{BOREL}$ if there is a Borel $\\Pi$-coloring of every such Borel graph $\\mathcal{G}$.\nMoreover, we say that $\\Pi$ is in the class $\\mathsf{MEASURE}$ if for every such Borel graph $\\mathcal{G}$ and for every \\emph{Borel probability measure} $\\mu$ on $(X,\\mathcal{B})$ there is a Borel $\\Pi$-coloring of $\\mathcal{G}$ on a $\\mu$-conull set.\nSimilarly, we say that $\\Pi$ is in the class $\\mathsf{BAIRE}$ if for every such Borel graph $\\mathcal{G}$ and for every \\emph{compatible Polish topology} $\\tau$ on $(X,\\mathcal{B})$ there is a Borel $\\Pi$-coloring of $\\mathcal{G}$ on a $\\tau$-comeager set.\nIt follows directly from the definition that $\\mathsf{BOREL}\\subseteq \\mathsf{MEASURE},\\mathsf{BAIRE}$.\n\\end{comment}\n\n\n\n\\begin{comment}\nNext we follow Gao et al. \\cite{GJKS}.\nDenote as $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ the free part of the Bernoulli shift $\\mathbb{Z}^d\\curvearrowright 2^{\\mathbb{Z}^d}$ endowed with the inherited product topology.\nThen it is easy to see that $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ is a Polish space and that the action is continuous.\nConsider the Borel graph $\\mathcal{G}$ on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ induced by the standard set of generators.\nThis graph is universal in the sense that an LCL $\\Pi$ is in the class $\\mathsf{BOREL}$ if and only if there is a Borel $\\Pi$-coloring of $\\mathcal{G}$, this follows from result of Seward and Tucker-Drob~\\cite{?}.\nWe say that an LCL $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ if there is a $\\Pi$-coloring of $\\mathcal{G}$ that is continuous.\nThis is the same as saying that that there is a continuous equivariant map from $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ to $X_\\Pi$.\nThe following is a corollary of the Twelve Tiles Theorem of Gao et al. \\cite{GJKS}.\n\n\\begin{theorem}\\cite[Theorem 5.5]{GJKS}\n$\\mathsf{CONTINUOUS}=\\mathsf{LOCAL}(O(\\log^* n))$.\n\\end{theorem}\n\nWe note that in the general case of all groups this equality was shown by Bernshteyn\\cite{?} and, independently, by Seward. \n\nIt remains to describe how toastable algorithms give Borel solutions.\nAs an immediate corollary we get that all the problems from Section~\\ref{sec:SpecificProblems} are in the class $\\mathsf{BOREL}$.\n\n\\begin{theorem}\n$\\LTOAST \\subseteq \\mathsf{BOREL}$.\n\\end{theorem}\n\\begin{proof}\nSuppose that $\\Pi$ is in the class $\\LTOAST$ and let $f$ be an toast algorithm with parameter $k$.\nIt follows from work of Gao et al \\cite{GJKS2}, see Marks and Unger \\cite{Circle} for proof, that every Borel graph $\\mathcal{G}$ (that is induced by a free action of $\\mathbb{Z}^d$) admits a Borel $k$-toast for every $k\\in \\mathbb{N}$.\nIt is a standard fact that using $f$ inductively on the depth of the toast produces a Borel $\\Pi$-coloring of $\\mathcal{G}$.\n\\end{proof}\n\\end{comment}\n\n\n\n\n\n\\subsection{Finitely Dependent Processes}\n\nIn this section we use the notation that was developed in \\cref{subsec:subshifts}, also we restrict our attention to LCLs without input.\nLet $\\Pi=(b,t,\\mathcal{P})$ be such an LCL.\nRecall that $X_\\Pi$ is the space of all $\\Pi$-colorings.\n\nWe say that a distribution $\\mu$ on $b^{\\mathbb{Z}^d}$ is \\emph{finitely dependent} if it is shift-invariant and there is $k\\in \\mathbb{N}$ such that the events $\\mu\\upharpoonright A$ and $\\mu \\upharpoonright B$ are independent for every finite set $A,B\\subseteq \\mathbb{Z}^d$ such that $d(A,B)\\ge k$.\nNote that shift-invariance is the same as saying that the distribution $\\mu\\upharpoonright A$ does not depend on the position of $A$ in $\\mathbb{Z}^d$ but rather on its shape.\n\nHere we study the following situation:\nSuppose that there is a finitely dependent process $\\mu$ that is concentrated on $X_\\Pi$.\nIn that case we say that $\\Pi$ is in the class $\\mathsf{FINDEP}$. \nIn other words, $\\Pi \\in \\mathsf{FINDEP}$ if there is a distribution from which we can sample solutions of $\\Pi$ such that if we restrict the distribution to any two sufficient far sets, the corresponding restrictions of the solution of $\\Pi$ that we sample are independent. \n\nIt is easy to see that all problems in $\\mathsf{LOCAL}(O(1))$ (that is, problems solvable by a block factor) are finitely dependent, but examples of finitely dependent processes outside of this class were rare and unnatural, see the discussion in \\cite{FinDepCol}, until Holroyd and Liggett \\cite{FinDepCol} provided an elegant construction for 3- and 4-coloring of a path and, more generally, for distance coloring of a grid with constantly many colors. This implies the following result.\n\n\\begin{theorem}[\\cite{FinDepCol}]\n$\\mathsf{LOCAL}(O(\\log^* n)) \\subseteq \\mathsf{FINDEP}$ (see \\cref{fig:big_picture_grids}).\n\\end{theorem} \n\nOn the other hand, recently Spinka \\cite{Spinka} proved that any finitely dependent process can be described as a finitary factor of iid process.\\footnote{His result works in a bigger generality for finitely generated amenable groups.}\nUsing \\cref{thm:toast_in_tail}, we sketch how this can be done, we keep the language of LCLs.\n\n\\begin{theorem}[\\cite{Spinka}]\nLet $\\Pi$ be an LCL on $\\mathbb{Z}^d$ and $\\mu$ be finitely dependent process on $X_\\Pi$, i.e., $\\Pi\\in \\mathsf{FINDEP}$.\nThen there is a finitary factor of iid $F$ (equivalently an oblivious local algorithm $\\mathcal{A}$) with tail decay $1\/r^{1- O(\\sqrt{\\log r})}$ such that the distribution induced by $F$ is equal to $\\mu$.\nIn fact, $\\Pi$ is in the class $\\TOAST$. \n\\end{theorem}\n\\begin{proof}[Proof Sketch]\nSuppose that every vertex is assigned uniformly two reals from $[0,1]$.\nUse the first string to produce a $k$-toast as in \\cref{thm:toast moment}, where $k$ is from the definition of $\\mu$.\n\nHaving this use the second random string to sample from $\\mu$ inductively along the constructed $k$-toast.\nIt is easy to see that this produces ffiid $F$ that induces back the distribution $\\mu$.\nIn fact, it could be stated separately that the second part describes a randomized $\\TOAST$ algorithm: always samples from $\\mu$ given the coloring on smaller pieces that was already fixed. The finitely dependent condition is exactly saying that the joint distribution over the solution on the small pieces of $\\mu$ is the same as the distribution we sampled from.\nNow use \\cref{thm:derandomOfToast} to show that $\\Pi\\in \\TOAST$. \n\\end{proof}\n\nIn our formalism, $\\mathsf{FINDEP} \\subseteq \\TOAST$, or equivalently we get $\\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r))) \\subseteq \\mathsf{FINDEP} \\subseteq \\mathsf{TAIL}(r^{1-o(1)})$.\nIt might be tempting to thing that perhaps $\\mathsf{FINDEP} =\\TOAST$.\nIt is however a result of Holroyd et al \\cite[Corollary~25]{HolroydSchrammWilson2017FinitaryColoring} that proper vertex $3$-coloring, $\\Pi^{d}_{3,\\chi}$, is not in $\\mathsf{FINDEP}$ for every $d>1$.\nThe exact relationship between these classes remain mysterious.\n\n\n\n\\begin{comment}\n\\begin{theorem}\nLet $\\Pi$ be an LCL on $\\mathbb{Z}^d$.\nSuppose that $\\Pi$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\nThen there is finitely dependent process $\\mu$ on $X_\\Pi$.\n\\end{theorem}\n\\begin{proof}\nIt follows from Theorem~\\ref{thm:classification_of_local_problems_on_grids} that there is $r,\\ell \\in \\mathbb{N}$ such that $\\ell\\le r-t$ and a rule $\\mathcal{A}$ of locality at most $\\ell$ that produces a $\\Pi$-coloring for every $(r+t)$-power proper vertex coloring of $\\mathbb{Z}^d$.\n\nSuppose that $\\mu_{(r+t)}$ is a finitely dependent $(r+t)$-power proper vertex coloring of $\\mathbb{Z}^d$ process with locality $k\\in \\mathbb{N}$.\nWe claim that composing $\\mu_{(r+t)}$ with $\\mathcal{A}$ yields a finitely dependent process with locality $(2\\ell+k)$.\nLet $A\\subseteq \\mathbb{Z}^d$ be a finite set.\nThen the output of $\\mathcal{A}$ depends on $\\mathcal{B}(A,\\ell)$.\nTherefore if $A,B\\subseteq \\mathbb{Z}^d$ satisfy $d(A,B)>2\\ell+k$, then\n$$d(\\mathcal{B}(A,\\ell),\\mathcal{B}(B,\\ell))>k$$\nand we conclude that the outputs of $\\mathcal{A}$ on $A$ and $B$ are independent.\n\nIt remains to find $\\mu_{r}$ finitely dependent $r$-power proper vertex coloring of $\\mathbb{Z}^d$ process for every $r>0$.\nThis is done by Holroyd and Liggett \\cite[Theorem~20]{FinDepCol}.\n{\\color{red} or something like that, they use a lot of colors fr $r$-power coloring...}\n\\end{proof}\n\nGiven a fiid $F$ we define $F^*\\lambda$ to be the push-forward of the product measure via $F$.\nSince $F$ is equivariant it follows that $F^*\\lambda$ is shift-invariant.\n\nThe following theorem was essentially proven by Spinka\\cite{?}. His result nicely fits to our definitions in \\cref{sec:Toast}. \n\\begin{theorem}\n\n\\end{theorem}\n\n\n\\begin{theorem}\nLet $\\Pi$ be an LCL problem on $\\mathbb{Z}^d$ and $\\mu$ be finitely dependent process on $X_\\Pi$.\nThen for every $\\epsilon>0$ there is a ffiid $F$ such that $F^*\\lambda=\\mu$ and the tail of $F$ has the $(1-\\epsilon)$-moment finite.\nMoreover, $\\Pi$ is in the class $\\TOAST$.\n\\end{theorem}\n\n\\begin{remark}\nMaybe the random toast construction works for any finitely dependent processes not just LCL.\n\nThere is the notion of strongly irreducible shifts of finite type, some comment about it...\n\nDefine $\\mathsf{FINDEP}$ to be the class of all LCL problems that admit a finitely dependent distribution $\\mu$.\nIt follows that $\\mathsf{TAIL}(\\tow(\\Omega(r)))\\subseteq \\mathsf{FINDEP} \\subseteq \\mathsf{TOAST}$ and we have $\\mathsf{FINDEP}\\not = \\TOAST$ because proper vertex $3$-coloring $\\Pi^d_{\\chi,3}$ is not in $\\mathsf{FINDEP}$ for every $d>1$, by \\cite[Corollary~25]{HolroydSchrammWilson2017FinitaryColoring}.\n\\end{remark}\n\n\n\\begin{proof}\nWe define a spontaneous randomized toast algorithm $\\Omega$ such that $\\Omega(\\mathcal{D},\\ell)$ is a $\\Pi$-coloring for every $k$-toast $\\mathcal{D}$ and every $\\ell\\in [0,1]^{\\mathbb{Z}^d}$.\nThis already implies the moreover part by Theorem~\\ref{thm:derandomOfToast}.\n\nLet $A\\subseteq \\mathbb{Z}^d$ be a finite subset of $\\mathbb{Z}^d$.\nWrite $\\mathbb{P}_A$ for the distribution of $\\Pi$-patterns of $\\mu$ on $A$, i.e., $\\mathbb{P}_A=\\mu\\upharpoonright A$.\nSuppose now that $(A,c,\\ell)$ are given, where $A\\subseteq \\mathbb{Z}^d$ is finite subset, $c$ is a partial coloring of $A$ and $\\ell\\in [0,1]^{A}$.\nConsider the conditional distribution $\\mathbb{P}_{A,c}$ of $\\mathbb{P}_A$ given the event $c$.\nIf $\\mathbb{P}_{A,c}$ is non-trivial, then define $\\Omega$ such that $\\Omega(A,c,{-})$ induces the same distribution as $\\mathbb{P}_{A,c}$ when $\\ell$ is fixed on $\\operatorname{dom}(c)$.\nOtherwise define $\\Omega$ arbitrarily.\n\nLet $\\mathcal{D}$ be a $k$-toast.\nWe show that $\\Omega(\\mathcal{D},\\ell)$ is a $\\Pi$-coloring for every $\\ell\\in [0,1]^{\\mathbb{Z}^d}$ and $\\Omega(\\mathcal{D},{-})$ produces $\\mu$.\nNote that the first claim shows that $\\Pi$ is in the class $\\RTOAST$, while the second produces $\\mu$ as ffiid as follows.\nSuppose that every $v\\in \\mathbb{Z}^d$ is assigned two independent reals $l_{v,0}$ and $l_{v,1}$.\nUse the first real to produce a $k$-toast as in Theorem~\\ref{thm:toast moment} and use the second real together with the produced toast as an input to $\\Omega$.\nIt follows from the independence of $l_{v,0}$ and $l_{v,1}$ that this produces $\\mu$.\n\nWe show that\n$$\\Omega(\\mathcal{D},{-})\\upharpoonright D=\\mu\\upharpoonright D$$\nfor every $D\\in \\mathcal{D}$ inductively on $\\mathfrak{r}_{\\mathcal{D}}$.\nIf $\\mathfrak{r}_{\\mathcal{D}}(D)=0$, then it follows from the definition.\nSuppose that $\\mathfrak{r}_{\\mathcal{D}}(D)>0$ and let $E_1,\\dots, E_t\\in \\mathcal{D}$ be an enumeration of $E\\subset D$.\nIt follows from the inductive hypothesis together with the assumption that $\\mu$ is $k$-independent and $\\mathcal{D}$ is a $k$-toast that \n$$\\Omega(\\mathcal{D},{-})\\upharpoonright \\bigcup_{i\\le t}E_i=\\mu\\upharpoonright \\bigcup_{i\\le t} E_i.$$\nNow when $\\ell$, and therefore a partial coloring $c$, is fixed on $\\bigcup_{i\\le t}E_i=\\operatorname{dom}(c)$, the definition $\\Omega(D,c,{-})$ extends $c$ in such a way that $\\mathbb{P}_{D,c}=\\mu\\upharpoonright \\operatorname{dom}(c)$.\nSince this holds for any such assignment $\\ell$ of real numbers on $\\bigcup_{i\\le t}E_i$ we deduce that\n$$\\Omega(\\mathcal{D},{-})\\upharpoonright D=\\mu\\upharpoonright D$$\nand the proof is finished.\n\\end{proof}\n\\end{comment}\n\n\n\n\n\n\n\\subsection{$\\mathsf{TAIL}$ Complexities in Distributed and Centralised Models}\n\\label{subsec:congestLCA}\n\nAlthough we are not aware of a systematic work on $\\mathsf{TAIL}$ complexities in the distributed algorithms community, similar notions like oblivious algorithms \\cite{Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring}, node-average complexity \\cite{feuilloley2017node_avg_complexity_of_col_is_const,barenboim2018avg_complexity,chatterjee2020sleeping_MIS} or energy complexity \\cite{chang2018energy_complexity_exp_separation,chang2020energy_complexity_bfs} are studied. \nThe breakthrough MIS algorithm of Ghaffari \\cite{ghaffari2016MIS} may have found many applications exactly because of the fact that its basis is an algorithm with exponential tail behavior. \n\nWe discuss two applications of algorithms with nontrivial $\\mathsf{TAIL}$ complexities. First, the node-average complexity literature studies the average coding radius that a node needs to answer. Whenever an oblivious algorithm has its first moment finite, we have an algorithm with constant average coding radius. \nWith a little more work, one can see that the result from \\cref{thm:3in toast}, that is $3$-coloring of grids has $\\mathsf{TAIL}$ complexity $r^{1-O(1\/\\sqrt{\\log r})}$ can be adapted to the setting of finite grids where it gives a $3$-coloring algorithm with average complexity $2^{O(\\sqrt{\\log n})}$. \nFor nonconstant $\\Delta$, the situation is more complicated. In \\cite{barenboim2018avg_complexity} it was shown that the Luby's $\\Delta+1$ coloring algorithm has tail $2^{\\Omega(r)}$ even for nonconstant $\\Delta$, but the question of whether MIS can have constant average coding radius in nonconstant degree graphs is open \\cite{chatterjee2020sleeping_MIS}. \n\nAnother application of the $\\mathsf{TAIL}$ bound approach comes from the popular centralised $\\mathsf{LCA}$ model \\cite{lca11,lca12}: there one has a huge graph and instead of solving some problem, e.g., coloring, on it directly, we are only required to answer queries about the color of some given nodes. It is known that, as in the $\\mathsf{LOCAL}$ model, one can solve $\\Delta+1$ coloring with $O(\\log^* n)$ queries on constant degree graphs by a variant of Linial's algorithm \\cite{even03_conflictfree}. But using the tail version of the Linial's algorithm instead, we automatically get that $O(\\log^* n)$ is only a worst-case complexity guarantee and on average we expect only $O(1)$ queries until we know the color of a given node. That is, the amortized complexity of a query is in fact constant. The same holds e.g. for the $O(\\log n)$ query algorithm for LLL from \\cite{brandt_grunau_rozhon2021LLL_in_LCA}. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Open Questions}\n\\label{sec:open_problems}\n\nHere we collect some open problems that we find interesting. \n\n\\begin{enumerate}\n \\item Is it true that $\\RLTOAST = \\LTOAST$? If yes, this implies $\\mathsf{TAIL} \\subseteq \\mathsf{BOREL}$. \n \\item Is it true that $\\RLTOAST = \\RTOAST$? If yes, this implies $\\mathsf{TAIL} = \\mathsf{TAIL}(r^{1-o(1)})$ and answers the fourth question of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n\t\\item Can the tail complexity $r \/ 2^{O(\\sqrt{\\log r})}$ in the construction of toast be improved to $r \/ \\operatorname{poly} \\log (r)$? The same can be asked for the complexity of intermediate problems from \\cref{sec:Inter}. \n\t\t\\item Is there a local coloring problem $\\Pi$ such that ${\\bf M}(\\Pi)$ is non-integer value or $0$, i.e., ${\\bf M}(\\Pi)\\in [0,d-1]\\setminus \\{1,2,\\dots, d-1\\}$?\n\t\t\\item Can you find an example of a local problem without input labels which does not belong to any class in \\cref{fig:big_picture_grids} but the solutions to this problem are not ``periodic'' (this cannot happen for $d=1$). Maybe (a version of) Penrose tilings can be helpful there?\n\t\\item What is the value ${\\bf M}(\\Pi^d_{pm})$, where $\\Pi^d_{pm}$ is the problem of finding a perfect matching of $\\mathbb{Z}^d$, where $d>2$?\n \\item Let $d>1$.\n\tFor what finite collections of $d$-dimensional rectangles $\\mathcal{R}$ is $\\Pi^d_{\\mathcal{R}}$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$? Is there a finite collection of $d$-dimensional rectangles $\\mathcal{R}$ that would satisfy $1< {\\bf M}(\\Pi^d_{\\mathcal{R}})\\le d-1$?\n\t\\item Is it true (on $\\mathbb{Z}^d$) that $\\mathsf{BOREL}=\\mathsf{MEASURE}$ or $\\mathsf{BOREL} = \\mathsf{BAIRE}$?\n\\end{enumerate}\n\n\\section*{Acknowledgement}\nWe would like to thank Ferenc Bencs, Anton Bernshteyn, Sebastian Brandt, Yi-Jun Chang, Mohsen Ghaffari, Aranka Hru\u0161kov\u00e1, Stephen Jackson, Oleg Pikhurko, Brandon Sewart, Yinon Spinka, L\u00e1szlo T\u00f3th, and Zolt\u00e1n Vidny\u00e1szky for many engaging discussions. \n\nThe first author was supported by Leverhulme Research Project Grant RPG-2018-424. \nThis project has received funding from the European Research Council (ERC) under the European\nUnions Horizon 2020 research and innovation programme (grant agreement No. 853109). \n\n\\bibliographystyle{alpha}\n\n\n\\section{Introduction}\n\\label{sec:intro}\nIn this work, we study local problems on regular trees from three different perspectives. \n\n\\textit{First,} we consider the perspective of distributed algorithms. In distributed computing, the studied setup is a network of computers where each computer can only communicate with its neighbors. Roughly speaking, the question of interest in this area is which problems can be solved with only a few rounds of communication in the underlying network. \n\n\\textit{Second}, we consider the perspective of (finitary) factors of iid processes.\nIn probability, random processes model systems that appear to vary in a random manner.\nThese include Bernoulli processes, Random walks etc.\nA particular, well-studied, example is the Ising model.\n\n\\textit{Third}, we investigate the perspective of descriptive combinatorics. The goal of this area is to understand which constructions on infinite graphs can be performed without using the so-called axiom of choice.\n\nAlthough many of the questions of interest asked in these three areas are quite similar to each other, no systematic connections were known until an insightful paper of Bernshteyn~\\cite{Bernshteyn2021LLL} who showed that results from distributed computing can automatically imply results in descriptive combinatorics.\nIn this work, we show that the connections between the three areas run much deeper than previously known, both in terms of techniques and in terms of complexity classes.\nIn fact, our work suggests that it is quite useful to consider all three perspectives as part of a common theory, and we will attempt to present our results accordingly.\nWe refer the reader to \\cref{fig:big_picture_trees} for a partial overview of the rich connections between the three perspectives, some of which are proven in this paper. \n\nIn this work, we focus on the case where the graph under consideration is a regular tree.\nDespite its simplistic appearance, regular trees play an important role in each of the three areas, as we will substantiate at the end of this section.\nTo already provide an example, in the area of distributed algorithms, almost all locality lower bounds are achieved on regular trees.\nMoreover, when regarding lower bounds, the property that they already apply on regular trees actually \\emph{strengthens} the result---a fact that is quite relevant for our work as our main contribution regarding the transfer of techniques between the areas is a new lower bound technique in the area of distributed computation that is an adaptation and generalization of a technique from descriptive combinatorics.\nRegarding our results about the relations between complexity classes from the three areas, we note that such connections are also studied in the context of paths and grids in two parallel papers \\cite{grebik_rozhon2021LCL_on_paths,grebik_rozhon2021toasts_and_tails}.\n\nIn the remainder of this section, we give a high-level overview of the three areas that we study.\nThe purpose of these overviews is to provide the reader with a comprehensive picture of the studied settings that can also serve as a starting point for delving deeper into selected topics in those areas---in order to follow our paper, it is not necessary to obtain a detailed understanding of the results and connections presented in the overviews. Necessary technical details will be provided in \\cref{sec:preliminaries}.\nMoreover, in \\cref{sec:contribution}, we present our contributions in detail.\n\n\n\n\n\n\\paragraph{Distributed Computing}\nThe definition of the $\\mathsf{LOCAL}$ model of distributed computing by Linial~\\cite{linial92LOCAL} was motivated by the desire to understand distributed algorithms in huge networks. \nAs an example, consider a huge network of wifi routers. Let us think of two routers as connected by an edge if they are close enough to exchange messages. \nIt is desirable that such close-by routers communicate with user devices on different channels to avoid interference.\nIn graph-theoretic language, we want to properly color the underlying network. \nEven if we are allowed a color palette with $\\Delta+1$ colors where $\\Delta$ denotes the maximum degree of the graph (which would admit a simple greedy algorithm in a sequential setting), the problem remains highly interesting in the distributed setting, as, ideally, each vertex decides on its output color after only a few rounds of communication with its neighbors, which does not allow for a simple greedy solution. \n\nThe $\\mathsf{LOCAL}$ model of distributed computing formalizes this setup: we have a large network, where each vertex knows the network's size, $n$, and perhaps some other parameters like the maximum degree $\\Delta$. In the case of randomized algorithms, each vertex has access to a private random bit string, while in the case of deterministic algorithms, each vertex is equipped with a unique identifier from a range polynomial in the size $n$ of the network. \nIn one round, each vertex can exchange any message with its neighbors and can perform an arbitrary computation. The goal is to find a solution to a given problem in as few communication rounds as possible. \nAs the allowed message size is unbounded, a $t$-round $\\mathsf{LOCAL}$ algorithm can be equivalently described as a function that maps $t$-hop neighborhoods to outputs---the output of a vertex is then simply the output its $t$-hop neighborhood\nmapped to by this function.\nAn algorithm is correct if and only if the collection of outputs at all vertices constitutes a correct solution to the problem. \n\nThere is a rich theory of distributed algorithms and the local complexity of many problems is understood. \nThe case of trees is a highlight of the theory: it is known that any local problem (a class of natural problems we will define later) belongs to one of only very few complexity classes.\nMore precisely, for any local problem, its randomized local complexity is either $O(1), \\Theta(\\log^* n), \\Theta(\\log\\log n), \\Theta(\\log n)$, or $\\Theta(n^{1\/k})$ for some $k \\in \\mathbb{N}$. Moreover, the deterministic complexity is always the same as the randomized one, except for the case $\\Theta(\\log\\log n)$, for which the corresponding deterministic complexity is $\\Theta(\\log n)$ (see \\cref{fig:big_picture_trees}). \n\n\n\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{fig\/big_picture_trees.pdf}\n \\caption{Complexity classes on regular trees considered in the three areas of distributed computing, factors of iid processes\/uniform algorithms, and descriptive combinatorics. The left part shows complexity classes of distributed computing. We use the shorthand $\\mathsf{LOCAL}$ if it does not matter whether we talk about the deterministic or randomized complexity. These two notions differ only for the class of problems of randomized local complexity $O(\\log\\log n)$, which have deterministic complexity $O(\\log n)$. \\\\\n The uniform complexity classes of sublogarithmic complexity are in correspondence to appropriate classes in the randomized local complexity model, as proven in \\cref{sec:separating_examples}. On the other hand, the class fiid is very similar to the class $\\mathsf{MEASURE}$ from descriptive combinatorics. The equivalence of the class $\\mathsf{CONTINUOUS}$ and $\\mathsf{LOCAL}(O(\\log^* n)) = \\mathsf{OLOCAL}(O(\\log^* 1\/\\varepsilon))$ is marked with a dashed arrow as it was proven in case the tree is generated by a group action (think of the tree being equipped with an additional $\\Delta$-edge coloring). The inclusion $\\mathsf{LOCAL}(O(\\log^* n)) \\subseteq \\mathsf{BOREL}$ however clearly holds also in our setting. The class $\\mathsf{BOREL}$ is incomparable with $\\mathsf{RLOCAL}(O(\\log\\log n))$, as proven in \\cref{sec:separating_examples}. }\n \\label{fig:big_picture_trees}\n\\end{figure}\n\n\n\\todo{thm 5.1 - 5.3}\n\n\n\n\n\\paragraph{(Finitary) Factors of iid Processes and Uniform Algorithms}\n\nIn recent years, \\emph{factors of iid (fiid) processes} on trees attracted a lot of attention in combinatorics, probability, ergodic theory and statistical physics \\cite{ABGGMP,ArankaFeriLaci,BackSzeg,BackVir,backhausz,backhausz2,BGHV,Ball,Bowen1,Bowen2,CPRR,csoka2015invariant,GabLyo,GamarnikSudan,GGP,HarangiVirag,HolroydPoisson,HLS,HPPS,HolroydPeres,HoppenWormald,Kun,RahmanPercolation,RahVir,LyonsTrees,Mester,schmidt2016circular,Timar1,Timar2}.\nIntuitively, factors of iid processes are randomized algorithms on, e.g., infinite $\\Delta$-regular trees, where each vertex outputs a solution to a problem after it explores random strings on vertices of the whole tree.\nAs an example, consider the \\emph{perfect matching} problem.\nAn easy parity argument shows that perfect matching cannot be solved by any local randomized algorithm on finite trees.\nHowever, if we allow a small fraction of vertices not to be matched, then, by a result of Nguyen and Onak \\cite{nguyen_onak2008matching} (see also \\cite{ElekLippner}), there is a constant-round randomized algorithm that produces such a matching on high-girth graphs (where the constant depends on the fraction of unmatched vertices that we allow).\nThis result can also be deduced from a result of Lyons and Nazarov \\cite{lyons2011perfect}, who showed that perfect matching can be described as a factor of iid process on an infinite $\\Delta$-regular tree.\nThe high-level idea behind this connection is that high-girth graphs approximate the infinite $\\Delta$-regular tree and constant-round local algorithms approximate factors of iid processes.\nThis correspondence is formalized in the notion of \\emph{Benjamini-Schramm} or \\emph{local-global} convergence \\cite{benjamini2011recurrence, hatamilovaszszegedy}.\nWe note that getting only ``approximate'' solutions, that is, solutions where a small fraction of vertices does not have to satisfy the constraints of a given problem, is intrinsic in this correspondence.\nRegardless, there are many techniques, such as entropy inequality \\cite{backhausz} or correlation decay \\cite{backhausz2}, and particular results such as the aforementioned perfect matching problem \\cite{lyons2011perfect} that provide lower and upper bounds, respectively, in our setting as well.\nWe refer the reader to \\cite{LyonsTrees,BGHV} for a comprehensive summary of the field. \n\n\nIn this paper, we mostly consider a stronger condition than fiid, namely so-called \\emph{finitary} factors of iid (ffiid) processes that are studied in the same context as fiid \\cite{holroyd2017one_dependent_coloring,HolroydSchrammWilson2017FinitaryColoring,Spinka}. \nPerhaps surprisingly, the notion of ffiid is identical to the notion of so-called uniform distributed randomized algorithms \\cite{Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring,grebik_rozhon2021toasts_and_tails} that we now describe. \nWe define an uniform local algorithm as a randomized local algorithm that does not know the size of the graph $n$ -- this enables us to run such an algorithm on infinite graphs, where there is no $n$. \nMore precisely, we require that each vertex eventually outputs a solution that is compatible with the output in its neighborhood, but the time until the vertex finishes is a potentially unbounded random variable. \nAs in the case of classical randomized algorithms, we can now measure the \\emph{uniform complexity} of an uniform local algorithm (known as the tail decay of ffiid \\cite{HolroydSchrammWilson2017FinitaryColoring}). \nThe uniform complexity of an algorithm is defined as the function $t(\\varepsilon)$ such that the probability that the algorithm run on a specific vertex needs to see outside its $t(\\varepsilon)$-hop neighborhood is at most $\\varepsilon$. \nAs in the case of classical local complexity, there is a whole hierarchy of possible uniform complexities (see \\cref{fig:big_picture_trees}). \n\nWe remark that uniform distributed local algorithms can be regarded as Las Vegas algorithms. The output will always be correct; there is however no fixed guarantee at what point all vertices have computed their final output. On the other hand, a randomized distributed local algorithm can be viewed as a Monte Carlo algorithm as it needs to produce an output after a fixed number of rounds, though the produced output might be incorrect. \n\n\n\n\n\n\n\n\n\\paragraph{Descriptive Combinatorics}\n \n \n\t\n\tThe Banach-Tarski paradox states that a three-dimensional ball of unit volume can be decomposed into finitely many pieces that can be moved by isometries (distance preserving transformations such as rotations and translations) to form two three-dimensional balls each of them with unit volume(!).\n The graph theoretic problem lurking behind this paradox is the following: fix finitely many isometries of $\\mathbb{R}^3$ and then consider a graph where $x$ and $y$ are connected if there is an isometry that sends $x$ to $y$.\n\tThen our task becomes to find a perfect matching in the appropriate subgraph of this graph -- namely, the bipartite subgraph where one partition contains points of the first ball and the other contains points of the other two balls. \n\tBanach and Tarski have shown that, with a suitably chosen set of isometries, the axiom of choice implies the existence of such a matching. \n\tIn contrast, since isometries preserve the Lebesgue measure, the pieces in the decomposition cannot be Lebesgue measurable. \n\tSurprisingly, Dougherty and Foreman \\cite{doughertyforeman} proved that the pieces in the Banach-Tarski paradox can have the \\emph{Baire property}. \n\tThe Baire property is a topological analogue of being Lebesgue measurable; a subset of $\\mathbb{R}^3$ is said to have the Baire property if its difference from some open set is topologically negligible.\n\t\n \n\t \n\t\n\t\n Recently, results similar to the Banach-Tarski paradox that lie on the border of combinatorics, logic, group theory, and ergodic theory led to an emergence of a new field often called \\textit{descriptive} or \\textit{measurable combinatorics}. \n The field focuses on the connection between the discrete and continuous and is largely concerned with the investigation of graph-theoretic concepts. \n The usual setup in descriptive combinatorics is that we have a graph with uncountably many connected components, each being a countable graph of bounded degree. \n For example, in case of the Banach-Tarski paradox, the vertices of the underlying graph are the points of the three balls, edges correspond to isometries, and the degree of each vertex is bounded by the number of chosen isometries. \n Some of the most beautiful results related to the field include \\cite{laczk, marksunger,measurablesquare,doughertyforeman,marks2016baire,gaboriau,KST,DetMarks,millerreducibility,conley2020borel,csokagrabowski,Bernshteyn2021LLL}, see \\cite{kechris_marks2016descriptive_comb_survey,pikhurko2021descriptive_comb_survey} for recent surveys. \n \n Importantly, in many of these results, including the Banach-Tarski paradox, graphs where each component is an infinite $\\Delta$-regular tree appear naturally. \n Oftentimes, questions considered in descriptive combinatorics lead to constructing a solution to a local problem in the underlying uncountable graph (in the case of Banach-Tarski, the local problem is perfect matching). \n The construction needs to be such that the solution of the problem has some additional regularity properties. For example in the case of Banach-Tarski, a solution is possible when the regularity condition is the Baire property, but not if it is Lebesgue measurability.\n In fact, together with Borel measurability these are the most prominent regularity conditions studied in descriptive combinatorics.\n \n The corresponding complexity classes of local problems that always admit a solution with the respective regularity property are $\\mathsf{BOREL},\\mathsf{MEASURE},\\mathsf{BAIRE}$ (See \\cref{fig:big_picture_trees}). \n In this paper, we moreover consider the setting where each connected component of the underlying graph is a $\\Delta$-regular tree. \n \n \n \n \n \n \n\t\n\t\n\t\t\n\t\n\t\t\n\t\t\n\n\t The connection between distributed computing and descriptive combinatorics arises from the fact that in descriptive combinatorics we care about constructions that do not use the axiom of choice. \n\t In the distributed language, the axiom of choice corresponds to leader election, that is, the constructions in descriptive combinatorics do not allow picking exactly one point in every component. \n\t \n\t To get some intuition about the power of the complexity class $\\mathsf{BOREL}$, we note that Borel constructions allow us to alternate countably many times the following two operations. \n\t First, any local algorithm with constant local complexity can be run. \n\t Second, we have an oracle that provides a maximal independent set (MIS) on any graph that can be constructed locally from the information computed so far \\cite{KST}. \n Note that from the speedup result of \\cite{chang_kopelowitz_pettie2019exp_separation} we get that every local problem with local complexity $O(\\log^* n)$ can be solved by constant local constructions and \\emph{one} call to such an MIS oracle. This implies the inclusion $\\mathsf{LOCAL}(O(\\log^* n)) \\subseteq \\mathsf{BOREL}$ in \\cref{fig:big_picture_trees} proven in the insightful paper of Bernshteyn \\cite{Bernshteyn2021LLL}. \n The relationship of the class $\\mathsf{MEASURE}$ (and of the class $\\mathsf{fiid}$ from the discussion of factors) to the class $\\mathsf{BOREL}$ is analogous to the relationship of randomized distributed algorithms to deterministic distributed algorithms. \n\n \n \n\n\n\n\n\n\n\t\n\n\t\n\t\n\t\n\n\n\n\n\n\n\\paragraph{Local Problems on Regular Trees}\n\nAfter introducing the three areas of interest in this work, we conclude the section by briefly discussing the kinds of problems we focus on, which are local problems on regular trees.\nMore precisely, we study \\emph{locally checkable labeling (LCL)} problems, which are a class of problems, where the correctness of the solution can be checked locally.\nExamples include classical problems from combinatorics such as proper vertex coloring, proper edge coloring, perfect matching, and maximal independent set.\nOne main goal of this paper is to understand possible complexity classes of LCLs without inputs on infinite $\\Delta$-regular trees and their finite analogues.\nWe refer the reader to \\cref{sec:preliminaries} for a precise definition of a finite $\\Delta$-regular tree.\n\nThe motivation for studying regular trees in this work stems from different sources: (a) infinite $\\Delta$-regular trees are studied in the area of ergodic theory \\cite{Bowen1,Bowen2}, random processes \\cite{backhausz,backhausz2,lyons2011perfect} and descriptive combinatorics \\cite{DetMarks,BrooksMeas}, (b) many lower bounds in distributed computing are proven in regular trees \\cite{balliu_et_al:binary_lcl,balliu2019LB,brandt_etal2016LLL,brandt19automatic,Brandt20tightLBmatching,ChangHLPU20,goeoes14_PODC}, and (c) connecting and comparing the techniques of the three areas in this simple setting reveals already deep connections, see \\cref{sec:contribution}.\n\n\n\n\n\n\n\n\n\n\\section{Our Contributions}\n\\label{sec:contribution}\n\nWe believe that our main contribution is presenting all three perspectives as part of a common theory. Our technical contribution is split into three main parts. \n\n\n\n\n\\subsection{Generalization of Marks' Technique}\\label{sec:intromarks}\nIn Section \\ref{sec:marks} we extend the Borel determinacy technique of Marks \\cite{DetMarks}, which was used to prove the nonexistence of Borel $\\Delta$-colorings and perfect matchings, to a broader class of problems, and \nadapt the extended technique to the distributed setting, thereby obtaining a simple method for proving distributed lower bounds.\nThis method is the first lower bound technique for distributed computing using ideas coming from descriptive combinatorics (see \\cite{BernshteynVizing} for a distributed computing upper bound motivated by descriptive combinatorics).\nMoreover, we show how to use the developed techniques to obtain both $\\mathsf{BOREL}$ and $\\mathsf{LOCAL}$ lower bounds for local problems from a natural class, called homomorphism problems.\nOur key technical ingredient for obtaining the mentioned techniques and results is a novel technique based on the notion of an \\emph{ID graph}.\nWe note that a very similar concept to the ID graph was independently discovered by \\cite{id_graph}. \n\n\\paragraph{Marks' technique}\nIn the following we give an introduction to Marks' technique by going through a variant of his proof~\\cite{DetMarks,Marks_Coloring} that shows that $\\Delta$-coloring has deterministic local complexity $\\Omega(\\log n)$. The proof already works in the case where the considered regular tree comes with an input $\\Delta$-edge coloring. In this case, the output color of a given vertex $u$ can be interpreted as that $u$ ``grabs'' the incident edge of that color. The problem is hence equivalent to the \\emph{edge grabbing} problem where every vertex is required to grab an incident edge such that no two vertices grab the same edge. \n\nWe first show the lower bound in the case that vertices do not have unique identifiers but instead are properly colored with $L > \\Delta$ colors. \nSuppose there is an algorithm $\\mathcal{A}$ solving the edge grabbing problem with local complexity $t(n) = o(\\log n)$, and consider a tree rooted at vertex $u$ of depth $t(n)$; such a tree has less than $n$ vertices, for large enough $n$.\nAssume that $u$ has input color $\\sigma \\in [L]$, and consider, for some fixed edge color $\\alpha$, the edge $e$ that is incident to $u$ and has color $\\alpha$.\nTwo players, Alice and Bob, are playing the following game. In the $i$-th round, Alice colors the vertices at distance $i$ from $u$ in the subtree reachable via edge $e$ with colors from $[L]$. Then, Bob colors all other vertices at distance $i$ from $u$ with colors from $[L]$ (see \\cref{fig:game}).\nConsider the output of $u$ when executing $\\mathcal{A}$ on the obtained colored tree.\nBob wins the game if $u$ grabs the edge $e$, and Alice wins otherwise. \n\nNote that either Alice or Bob has a winning strategy. \nGiven the color $\\sigma$ of $u$, if, for each edge color $\\alpha$, Alice has a winning strategy in the game corresponding to the pair $(\\sigma, \\alpha)$, then we can create $\\Delta$ copies of Alice and let them play their strategy on each subtree of $u$, telling them that the colors chosen by the other Alices are what Bob played. The result is a coloring of the input tree such that $u$, by definition, does not pick any edge, contradicting the fact that $\\mathcal{A}$ provides a valid solution!\nSo for every $\\sigma$ there is at least one $\\alpha$ such that Bob has a winning strategy for the game corresponding to $(\\sigma, \\alpha)$. By the pigeonhole principle, there are two colors $\\sigma_1, \\sigma_2$, such that Bob has a winning strategy for both pairs $(\\sigma_1, \\alpha)$ and $(\\sigma_2, \\alpha)$. \nBut now we can imagine a tree rooted in an edge between vertices $u_1, u_2$ that are colored with colors $\\sigma_1, \\sigma_2$. We can now take two copies of Bob, one playing at $u_1$ and the other playing at $u_2$ and let them battle it out, telling each copy that the other color from $\\{\\sigma_1, \\sigma_2\\}$ and whatever the other copy plays are the moves of Alice. The resulting coloring has the property that both $u_1$ and $u_2$, when executing $\\mathcal{A}$ on the obtained colored tree, grab the edge between them, a contradiction that finishes the proof!\n\n\\paragraph{The ID graph}\nThe downside of the proof is that it does not work in the model with unique identifiers (where the players' moves consist in assigning identifiers instead of colors), since gluing copies of the same player could result in an identifier assignment where the identifiers are not unique. \nOne possible remedy is to conduct the whole proof in the context of Borel graphs as was done by Marks. This proves an even stronger statement, namely that $\\Delta$-coloring is not in the class $\\mathsf{BOREL}$, but requires additional ad-hoc tricks and a pretty heavy set theoretic tool---Martin's celebrated Borel determinacy theorem~\\cite{martin} stating that even for infinite two-player games one of the players has to have a winning strategy if the payoff set is Borel. \nThe ID graph enables us to adapt the proof (and its generalization that we develop in \\cref{sec:marks}) to the distributed setting, where the fact that one of the players has a winning strategy is obvious. \nMoreover, we use an infinite version of the ID graph to generalize Marks' technique also in the Borel setting. \n\nHere is how it works: \nThe ID graph is a specific graph whose vertices are the possible unique input identifiers (for input graphs of size $n$), that is, numbers from $[n^{O(1)}]$. Its edges are colored with colors from $[\\Delta]$ and its girth is $\\Omega(\\log n)$. When we define the game between Alice and Bob, we require them to label vertices with identifiers in such a way that whenever a new vertex is labeled with identifier $i$, and its already labeled neighbor has identifier $j$, then $ij$ is an edge in the ID graph.\nMoreover, the color of edge $ij$ in the ID graph is required to be the same as the color of the edge between the vertices labeled $i$ and $j$ in the tree where the game is played. \nIt is straightforward to check that these conditions enforce that even if we let several copies of the same player play, the resulting tree is labeled with unique identifiers. Hence, the same argument as above now finally proves that the deterministic local complexity of $\\Delta$-coloring is $\\Omega (\\log n)$. \n\nWe note that our ID graph technique is of independent interest and may have applications in many different contexts. \nTo give an example from distributed computing, consider the proof of the deterministic $\\Omega(\\log n)$-round lower bound for $\\Delta$-coloring developed by the distributed community~\\cite{brandt_etal2016LLL,chang_kopelowitz_pettie2019exp_separation}, which is based on the celebrated round elimination technique. \nEven though the result is deterministic, the proof is quite technical due to the fact that it relies on examining randomized algorithms, for reasons similar to the reasons why Marks' proof does not apply directly to the setting with unique identifiers. \nFortunately, it can be again streamlined with the use of the ID graph technique. \nMoreover, in a follow-up work, the ID graph technique has already led to a new lower bound for the Lov\u00e1sz local lemma~\\cite{brandt_grunau_rozhon2021LLL_in_LCA} in the area of Local Computation Algorithms (which is part of the realm of sequential computation), thereby giving further evidence for the versatility of the technique. \n\n\n\\paragraph{Marks vs. Round Elimination}\nIt is quite insightful to compare Marks' technique (and our generalization of it) with the powerful round elimination technique~\\cite{brandt19automatic}, which has been responsible for all locality lower bounds of the last years~\\cite{brandt_etal2016LLL, balliu2019LB, brandt19automatic, balliu2019hardness_homogeneous, balliu_et_al:binary_lcl, Brandt20tightLBmatching, balliu20rulingset, ChangHLPU20, balliu21dominatingset}. \nWhile, on the surface, Marks' approach developed for the Borel world may seem quite different from the round elimination technique, there are actually striking similarities between the two methods.\nOn a high level, in the round elimination technique, the following argument is used to prove lower bounds in the LOCAL model:\nIf a $T$-round algorithm exists for a problem $\\Pi_0$ of interest, then there exists a $(T-1)$-round algorithm for some problem $\\Pi_1$ that can be obtained from $\\Pi_0$ in a mechanical manner.\nBy applying this step iteratively, we obtain a problem $\\Pi_t$ that can be solved in $0$ rounds; by showing that there is no $0$-algorithm for $\\Pi_t$ (which is easy to do if $\\Pi_t$ is known), a $(T+1)$-round lower bound for $\\Pi_0$ is obtained.\n\nThe interesting part regarding the relation to Marks' technique is how the ($T-i-1$)-round algorithms $\\mathcal{A}'$ are obtained from the ($T-i$)-round algorithms $\\mathcal{A}$ in the round elimination framework: in order to execute $\\mathcal{A}'$, each vertex $v$, being aware of its ($T-i-1$)-hop neighborhood, essentially asks whether, for all possible extensions of its view by one hop along a chosen incident edge, there exists some extension of its view by one hop along all other incident edges such that $\\mathcal{A}$, executed on the obtained ($T-i$)-hop neighborhood, returns a certain output at $v$, and then bases its output on the obtained answer.\nIt turns out that the vertex sets corresponding to these two extensions correspond precisely to two moves of the two players in the game(s) played in Marks' approach: more precisely, in round $T-i$ of a game corresponding to the considered vertex $v$ and the chosen incident edge, the move of Alice consists in labeling the vertices corresponding to the first extension, and the move of Bob consists in labeling the vertices corresponding to the second extension.\n\nHowever, despite the similarities, the two techniques (at least in their current forms) have their own strengths and weaknesses and are interestingly different in that there are local problems that we know how to obtain good lower bounds for with one technique but not the other, and vice versa.\nFinding provable connections between the two techniques is an exciting research direction that we expect to lead to a better understanding of the possibilities and limitations of both techniques.\n\nIn \\cref{sec:marks} we use our generalized and adapted version of Marks' technique to prove new lower bounds for so-called homomorphism problems. \nHomomorphism problems are a class of local problems that generalizes coloring problems---each vertex is to be colored with some color and there are constraints on which colors are allowed to be adjacent. \nThe constraints can be viewed as a graph---in the case of coloring this graph is a clique. \nIn general, whenever the underlying graph of the homomorphism problem is $\\Delta$-colorable, its deterministic local complexity is $\\Omega(\\log n)$, because solving the problem would imply that we can solve $\\Delta$-coloring too (in the same runtime). \nIt seems plausible that homomorphism problems of this kind are the only hard, i.e., $\\Omega(\\log n)$, homomorphism problems. However, our generalization of Marks' technique asserts that this is not true. \n\n\\begin{theorem}\nThere are homomorphism problems whose deterministic local complexity on trees is $\\Omega(\\log n)$ such that the chromatic number of the underlying graph is $2\\Delta-2$.\n\\end{theorem}\n\nIt is not known how to prove the same lower bounds using round elimination\\footnote{Indeed, the descriptions of the problems have comparably large numbers of labels and do not behave like so-called ``fixed points'' (i.e., nicely) under round elimination, which suggests that it is hard to find a round elimination proof with the currently known approaches.}; in fact, as far as we know, these problems are the only known examples of problems on $\\Delta$-regular trees for which a lower bound is known to hold but currently not achievable by round elimination. \nProving the same lower bounds via round elimination is an exciting open problem. \n\n\n\n\n\n\n\n\n\n\n\\subsection{Separation of Various Complexity Classes}\n\\label{sec:introseparation}\n\\paragraph{Uniform Complexity Landscape}\nWe investigate the connection between randomized and uniform distributed local algorithms, where uniform algorithms are equivalent to the studied notion of finitary factors of iid. \nFirst, it is simple to observe that local problems with uniform complexity $t(\\varepsilon)$ have randomized complexity $t(1 \/ n^{O(1)})$ -- by definition, every vertex knows its local output after that many rounds with probability $1 - 1\/n^{O(1)}$. The result thus follows by a union bound over the $n$ vertices of the input graph.\n\nOn the other hand, we observe that on $\\Delta$-regular trees the implication also goes in the opposite direction in the following sense. Every problem that has a randomized complexity of $t(n) = o(\\log n)$ has an uniform complexity of $O(t(1\/\\varepsilon))$. \n\nOne could naively assume that this equivalence also holds for higher complexities, but this is not the case. Consider for example the $3$-coloring problem.\nIt is well-known in the distributed community that $3$-coloring a tree can be solved deterministically in $O(\\log n)$ rounds using the rake-and-compress decomposition~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup,MillerR89}.\nOn the other hand, there is no uniform algorithm for $3$-coloring a tree. \nIf there were such an uniform algorithm, we could run it on any graph with large enough girth and color $99\\%$ of its vertices with three colors. This in turn would imply that the high-girth graph has an independent set of size at least $0.99 \\cdot n\/3$. This is a contradiction with the fact that there exists high-girth graphs with a much smaller independence number~\\cite{bollobas}. \n\nInterestingly, the characterization of Bernshteyn \\cite{Bernshteyn_work_in_progress} implies that \\emph{any} uniform distributed algorithm can be ``sped up'' to a deterministic local $O(\\log n)$ complexity, as we prove in~\\cref{thm:MainBaireLog}. \n\nWe show that there are local problems that can be solved by an uniform algorithm but only with a complexity of $\\Omega(\\log 1\/\\varepsilon)$. Namely, the problem of constructing a $2$-hop perfect matching on infinite $\\Delta$-regular trees for $\\Delta \\ge 3$ has an uniform local complexity between $\\Omega(\\log 1\/\\varepsilon)$ and $O(\\operatorname{poly}\\log 1\/\\varepsilon)$. \nFormally, this proves the following theorem. \n\n\\begin{theorem}\n$\\mathsf{OLOCAL}(O(\\log\\log 1\/\\varepsilon)) \\subsetneq \\mathsf{OLOCAL}(O(\\operatorname{poly}\\log 1\/\\varepsilon))$. \n\\end{theorem}\n\nThe uniform algorithm for this problem is based on a so-called one-ended forest decomposition introduced in \\cite{BrooksMeas} in the descriptive combinatorics context. In a one-ended forest decomposition, each vertex selects exactly one of its neighbors as its parent by orienting the corresponding edge outwards. This defines a decomposition of the vertices into infinite trees. We refer to such a decomposition as a one-ended forest decomposition if the subtree rooted at each vertex only contains finitely many vertices. \nHaving computed such a decomposition, $2$-hop perfect matching can be solved inductively starting from the leaf vertices of each tree.\n\nWe leave the understanding of the uniform complexity landscape in the regime $\\Omega(\\log 1\/\\varepsilon)$ as an exciting open problem. In particular, does there exist a function $g(\\varepsilon)$ such that each local problem that can be solved by an uniform algorithm has an uniform complexity of $O(g(\\varepsilon))$?\n\n\n\n\\paragraph{Relationship of Distributed Classes with Descriptive Combinatorics}\n\nBernshteyn recently proved that $\\mathsf{LOCAL}(\\log^* n) \\subseteq \\mathsf{BOREL}$ \\cite{Bernshteyn2021LLL}. That is, each local problem with a deterministic $\\mathsf{LOCAL}$ complexity of $O(\\log^* n)$ also admits a Borel-measurable solution. A natural question to ask is whether the converse also holds. \nIndeed, it is known that $\\mathsf{LOCAL}(\\log^* n) = \\mathsf{BOREL}$ on paths with no additional input \\cite{grebik_rozhon2021LCL_on_paths}. \nWe show that on regular trees the situation is different. \nOn one hand, a characterization of Bernshteyn \\cite{Bernshteyn_work_in_progress} implies that $\\mathsf{BOREL} \\subseteq \\mathsf{BAIRE} \\subseteq \\mathsf{LOCAL}(O(\\log n))$. \nOn the other hand, we show that this result cannot be strengthened by proving the following result. \n\n\\begin{theorem}\n$\\mathsf{BOREL} \\subsetneq \\mathsf{RLOCAL}(o(\\log n))$.\n\\end{theorem}\n\nThat is, there exists a local problem that admits a Borel-measurable solution but cannot be solved with a (randomized) $\\mathsf{LOCAL}$ algorithm running in a sublogarithmic number of rounds. \n\nLet us sketch a weaker separation, namely that $\\mathsf{BOREL} \\setminus \\mathsf{LOCAL}(O(\\log^* n)) \\neq \\emptyset$. Consider a version of $\\Delta$-coloring where a subset of vertices can be left uncolored. However, the subgraph induced by the uncolored vertices needs to be a collection of doubly-infinite paths (in finite trees, this means each path needs to end in a leaf vertex). \nThe nonexistence of a fast distributed algorithm for this problem essentially follows from the celebrated $\\Omega(\\log n)$ deterministic lower bound for $\\Delta$-coloring of~\\cite{brandt_etal2016LLL}. \nOn the other hand, the problem allows a Borel solution. First, sequentially compute $\\Delta-2$ maximal independent sets, each time coloring all vertices in the MIS with the same color, followed by removing all the\ncolored vertices from the graph. In that way, a total of $\\Delta-2$ colors are used. Moreover, each\nuncolored vertex has at most $2$ uncolored neighbors. \nThis implies that the set of uncolored vertices forms a disjoint union of finite paths, one ended\ninfinite paths and doubly infinite paths. The first two classes can be colored inductively with two\nadditional colors, starting at one endpoint of each path in a Borel way (namely it can be done by making use of the countably many MISes in larger and larger powers of the input graph). \nHence, in the end only doubly infinite paths are left uncolored, as desired. \n\nTo show the stronger separation between the classes $\\mathsf{BOREL}$ and $\\mathsf{RLOCAL}(o(\\log n))$ we use a variation of the\n$2$-hop perfect matching problem. In this variation, some of the vertices can be left unmatched, but similar as in the variant of the $\\Delta$-coloring problem described above, the graph induced by all the unmatched vertices needs to satisfy some additional constraints.\n\nWe conclude the paragraph by noting that the separation between the classes $\\mathsf{BOREL}$ and $\\mathsf{LOCAL}(O(\\log^* n))$ is not as simple as it may look in the following sense. \nThis is because problems typically studied in the $\\mathsf{LOCAL}$ model with a $\\mathsf{LOCAL}$ complexity of $\\omega(\\log^* n)$ like $\\Delta$-coloring and perfect matching also do not admit a Borel-measurable solution due to the technique of Marks \\cite{DetMarks} that we discussed in \\cref{sec:intromarks}.\n\n\n\n\n\n\\subsection{$\\mathsf{LOCAL}(O(\\log n)) = \\mathsf{BAIRE}$} \n\nWe already discussed that one of complexity classes studied in descriptive combinatorics is the class $\\mathsf{BAIRE}$.\nRecently, Bernshteyn proved \\cite{Bernshteyn_work_in_progress} that all local problems that are in the complexity class $\\mathsf{BAIRE}, \\mathsf{MEASURE}$ or $\\mathsf{fiid}$ have to satisfy a simple combinatorial condition which we call being $\\ell$-full. \nOn the other hand, all $\\ell$-full problems allow a $\\mathsf{BAIRE}$ solution \\cite{Bernshteyn_work_in_progress}.\nThis implies a complete combinatorial characterization of the class $\\mathsf{BAIRE}$. \nWe defer the formal definition of $\\ell$-fullness to Section \\ref{sec:baire} as it requires a formal definition of a local problem. \nInformally speaking, in the context of vertex labeling problems, a problem is $\\ell$-full if we can choose a subset $S$ of the labels with the following property.\nWhenever we label two endpoints of a path of at least $\\ell$ vertices with two labels from $S$, we can extend the labeling with labels from $S$ to the whole path such that the overall labeling is valid.\nFor example, proper $3$-coloring is $3$-full with $S=\\{1,2,3\\}$ because for any path of three vertices such that its both endpoints are colored arbitrarily, we can color the middle vertex so that the overall coloring is proper. \nOn the other hand, proper $2$-coloring is not $\\ell$-full for any $\\ell$. \n\nWe complement this result as follows. \nFirst, we prove that any $\\ell$-full problem has local complexity $O(\\log n)$, thus proving that all complexity classes considered in the areas of factors of iids and descriptive combinatorics from \\cref{fig:big_picture_trees} are contained in $\\mathsf{LOCAL}(O(\\log n))$. \nIn particular, this implies that the existence of \\emph{any} uniform algorithm implies a local distributed algorithm for the same problem of local complexity $O(\\log n)$. \nWe obtain this result via the well-known rake-and-compress decomposition~\\cite{MillerR89}. \n\n\nOn the other hand, we prove that any problem in the class $\\mathsf{LOCAL}(O(\\log n))$ satisfies the $\\ell$-full condition. \nThe proof combines a machinery developed by Chang and Pettie~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}\nwith additional nontrivial ideas.\nIn this proof we construct recursively a sequence of sets of rooted, layered, and partially labeled trees, where the partial labeling is computed by simulating any given $O(\\log n)$-round distributed algorithm, \nand then the set $S$ meeting the $\\ell$-full condition is constructed by considering all possible extensions of the partial labeling to complete correct labeling of these trees.\n\nThis result implies the following equality:\n\\begin{theorem}\n$\\mathsf{LOCAL}(O(\\log n)) = \\mathsf{BAIRE}$. \n\\end{theorem}\n\nThis equality is surprising in that the definitions of the two classes do not seem to have much in common on the first glance!\nMoreover, the proof of the equality relies on nontrivial results in both distributed algorithms (the technique of Chang and Pettie~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}) and descriptive combinatorics (the fact that a hierarchical decomposition, so-called toast, can be constructed in $\\mathsf{BAIRE}$, \\cite{conleymillerbound}, see Proposition \\ref{pr:BaireToast}). \n\n\n\n\n\n\n\n\n\n\n\n\nThe combinatorial characterization of the local complexity class $\\mathsf{LOCAL}(O(\\log n))$ on $\\Delta$-regular trees is interesting from the perspective of distributed computing alone. \nThis result can be seen as a part of a large research program aiming at classification of possible local complexities on various graph classes~\\cite{balliu2020almost_global_problems,brandt_etal2016LLL,chang_kopelowitz_pettie2019exp_separation,chang_pettie2019time_hierarchy_trees_rand_speedup,chang2020n1k_speedups,Chang_paths_cycles,balliu2021_rooted_trees,balliu_et_al:binary_lcl,balliu2018new_classes-loglog*-log*}. \nThat is, we wish not only to understand possible complexity classes (see the left part of \\cref{fig:big_picture_trees} for possible local complexity classes on regular trees), but also to find combinatorial characterizations of problems in those classes that allow us to efficiently decide for a given problem which class it belongs to. \nUnfortunately, even for grids with input labels, it is \\emph{undecidable} whether a given local problem can be solved in $O(1)$ rounds~\\cite{naorstockmeyer,brandt_grids}, since\nlocal problems on grids can be used to simulate a Turing machine.\nThis undecidability result does not apply to paths and trees, hence for these graph classes it is still hopeful that we can find simple and useful characterizations for different classes of distributed problems. \n\nIn particular, on paths it is decidable what classes a given local problem belongs to, for all classes coming from the three areas considered here, and this holds even if we allow inputs~\\cite{Chang_paths_cycles,grebik_rozhon2021LCL_on_paths}. \nThe situation becomes much more complicated when we consider trees. \nRecently, classification results on trees were obtained for so-called binary-labeling problems \\cite{balliu_et_al:binary_lcl}. \nMore recently, a complete classification was obtained in the case of \\emph{rooted regular trees}~\\cite{balliu2021_rooted_trees}. \nAlthough their algorithm takes exponential time in the worst case, the authors provided a practical implementation fast enough to classify many typical problems of interest. \n\nMuch less is known for general, \\emph{unoriented} trees, with an arbitrary number of labels. \nIn general, deciding the optimal distributed complexity for a local problem on bounded-degree trees is $\\mathsf{EXPTIME}$-hard~\\cite{chang2020n1k_speedups}, such a hardness result does not rule out the possibility for having a simple and polynomial-time characterization for the case of \\emph{regular trees}, where there is no input and the constraints are placed only on degree-$\\Delta$ vertices. \nIndeed, it was stated in~\\cite{balliu2021_rooted_trees} as an open question to find such a characterization. Our characterization of $\\mathsf{LOCAL}(O(\\log n)) = \\mathsf{BAIRE}$ by $\\ell$-full problems makes progress in better understanding the distributed complexity classes on trees and towards answering this open question. \n\n\n\n\n\n\n\n\\paragraph{Roadmap}\n\\hypertarget{par:roadmap}{}\nIn \\cref{sec:preliminaries}, we define formally all the three setups we consider in the paper. \nIn \\cref{sec:marks} we discuss the lower bound technique of Marks and the new concept of an ID graph.\nNext, in \\cref{sec:separating_examples} we prove some basic results about the uniform complexity classes and give examples of problems separating some classes from \\cref{fig:big_picture_trees}. \nFinally, in \\cref{sec:baire} we prove that a problem admits a Baire measurable solution if and only if it admits a distributed algorithm of local complexity $O(\\log n)$.\n\nThe individual sections can be read largely independently of each other.\nMoreover, most of our results that are specific to only one of the three areas can be understood without reading the parts of the paper that concern the other areas.\nWe encourage the reader interested mainly in one of the areas to skip the respective parts.\n\n\n\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\n\nIn this section, we explain the setup we work with, the main definitions and results. \nThe class of graphs that we consider in this work are either infinite $\\Delta$-regular trees, or their finite analogue that we define formally in \\cref{subsec:local_problems}.\n\nWe sometimes explicitly assume $\\Delta > 2$. \nThe case $\\Delta = 2$, that is, studying paths, behaves differently and seems much easier to understand \\cite{grebik_rozhon2021LCL_on_paths}. \nUnless stated otherwise, we do not consider any additional structure on the graphs, but sometimes it is natural to work with trees with an input $\\Delta$-edge-coloring. \n\n\n\\subsection{Local Problems on $\\Delta$-regular trees}\n\\label{subsec:local_problems}\n\nThe problems we study in this work are locally checkable labeling (LCL) problems, which, roughly speaking, are problems that can be described via local constraints that have to be satisfied in a suitable neighborhood of each vertex.\nIn the context of distributed algorithms, these problems were introduced in the seminal work by Naor and Stockmeyer~\\cite{naorstockmeyer}, and have been studied extensively since.\nIn the modern formulation introduced in~\\cite{brandt19automatic}, instead of labeling vertices or edges, LCL problems are described by labeling half-edges, i.e., pairs of a vertex and an incident edge.\nThis formulation is very general in that it not only captures vertex and edge labeling problems, but also others such as orientation problems, or combinations of all of these types.\nBefore we can provide this general definition of an LCL, we need to introduce some definitions.\nWe start by formalizing the notion of a half-edge.\n\n\\begin{definition}[Half-edge]\n\\label{def:half-edge}\n A \\emph{half-edge} is a pair $(v, e)$ where $v$ is a vertex, and $e$ an edge incident to $v$.\n We say that a half-edge $(v, e)$ is \\emph{incident} to a vertex $w$ if $w = v$, we say that a vertex $w$ is \\emph{contained} in a half edge $(v,e)$ if $w=v$, and we say that $(v, e)$ \\emph{belongs} to an edge $e'$ if $e' = e$.\n We denote the set of all half-edges of a graph $G$ by $H(G)$.\n A \\emph{half-edge labeling} is a function $c \\colon H(G) \\to \\Sigma$ that assigns to each half-edge an element from some label set $\\Sigma$.\n\\end{definition}\n\nIn order to speak about finite $\\Delta$-regular trees, we need to consider slightly modified definition of a graph.\nWe think of each vertex to be contained in $\\Delta$-many half-edges, however, not every half edge belongs to an actual edge of the graph.\nThat is half-edges are pairs $(v,e)$, but $e$ is formally not a pair of vertices.\nSometimes we refer to these half-edges as \\emph{virtual} half-edges.\nWe include a formal definition to avoid confusions. See also \\cref{fig:Delta_reg_tree}. \n\n\n\\begin{definition}[$\\Delta$-regular trees]\n A tree $T$, finite or infinite is a \\emph{$\\Delta$-regular tree} if either it is infinite and $T=T_\\Delta$, where $T_\\Delta$ is the unique infinite $\\Delta$-regular tree, that is each vertex has exactly $\\Delta$-many neighbors, or it is finite of maximum degree $\\Delta$ and each vertex $v\\in T$ of degree $d\\le \\Delta$ is contained in $(\\Delta-d)$-many virtual half-edges.\n \n \n Formally, we can view $T$ as a triplet $(V(T), E(T), H(T))$, where $(V(T),E(T))$ is a tree of maximum degree $\\Delta$ and $H(T)$ consists of real half-edges, that is pairs $(v,e)$, where $v\\in V(T)$, $e\\in E(T)$ and $e$ is incident to $v$, together with some virtual edges, in the case when $T$ is finite, such that each vertex $v\\in V(T)$ is contained in exactly $\\Delta$-many half-edges (real or virtual).\n \n \n\\end{definition}\n\n\\begin{figure}\n \\centering\n \\includegraphics{fig\/Delta_reg_tree.pdf}\n \\caption{A $3$-regular tree on 15 vertices. Every vertex has $3$ half-edges but not all half-edges lead to a different vertex. }\n \\label{fig:Delta_reg_tree}\n\\end{figure}\n\nAs we are considering trees in this work, each LCL problem can be described in a specific form that provides two lists, one describing all label combinations that are allowed on the half-edges incident to a vertex, and the other describing all label combinations that are allowed on the two half-edges belonging to an edge.\\footnote{Every problem that can be described in the form given by Naor and Stockmeyer~\\cite{naorstockmeyer} can be equivalently described as an LCL problem in this list form, by simply requiring each output label on some half-edge $h$ to encode all output labels in a suitably large (constant) neighborhood of $h$ in the form given in~\\cite{naorstockmeyer}.}\nWe arrive at the following definition for LCLs on $\\Delta$-regular trees.\\footnote{Note that the defined LCL problems do not allow the correctness of the output to depend on input labels.}\n\n\\begin{definition}[LCLs on $\\Delta$-regular trees]\\label{def:LCL_trees}\nA \\emph{locally checkable labeling problem}, or \\emph{LCL} for short, is a triple $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$, where $\\Sigma$ is a finite set of labels, $\\mathcal{V}$ is a subset of unordered cardinality-$\\Delta$ multisets\\footnote{Recall that a multiset is a modification of the concept of sets, where repetition is allowed.} of labels from $\\Sigma$ , and $\\mathcal{E}$ is a subset of unordered cardinality-$2$ multisets of labels from $\\Sigma$.\n\nWe call $\\mathcal{V}$ and $\\mathcal{E}$ the \\emph{vertex constraint} and \\emph{edge constraint} of $\\Pi$, respectively.\nMoreover, we call each multiset contained in $\\mathcal{V}$ a \\emph{vertex configuration} of $\\Pi$, and each multiset contained in $\\mathcal{E}$ an \\emph{edge configuration} of $\\Pi$.\n\nLet $T$ be a $\\Delta$-regular tree and $c:H(T)\\to \\Sigma$ a half-edge labeling of $T$ with labels from $\\Sigma$.\nWe say that $c$ is a \\emph{$\\Pi$-coloring}, or, equivalently, a correct solution for $\\Pi$, if, for each vertex $v$ of $T$, the multiset of labels assigned to the half-edges incident to $v$ is contained in $\\mathcal{V}$, and, for each edge $e$ of $T$, the cardinality-$2$ multiset of labels assigned to the half-edges belonging to $e$ is an element of $\\mathcal{E}$.\n\\end{definition}\n\nAn equivalent way to define our setting would be to consider $\\Delta$-regular trees as commonly defined, that is, there are vertices of degree $\\Delta$ and vertices of degree $1$, i.e., leaves.\nIn the corresponding definition of LCL one would consider leaves as unconstrained w.r.t. the vertex constraint, i.e., in the above definition of a correct solution the condition ``for each vertex $v$'' is replaced by ``for each non-leaf vertex $v$''.\nEquivalently, we could allow arbitrary trees of maximum degree $\\Delta$ as input graphs, but, for vertices of degree $< \\Delta$, we require the multiset of labels assigned to the half-edges to be extendable to some cardinality-$\\Delta$ multiset in $\\mathcal{V}$.\nWhen it helps the exposition of our ideas and is clear from the context, we may make use of these different but equivalent perspectives.\n\nWe illustrate the difference between our setting and ``standard setting'' without virtual half-edges on the perfect matching problem.\nA standard definition of the perfect matching problem is that some edges are picked in such a way that each vertex is covered by exactly one edge.\nIt is easy to see that there is no local algorithm to solve this problem on the class of finite trees (without virtual half-edges), this is a simple parity argument.\nHowever, in our setting, every vertex needs to pick exactly one half-edge (real or virtual) in such a way that both endpoints of each edge are either picked or not picked.\nWe remark that in our setting it is not difficult to see that (if $\\Delta>2$), then this problem can be solved by a local deterministic algorithm of local complexity $O(\\log(n))$.\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The $\\mathsf{LOCAL}$ model}\n\nIn this section, we define local algorithms and local complexity.\nWe discuss the general relation between the classical local complexity and the uniform local complexity. Recall that when we talk about distributed algorithm on $\\Delta$-regular trees, the algorithm has access to $n$, the size of the tree. The measure of complexity is the classical local complexity. On the other hand, when we talk about uniform distributed algorithms on $\\Delta$-regular trees, we talk about an infinite $\\Delta$-regular tree and the measure of complexity is the uniform local complexity. \nWe start with the classical notions.\nRecall that $B(v,t)$ is the $t$-hop neighborhood of a vertex $v$ in an underlying graph $G$.\n\n\\begin{definition}[Local algorithm]\n\\label{def:local_algorithm}\nA \\emph{distributed local algorithm} $\\mathcal{A}$ of local complexity $t(n)$ is a function defined on all possible $t(n)$-hop neighborhoods of a vertex. Applying an algorithm $\\mathcal{A}$ on an input graph $G$ means that the function is applied to a $t(n)$-hop neighborhood $B(u, t(n))$ of each vertex $u$ of $G$. The output of the function is a labeling of the half-edges around the given vertex. The algorithm also takes as input the size of the input graph $n$. \n\\end{definition}\n\n\\begin{definition}[Local complexity]\n\\label{def:local_complexity}\nWe say that an LCL problem $\\Pi$ has a deterministic local complexity $t(n)$ if there is a local algorithm $\\mathcal{A}$ of local complexity $t(n)$ such that when run on the input graph $G$, with each of its vertices having a unique identifier from $[n^{O(1)}]$, $\\mathcal{A}$ always returns a valid solution to $\\Pi$. \nWe also say $\\Pi \\in \\mathsf{DLOCAL}( O(t(n))$. \n\nThe randomized complexity is define analogously, but instead of unique identifiers, each vertex of $G$ has an infinite random string. The solution of $\\mathcal{A}$ needs to be a valid solution with probability $1 - 1\/n$. \nWe also say $\\Pi \\in \\mathsf{RLOCAL}( O(t(n))$. \n\\end{definition}\n\nWhenever we talk about \\emph{local} complexity on $\\Delta$-regular trees, we always tacitly think about the class of finite $\\Delta$-regular trees. We use the notation $\\mathsf{LOCAL}(O(t(n)))$ whenever the deterministic and the randomized complexity of the problems are the same up to constant factor (cf. \\cref{thm:basicLOCAL}). \n\n\\paragraph{Classification of Local Problems on $\\Delta$-regular Trees}\n\nThere is a lot of work aiming to classify all possible local complexities of LCL problems on bounded degree trees. This classification was recently finished.\nEven though the results in the literature are stated for the classical notion of finite trees, we note that it is easy to check that the same picture emerges if we restrict ourselves to $\\Delta$-regular trees according to our definition.\n\n\\begin{theorem}[Classification of local complexities of LCLs on $\\Delta$-regular trees \\cite{naorstockmeyer,chang_kopelowitz_pettie2019exp_separation,chang_pettie2019time_hierarchy_trees_rand_speedup,chang2020n1k_speedups,balliu2020almost_global_problems,balliu2019hardness_homogeneous,brandt_grunau_rozhon2021classification_of_lcls_trees_and_grids}]\n\\label{thm:basicLOCAL}\nLet $\\Pi$ be an LCL problem. Then the deterministic\/randomized local complexity of $\\Pi$ on $\\Delta$-regular trees is one of the following:\n\\begin{enumerate}\n \\item $O(1)$,\n \\item $\\Theta(\\log^*n)$,\n \\item $\\Theta(\\log n)$ deterministic and $\\Theta(\\log\\log n)$ randomized,\n \\item $\\Theta(\\log n)$,\n \\item $\\Theta(n^{1\/k})$ for $k \\in \\mathbb{N}$.\n\\end{enumerate}\n\\end{theorem}\n\nWe do not know whether the complexity classes $\\Theta(n^{1\/k})$ are non-empty on \\emph{$\\Delta$-regular} trees. The $2 \\frac{1}{2}$ coloring problems $\\mathcal{P}_k$ are known to have complexity $\\Theta(n^{1\/k})$ on \\emph{bounded-degree} trees~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}. However, as the correctness criterion of $\\mathcal{P}_k$ depends on the degree of vertices, it does not fit into the class of LCL problems that we consider in \\cref{def:LCL_trees}. \nNevertheless, as can be seen in \\cref{fig:big_picture_trees}, all classes of factors of iid and from descriptive combinatorics are already contained in the class $\\mathsf{LOCAL}(O(\\log n))$. \n\n\n\n\\paragraph{Uniform Algorithms}\nAs we mentioned before, when talking about local complexities, we always have in mind that the underlying graph is finite.\nIn particular, the corresponding algorithm knows the size of the graph.\nOn infinite $\\Delta$-regular trees, or infinite graphs in general, we use the following notion \\cite{HolroydSchrammWilson2017FinitaryColoring,Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring}. \n\n\\begin{definition}\nAn \\emph{uniform local algorithm} $\\mathcal{A}$ is a function that is defined on all possible (finite) neighborhoods of a vertex.\nFor some neighborhoods it outputs a special symbol $\\emptyset$ instead of a labeling of the half-edges around the central vertex.\nApplying $\\mathcal{A}$ on a graph $G$ means that for each vertex $u$ of $G$ the function is applied to $B(u,t)$, where $t$ is the minimal number such that $\\mathcal{A}(u,t)\\not=\\emptyset$.\nWe call $t$ the coding radius of $\\mathcal{A}$, and denote it, as a function on vertices, as $R_\\mathcal{A}$.\n\\end{definition}\n\nWe define the corresponding notion of \\emph{uniform local complexity} for infinite $\\Delta$-regular trees where each vertex is assigned an infinite random string.\n\n\\begin{definition}[Uniform local complexity \\cite{HolroydSchrammWilson2017FinitaryColoring}]\n\\label{def:uniform_complexity}\nWe say that the \\emph{uniform local (randomized) complexity} of an LCL problem $\\Pi$ is $t(\\varepsilon)$ if there is an uniform local algorithm $\\mathcal{A}$ such that the following hold on the infinite $\\Delta$-regular tree.\nRecall that $R_\\mathcal{A}$ is the random variable measuring the coding radius of a vertex $u$, that is, the distance $\\mathcal{A}$ needs to look at to decide the answer for $u$. Then, for any $0 < \\varepsilon < 1$:\n\\[\n\\P(R_\\mathcal{A} \\ge t(\\varepsilon)) \\leq \\varepsilon.\n\\]\nWe also say $\\Pi \\in \\mathsf{OLOCAL}(O(t(\\varepsilon))$.\n\\end{definition}\n\nWe finish by stating the following lemma that bounds the uniform complexity of concatenation of two uniform algorithm (we need to be little bit more careful and cannot just add the complexites up). \n\n\\begin{lemma}[Sequential Composition]\n\\label{lem:sequential_composition}\nLet $A_1$ and $A_2$ be two distributed uniform algorithms with an uniform local complexity of $t_1(\\varepsilon)$ and $t_2(\\varepsilon)$, respectively. \nLet $A$ be the sequential composition of $A_1$ and $A_2$. That is, $A$ needs to know the output that $A_2$ computes when the local input at every vertex is equal to the output of the algorithm $A_1$ at that vertex. Then, $t_A(\\varepsilon) \\leq t_{A_1} \\left( \\frac{\\varepsilon\/2}{\\Delta^{t_{A_2}(\\varepsilon\/2) + 1}}\\right) + t_{A_2}(\\varepsilon\/2)$.\n\\end{lemma}\n\\begin{proof}\nConsider some arbitrary vertex $u$. Let $E_1$ denote the event that the coding radius of $\\mathcal{A}_1$ is at most $t_{A_1} \\left( \\frac{\\varepsilon\/2}{\\Delta^{t_{A_2}(\\varepsilon\/2) + 1}}\\right)$ for all vertices in the $t_{A_2}(\\varepsilon\/2)$-hop neighborhood around $u$. As the $t_{A_2}(\\varepsilon\/2)$-hop neighborhood around $u$ contains at most $\\Delta^{t_{A_2}(\\varepsilon\/2) + 1}$ vertices, a union bound implies that $P(E_1) \\geq 1 - \\varepsilon\/2$. Moreover, let $E_2$ denote the event that the coding radius of algorithm $\\mathcal{A}_2$ at vertex $u$ is at most $t_{A_2}(\\varepsilon\/2)$. By definition, $P(E_2) \\geq 1 - \\varepsilon\/2$. Moreover, if both events $E_1$ and $E_2$ occur, which happens with probability at least $1 - \\varepsilon$, then the coding radius of algorithm $\\mathcal{A}$ is at most $t_{A_1} \\left( \\frac{\\varepsilon\/2}{\\Delta^{t_{A_2}(\\varepsilon\/2) + 1}}\\right) + t_{A_2}(\\varepsilon\/2)$, thus finishing the proof.\n \\end{proof}\n\n\n\\subsection{Descriptive combinatorics}\n\\label{subsec:descriptivecombinatorics}\n\nBefore we define formally the descriptive combinatorics complexity classes, we give a high-level overview on their connection to distributing computing for the readers more familiar with the latter.\n\nThe complexity class that partially captures deterministic local complexity classes is called $\\mathsf{BOREL}$ (see also Remark \\ref{rem:cont}).\nFirst note that by a result of Kechris, Solecki and Todor\\v{c}evi\\'c \\cite{KST} the \\emph{maximal independent set problem} (with any parameter $r\\in \\mathbb{N}$) is in this class for any bounded degree graph.\\footnote{That is, it is possible to find a Borel maximal independent set, i.e., a maximal independent set which is, moreover, a Borel subset of the vertex set.}\nIn particular, this yields that $\\mathsf{BOREL}$ contains the class $\\mathsf{LOCAL}(O(\\log^* n))$ by the characterization of \\cite{chang_kopelowitz_pettie2019exp_separation}, see \\cite{Bernshteyn2021LLL}. Moreover, as mentioned before, $\\mathsf{BOREL}$ is closed under countably many iterations of the operations of finding maximal independent set (for some parameter that might grow) and of applying a constant local rule that takes into account what has been constructed previously.\\footnote{It is in fact an open problem, whether this captures fully the class $\\mathsf{BOREL}$. However, note that an affirmative answer to this question would yield that problems can be solved in an ``effective\" manner in the Borel context, which is known not to be the case in unbounded degree graphs \\cite{todorvcevic2021complexity}.}\n\nTo get a better grasp of what this means, consider for example the proper vertex $2$-coloring problem on half-lines. It is clear that no local algorithm can solve this problem.\nHowever, as it is possible to determine the starting vertex after countably many iterations of the maximal independent set operation, we conclude that this problem is in the class $\\mathsf{BOREL}$.\nThe idea that $\\mathsf{BOREL}$ can compute some unbounded, global, information will be implicitly used in all the construction in \\cref{sec:separating_examples} that separate $\\mathsf{BOREL}$ from local classes.\n\nThe intuition behind the class $\\mathsf{MEASURE}$ is that it relates in the same way to the class $\\mathsf{BOREL}$, as randomized local algorithms relate to deterministic ones.\nIn particular, the operations that are allowed in the class $\\mathsf{MEASURE}$ are the same as in the class $\\mathsf{BOREL}$ but the solution of a given LCL can be incorrect on a measure zero set.\n\nThe class $\\mathsf{BAIRE}$ can be considered as a topological equivalent of the measure theoretic class $\\mathsf{MEASURE}$, that is, a solution can be incorrect on a topologically negligible set.\nThe main difference between the classes $\\mathsf{MEASURE}$ and $\\mathsf{BAIRE}$ is that in the later there is a hierarchical decomposition that is called \\emph{toast}.\n(Note that this phenomenon is present in the case of $\\mathsf{MEASURE}$ exactly on so-called amenable graphs. It is also tightly connected with the notion of hyperfiniteness \\cite{conleymillerbound,gao2015forcing}.)\nThe independence of colorings on a tree together with this structure allows for a combinatorial characterization of the class $\\mathsf{BAIRE}$, which was proven by Bernshteyn \\cite{Bernshteyn_work_in_progress}, see also \\cref{sec:baire}.\n\n\n\n\nNext we formulate the precise definitions.\nWe refer the reader to \\cite{pikhurko2021descriptive_comb_survey,kechris_marks2016descriptive_comb_survey,Bernshteyn2021LLL,kechrisclassical}, or to \\cite[Introduction,~Section~4.1]{grebik_rozhon2021LCL_on_paths} and \\cite[Section~7.1,~7.2]{grebik_rozhon2021toasts_and_tails} for intuitive examples and standard facts of descriptive set theory.\nIn particular, we do not define here the notion standard Borel\/probability space, a Polish topology, a Borel probability measure, Baire property etc.\n\nLet $\\mathcal{G}$ be a Borel graph of bounded maximum degree on a standard Borel space $X$.\nIn this paper we consider exclusively acyclic $\\Delta$-regular Borel graphs and we refer to them as \\emph{$\\Delta$-regular Borel forests}.\nIt is easy to see that the set of half-edges (see Definition \\ref{def:half-edge}) is naturally a standard Borel space, we denote this set by $H(\\mathcal{G})$.\nThus, it makes sense to consider Borel labelings of $H(\\mathcal{G})$.\nMoreover, if $\\mathcal{G}$ is a $\\Delta$-regular Borel forest and $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ is an LCL, we can also decide whether a coloring $f:\\mathcal{H}(\\mathcal{G})\\to \\Sigma$ is a solution to $\\Pi$ as in Definition \\ref{def:LCL_trees}. \nSimilarly, we say that the coloring $f$ solves $\\Pi$, e.g., on a $\\mu$-conull set for some Borel probability measure $\\mu$ on $X$ if there is a Borel set $C\\subseteq X$ such that $\\mu(C)=1$, the vertex constraints are satisfied around every $x\\in C$ and the edge constraints are satisfied for every $x,y\\in C$ that form an edge in $\\mathcal{G}$.\n\n\\begin{definition}[Descriptive classes]\n\\label{def:descriptive}\nLet $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ be an LCL.\nWe say that $\\Pi$ is in the class $\\mathsf{BOREL}$ if for every acyclic $\\Delta$-regular Borel graph $\\mathcal{G}$ on a standard Borel space $X$, there is a Borel function $f:H(\\mathcal{G})\\to \\Sigma$ that is a $\\Pi$-coloring of $\\mathcal{G}$.\n\nWe say that $\\Pi$ is in the class $\\mathsf{BAIRE}$ if for every acyclic $\\Delta$-regular Borel graph $\\mathcal{G}$ on a standard Borel space $X$ and every compatible Polish topology $\\tau$ on $X$, there is a Borel function $f:H(\\mathcal{G})\\to \\Sigma$ that is a $\\Pi$-coloring of $\\mathcal{G}$ on a $\\tau$-comeager set.\n\nWe say that $\\Pi$ is in the class $\\mathsf{MEASURE}$ if for every acyclic $\\Delta$-regular Borel graph $\\mathcal{G}$ on a standard Borel space $X$ and every Borel probability measure $\\mu$ on $X$, there is a Borel function $f:H(\\mathcal{G})\\to \\Sigma$ that is a $\\Pi$-coloring of $\\mathcal{G}$ on a $\\mu$-conull set.\n\\end{definition}\n\nThe following claim follows directly from the definition.\n\n\\begin{claim}\\label{cl:basicDC}\nWe have $\\mathsf{BOREL}\\subseteq \\mathsf{MEASURE}, \\mathsf{BAIRE}$.\n\\end{claim}\n\nRecently Bernshteyn \\cite{Bernshteyn2021LLL,Bernshteyn2021local=cont} proved several results that connect distributed computing with descriptive combinatorics.\nUsing the language of complexity classes we can formulate some of the result as inclusions in \\cref{fig:big_picture_trees}.\n\n\\begin{theorem}[\\cite{Bernshteyn2021LLL}]\nWe have\n\\begin{itemize}\n \\item $\\mathsf{LOCAL}(O(\\log^*(n)))\\subseteq \\mathsf{BOREL}$,\n \\item $\\mathsf{RLOCAL}(o(\\log(n))) \\subseteq \\mathsf{MEASURE}, \\mathsf{BAIRE}$.\n\\end{itemize}\nIn fact, these inclusions hold for any reasonable class of bounded degree graphs.\n\\end{theorem}\n\n\\begin{remark}\n\\label{rem:cont}\nThe ``truly'' local class in descriptive combinatorics is the class $\\mathsf{CONTINUOUS}$.\nEven though we do not define this class here\\footnote{To define the class $\\mathsf{CONTINUOUS}$, rather than asking for a continuous solution on all possible $\\Delta$-regular Borel graphs, one has to restrict to a smaller family of graphs, otherwise the class trivializes. To define precisely this family is somewhat inconvenient, and not necessary for our arguments.}, and we refer the reader to \\cite{Bernshteyn2021local=cont,GJKS,grebik_rozhon2021toasts_and_tails} for the definition and discussions in various cases, we mention that the inclusion \\ \\[\\mathsf{LOCAL}(O(\\log^* n))\\subseteq \\mathsf{CONTINUOUS} \\tag{*}\\] holds in most reasonable classes of bounded degree graphs, see \\cite{Bernshteyn2021LLL}.\nThis also applies to our situation\nRecently, it was shown by Bernshteyn \\cite{Bernshteyn2021local=cont}, and independently by Seward \\cite{Seward_personal}, that $(*)$ can be reversed for Cayley graphs of finitely generated groups.\nThis includes, e.g., natural $\\Delta$-regular trees with proper edge $\\Delta$-coloring, as this is a Cayely graph of free product of $\\Delta$-many $\\mathbb{Z}_2$ induced by the standard generating set.\nIt is, however, not clear whether it $(*)$ can be reversed in our situation, i.e., $\\Delta$-regular trees without any additional labels.\n\n\n\n\\end{remark}\n\n\n\n\n\n\n\\subsection{Random processes}\n\nWe start with an intutitve description of fiid processes. Let $T_\\Delta$ be the infinite $\\Delta$-regular tree.\nInformally, \\emph{factors of iid processes (fiid)} on $T_\\Delta$ are infinite analogues of local randomized algorithms in the following way.\nLet $\\Pi$ be an LCL and $u\\in T_\\Delta$.\nIn order to solve $\\Pi$, we are allowed to explore the whole graph, and the random strings that are assigned to vertices, and then output a labeling of half-edges around $u$.\nIf such an assignment is a measurable function and produces a $\\Pi$-coloring almost surely, then we say that $\\Pi$ is in the class $\\mathsf{fiid}$.\nMoreover, if every vertex needs to explore only finite neighborhood to output a solution, then we say that $\\Pi$ is in the class $\\mathsf{ffiid}$.\nSuch processes are called \\emph{finitary} fiid (\\emph{ffiid}). There is also an intimate connection between ffiid and uniform algorithms.\nThis is explained in \\cite[Section 2.2]{grebik_rozhon2021toasts_and_tails}.\nInformally, an ffiid process that almost surely solves $\\Pi$ is, in the language of distributed computing, an uniform local algorithm that solves $\\Pi$.\nThis allows us to talk about uniform local complexity of an ffiid.\nIn the rest of the paper we interchange both notions freely with slight preference for the distributed computing language.\n\nNow we define formally these classes, using the language of probability.\nWe denote by $\\operatorname{Aut}(T_\\Delta)$ the automorphism group of $T_\\Delta$.\nAn \\emph{iid process} on $T_\\Delta$ is a collection of iid random variables $Y=\\{Y_v\\}_{v\\in V(T_\\Delta)}$ indexed by vertices, or edges, half-edges etc, of $T_\\Delta$ such that their joint distribution is invariant under $\\operatorname{Aut}(T_\\Delta)$.\nWe say that $X$ is a \\emph{factor of iid process (fiid)} if $X=F(Y)$, where $F$ is a measurable $\\operatorname{Aut}(T_\\Delta)$-invariant map and $Y$ is an iid process on $T_\\Delta$.\\footnote{We always assume $Y=(2^\\mathbb{N})^{T_\\Delta}$ endowed with the product Lebesgue measure.}\nMoreover, we say that $X$ is a \\emph{finitary factor if iid process (ffiid)} if $F$ depends with probability $1$ on a finite (but random) radius around each vertex.\nWe denote as $R_F$ the random variable that assigns minimal such radius to a given vertex, and call it the \\emph{coding radius} of $F$.\nWe denote the space of all $\\Pi$-colorings of $T_\\Delta$ as $X_\\Pi$.\nThis is a subspace of $\\Sigma^{H(T_\\Delta)}$ that is invariant under $\\operatorname{Aut}(T_\\Delta)$.\n\n\\begin{definition}\nWe say that an LCL $\\Pi$ is in the class $\\mathsf{fiid}$ ($\\mathsf{ffiid}$) if there is an fiid (ffiid) process $X$ that almost surely produces elements of $X_\\Pi$.\n\nEquivalently, we can define $\\mathsf{ffiid} = \\bigcup_f \\mathsf{OLOCAL}(f(\\varepsilon))$ where $f$ ranges over all functions. That is, $\\mathsf{ffiid}$ is the class of problems solvable by any uniform distributed algorithm. \n\n\\end{definition}\n\n\nIt is obvious that $\\mathsf{ffiid}\\subseteq \\mathsf{fiid}$.\nThe natural connection between descriptive combinatorics and random processes is formulated by the inclusion $\\mathsf{MEASURE}\\subseteq \\mathsf{fiid}$.\nWhile this inclusion is trivially satisfied, e.g., in the case of $\\Delta$-regular trees with proper edge $\\Delta$-coloring, in our situation we need a routine argument that we include for the sake of completeness in \\cref{app:ktulu}.\n\n\\begin{restatable}{lemma}{FIIDinMEASURE}\\label{thm:FIIDinMEASURE}\nLet $\\Pi$ be an LCL such that $\\Pi\\in \\mathsf{MEASURE}$.\nThen $\\Pi\\in \\mathsf{fiid}$.\n\\end{restatable}\n\n\n\n\n\\subsection{Specific Local Problems}\n\nHere we list some well-known local problems that were studied in all three considered areas. We describe the results known about these problems with respect to classes from \\cref{fig:big_picture_trees}. \n\n\\paragraph{Edge Grabbing}\nWe start by recalling the well-known problem of edge grabbing (a close variant of the problem is known as sinkless orientation\\cite{brandt_etal2016LLL}). \nIn this problem, every vertex should mark one of its half-edges (that is, grab an edge) in such a way that no edge can be marked by two vertices. \nIt is known that $\\Pi_{\\text{edgegrab}} \\not\\in \\mathsf{LOCAL}(O(\\log^* n))$ \\cite{brandt_etal2016LLL} but $\\Pi_{\\text{edgegrab}} \\in \\mathsf{RLOCAL}(O(\\log\\log n))$.\n\nSimilarly, $\\Pi_{\\text{edgegrab}} \\not \\in \\mathsf{BOREL}$ by \\cite{DetMarks}, but $\\Pi_{\\text{edgegrab}}\\in \\mathsf{MEASURE}$ by \\cite{BrooksMeas}: to see the former, just note that if a $\\Delta$-regular tree admits proper $\\Delta$-colorings of both edges and vertices, every vertex can grab an edge of the same color as the color of the vertex. Thus, $\\Pi_{\\text{edgegrab}}\\in \\mathsf{BOREL}$ would yield that $\\Delta$-regular Borel forests with Borel proper edge-colorings admit a Borel proper $\\Delta$-coloring, contradicting \\cite{DetMarks}.\n\nThis completes the complexity characterization of $\\Pi_{\\text{edgegrab}}$ as well as the proper vertex $\\Delta$-coloring with respect to classes in \\cref{fig:big_picture_trees}. \n\n\n\n\n\n\\paragraph{Perfect Matching}\nAnother notorious LCL problem, whose position in \\cref{fig:big_picture_trees} is, however, not completely understood, is the perfect matching problem $\\Pi_{\\text{pm}}$.\nRecall that the perfect matching problem $\\Pi_{\\text{pm}}$ asks for a matching that covers all vertices of the input tree.\\footnote{In our formalism this means that around each vertex exactly one half-edge is picked.}\nIt is known that $\\Pi_{\\text{pm}} \\in \\mathsf{fiid}$ \\cite{lyons2011perfect}, and it is easy to see that $\\Pi_{\\text{pm}} \\not\\in \\mathsf{RLOCAL}(O(\\log\\log(n)))$ (we will prove a stronger result in \\cref{pr:AddingLineLLL}).\nMarks proved \\cite{DetMarks} that it is not in $\\mathsf{BOREL}$, even when the underlying tree admits a Borel bipartition.\nIt is not clear if $\\Pi_{\\text{pm}}$ is in $\\mathsf{ffiid}$, nor whether it is in $\\mathsf{MEASURE}$.\n\n\n\n\n\n\\paragraph{Graph Homomorphism}\nWe end our discussion with LCLs that correspond to graph homomorphisms (see also the discussion in Section \\ref{sec:marks}).\nThese are also known as edge constraint LCLs.\nLet $G$ be a finite graph.\nThen we define $\\Pi_G$ to be the LCL that asks for a homomorphism from the input tree to $G$, that is, vertices are labeled with vertices of $G$ in such a way that edge relations are preserved.\nThere are not many positive results except for the class $\\mathsf{BAIRE}$.\nIt follows from the result of Bernshteyn \\cite{Bernshteyn_work_in_progress} (see \\cref{sec:baire}) that $\\Pi_G\\in \\mathsf{BAIRE}$ if and only if $G$ is not bipartite.\nThe main negative results can be summarized as follows.\nAn easy consequence of the result of Marks \\cite{DetMarks} is that if $\\chi(G)\\le \\Delta$, then $\\Pi_G\\not \\in \\mathsf{BOREL}$.\nIn this paper, we describe examples of graphs of chromatic number up to $2\\Delta-2$ such that the corresponding homomorphism problem is not in $\\mathsf{BOREL}$, see \\cref{sec:marks}.\nThe theory of \\emph{entropy inequalities} see \\cite{backhausz} implies that if $G$ is a cycle on more than $9$ vertices, then $\\Pi_G\\not\\in \\mathsf{fiid}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Generalization of Marks' technique}\n\\label{sec:marks}\n\nIn this section, we first develop a new way of proving lower bounds in the $\\mathsf{LOCAL}$ model based on a generalization of a technique of Marks \\cite{DetMarks}. Then, we use ideas arising in the finitary setting---connected to the adaptation of Marks' technique---to obtain new results back in the Borel context.\nFor an introduction to Marks' technique and a high-level discussion about the challenges in adapting the technique to the standard distributed setting as well as our solution via the new notion of an ID graph, we refer the reader to \\cref{sec:intromarks}.\n\n\\begin{comment}\nIn \\cite{DetMarks}, Marks showed that there exist $\\Delta$-regular Borel forests that do not admit a Borel proper $\\Delta$-coloring. As it was indicated above, this is a striking difference between $\\mathsf{MEASURE}$ or $\\mathsf{BAIRE}$ and $\\mathsf{BOREL}$, as such colorings exist in the former classes, see \\cite{BrooksMeas}. Since prior to Marks' breakthrough (essentially) any lower bound on Borel chromatic numbers utilized measure theoretic or Baire category arguments, his approach was (and had to be) based on a new technique: \\emph{determinacy of infinite games}. \n\nFix a payoff set $A \\subseteq \\mathbb{R}$. Consider the infinite game in which two players alternatingly determine digits of a real number. The first player wins iff the resulting real is in $A$. A rather counter-intuitive fact of set theory is that there exists a set $A$ for which none of the players has a winning strategy, i.e., the game is not determined. However, a deep theorem of Martin \\cite{martin} says that if $A$ is Borel, then one of the players does have such a strategy. \nMarks has constructed $\\Delta$-regular Borel forests and games with Borel payoff sets, for which the existence of a winning strategy for either of the players contradicts the existence of a Borel proper $\\Delta$-coloring. \n\nIn what follows, we generalize and adapt his games to the finitary setting, thus obtaining new lower bounds for LOCAL algorithms; then we use ideas inspired by the LOCAL technique of ID graphs to prove a new theorem in the Borel context. \n\nNote that Marks considered Borel versions of certain locally checkable problems (e.g., coloring, edge grabbing, perfect matching). As discussed above, for these problems, lower bounds were independently obtained via the round elimination technique in the LOCAL context, see e.g., \\cite{brandt19automatic}\\todo{cite more stuff??-Z}. Remarkably, the core ideas behind Brandt's and Marks' argument exhibit a surprising degree of similarity. \n\n\\todo{add the story about going from borel to local, and then back from local to borel}\n\nHowever, our extension of Marks' technique yields lower bounds for fairly natural problems that were resistant to round elimination.\nMoreover, as with our approach the lower bound is obtained by checking whether the considered LCL satisfies a certain combinatorial property called \\emph{playability condition} (see Definition~\\ref{def:Playability}), we can use it to obtain lower bounds for problems that have complicated descriptions---something that the round elimination technique in its current form is ill-suited for. \\todo{Maybe this sentence should rather be somewhere in the intro?}\n\n\n\n\\end{comment}\n\n\n\n\n\n\n\nThe setting we consider in this section is $\\Delta$-regular trees that come with a proper $\\Delta$-edge coloring with colors from $[\\Delta]$.\nAll lower bounds discussed already hold under these restrictions (and therefore also in any more general setting).\n\nRecall that an LCL $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ is given by specifying a list of allowed vertex configurations and a list of allowed edge configurations (see Definition \\ref{def:LCL_trees}).\nTo make our lower bounds as widely applicable as possible, we replace the latter list by a separate list for each of the $\\Delta$ edge colors; in other words, we consider LCLs where the correctness of the output is allowed to depend on the input that is provided by the edge colors.\nHence, formally, in this section, an LCL is more generally defined: it is a tuple $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ where $\\Sigma$ and $\\mathcal{V}$ are as before, while $\\mathcal{E} = (\\mathcal{E}_{\\alpha})_{\\alpha \\in [\\Delta]}$ is now a $\\Delta$-tuple of sets $\\mathcal{E}_{\\alpha}$ consisting of cardinality-$2$ multisets.\nSimilarly as before, a half-edge labeling (see Definition \\ref{def:descriptive} for the Borel context) with labels from $\\Sigma$ is a correct solution for $\\Pi$ if, for each vertex $v$, the multiset of labels assigned to the half-edges incident to $v$ is contained in $\\mathcal{V}$, and, for each edge $e$, the multiset of labels assigned to the half-edges belonging to $e$ is contained in $\\mathcal{E}_{\\alpha}$, where $\\alpha$ is the color of the edge $e$.\n\nThe idea of our approach is to identify a condition that an LCL $\\Pi$ necessarily has to satisfy if it is solvable in $O(\\log^* n)$ rounds in the $\\mathsf{LOCAL}$ model.\nShowing that $\\Pi$ does not satisfy this condition then immediately implies an $\\Omega(\\log n)$ deterministic and $\\Omega(\\log \\log n)$ randomized lower bound, by Theorem~\\ref{thm:basicLOCAL}.\n\nIn order to define our condition we need to introduce the notion of a configuration graph: a \\emph{configuration graph} is a $\\Delta$-tuple $\\mathbb{P}=(\\mathbb{P}_\\alpha)_{\\alpha\\in \\Delta}$ of graphs, where the vertex set of each of the graphs $\\mathbb{P}_\\alpha$ is the set of subsets of $\\Sigma$, and there is an edge in $\\mathbb{P}_\\alpha$ connecting two vertices $S,T$ if and only if there are $\\mathtt{a}\\in S$ and $\\mathtt{b}\\in T$ such that $\\{\\mathtt{a},\\mathtt{b}\\}\\in {\\mathcal{E}}_\\alpha$, note that loops are allowed.\n(Naturally, we will consider any two vertices of different $\\mathbb{P}_\\alpha$ to be distinct, even if they correspond to the same subset of $\\Sigma$.)\n\nNow we are set to define the aforementioned condition. The intuition behind the playability condition is the following: assume that there exists a local algorithm $\\mathcal{A}$ that solves $\\Pi$ using the $t$ neighbourhood of a given vertex. We are going to define a family of two player games. The game will depend on some $S \\subseteq \\Sigma$. $\\operatorname{Alice}$ and $\\operatorname{Bob}$ will assign labels (or IDs) to the vertices in the $t$-neighbourhood (in some way specified later on, depending on $\\alpha \\in [\\Delta]$). When the assignment is complete, we evaluate $\\mathcal{A}$ on the root of the obtained labelled graph, this way obtaining an element $s$ of $\\Sigma$, and decide who is the winner based on $s \n\\in S$ or not. Naturally, it depends on $S$ and $\\alpha$, which player has a winning strategy. This gives rise to a two coloring of vertices of $\\mathbb{P}_\\alpha$ by colors $\\operatorname{Alice}$ and $\\operatorname{Bob}$. The failure of the playability condition will guarantee that using a strategy stealing argument one can derive a contradiction.\n\n\n\n\n\\begin{definition}[Playability condition]\\label{def:Playability}\nWe say that an LCL $\\Pi$ is \\emph{playable} if for every $\\alpha\\in [\\Delta]$ there is a coloring $\\Lambda_{\\alpha}$ of the vertices of $\\mathbb{P}_{\\alpha}$ with two colors $\\{\\operatorname{Alice},\\operatorname{Bob}\\}$ such that the following conditions are satisfied:\n\\begin{enumerate}\n \\item [(A)] For any tuple $(S_{\\alpha})_{\\alpha \\in [\\Delta]} \\in V(\\mathbb{P}_1) \\times \\dots \\times V(\\mathbb{P}_{\\Delta})$ satisfying $\\Lambda_\\alpha(S_\\alpha)=\\operatorname{Alice}$ for each $\\alpha \\in [\\Delta]$, there exist $\\mathtt{a}_\\alpha\\not\\in S_\\alpha$ such that $\\{\\mathtt{a}_\\alpha\\}_{\\alpha\\in \\Delta}\\in {\\mathcal{V}}$, and\n \\item [(B)] for any $\\alpha \\in [\\Delta]$, and any tuple $(S,T) \\in V(\\mathbb{P}_{\\alpha}) \\times V(\\mathbb{P}_{\\alpha})$ satisfying $\\Lambda_\\alpha(S)=\\Lambda_\\alpha(T)=\\operatorname{Bob}$, we have that $(S,T)$ is an edge of $\\mathbb{P}_{\\alpha}$.\n\\end{enumerate}\n\n\\end{definition}\n\nOur aim in this section is to show the following general results.\n\n\\begin{theorem}\\label{th:MainLOCAL}\nLet $\\Pi$ be an LCL that is not playable.\nThen $\\Pi$ is not in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\n\\end{theorem}\n\nUsing ideas from the proof of this result, we can formulate an analogous theorem in the Borel context.\nLet us mention that while \\cref{th:MainLOCAL} is a consequence of \\cref{thm:MainBorel} by \\cite{Bernshteyn2021LLL}, we prefer to state the theorems separately.\nThis is because the proof of the Borel version of the theorem uses heavily complex results such as the Borel determinacy theorem, theory of local-global limits and some set-theoretical considerations about measurable sets.\nThis is in a stark contrast with the proof of \\cref{th:MainLOCAL} that uses `merely' the existence of large girth graphs and determinacy of finite games.\nHowever, the ideas surrounding these concepts are the same in both cases.\n\n\\begin{theorem}\\label{thm:MainBorel}\nLet $\\Pi$ be an LCL that is not playable\nThen $\\Pi$ is not in the class $\\mathsf{BOREL}$.\n\\end{theorem}\n\nThe main application of these abstract results is to find lower bounds for \\emph{(graph) homomorphism LCLs} aka \\emph{edge constraint LCLs}.\nWe already defined this class in \\cref{subsec:local_problems} but we recall the definition in the formalism of this section to avoid any confusion.\nLet $G$ be a finite graph.\nThen $\\Pi_G=(\\Sigma,\\mathcal{V},\\mathcal{E})$ is defined by letting\n\\begin{enumerate}\n\\item \\label{p:vertex} ${\\mathcal{V}}=\\{({\\mathtt{a}},\\dots,{\\mathtt{a}}):\\mathtt{a}\\in \\Sigma\\}$,\n\\item $\\Sigma=V(G)$,\n\\item $\\forall \\alpha \\in [\\Delta] \\ (\\mathcal{E}_\\alpha=E(G)).$\n\\end{enumerate}\nAn LCL for which \\eqref{p:vertex} holds is called a \\emph{vertex-LCL}. There is a one-to-one correspondence between vertex-LCLs for which $\\forall \n\\alpha,\\beta \\ (\\mathcal{E}_\\alpha=\\mathcal{E}_\\beta)$ and LCLs of the form $\\Pi_G$. (Indeed, to vertex-LCLs with this property one can associate a graph $G$ whose vertices are the labels in $\\Sigma$ and where two vertices $\\ell, \\ell'$ are connected by an edge if and only if $\\{ \\ell, \\ell' \\} \\in \\mathcal{E}$.)\nNote that if $\\Pi$ is a vertex-LCL, then condition (A) in Definition~\\ref{def:Playability} is equivalent to the statement that no tuple $\\{S_\\alpha\\}_{\\alpha\\in \\Delta}$ that satisfies the assumption of (A) can cover $\\Sigma$, i.e., that $\\Sigma \\nsubseteq S_1 \\cup \\dots \\cup S_{\\Delta}$ for such a tuple.\nMoreover, if $\\Pi_G$ is a homomorphism LCL, then $\\mathbb{P}_\\alpha=\\mathbb{P}_{\\beta}$ for every $\\alpha,\\beta\\in \\Delta$.\n\nThe simplest example of a graph homomorphism LCL is given by setting $G$ to be the clique on $k$ vertices; the associated LCL $\\Pi_G$ is simply the problem of finding a proper $k$-coloring of the vertices of the input graph.\nNow we can easily derive Marks' original result from our general theorem.\n\n\\begin{corollary}[Marks \\cite{DetMarks}]\\label{cor:detmarks}\nLet $G$ be a finite graph that has chromatic number at most $\\Delta$. Then $\\Pi_G$ is not playable. In particular, there is a $\\Delta$-regular Borel forest that have Borel chromatic number $\\Delta+1$. \n\\end{corollary}\n\\begin{proof}\nLet $A_1,\\dots, A_\\Delta$ be some independent sets that cover $G$, and $\\Lambda_1, \\dots, \\Lambda_{\\Delta}$ arbitrary colorings of the vertices of $\\mathbb{P}_1, \\dots, \\mathbb{P}_{\\Delta}$, respectively, with colors from $\\{\\operatorname{Alice},\\operatorname{Bob}\\}$.\nIt follows that $\\Lambda_\\alpha(A_\\alpha)=\\operatorname{Alice}$ for every $\\alpha\\in \\Delta$, since otherwise condition (B) in Definition~\\ref{def:Playability} is violated with $S=T=A_\\alpha$.\nBut then condition (A) does not hold.\n\\end{proof}\n\nAs our main application we describe graphs with chromatic number larger than $\\Delta$ such that $\\Pi_G$ is not playable.\nThis rules out the hope that the complexity of $\\Pi_G$ is connected with chromatic number being larger than $\\Delta$.\nIn Section \\ref{subsec:homlcls} we show the following.\nNote that the case $k=\\Delta$ is \\cref{cor:detmarks}.\n\n\\begin{theorem}\n\\label{th:homomorphism}\nLet $\\Delta > 2$ and $\\Delta < k \\leq 2\\Delta-2$. There exists a graph $G_k$ with $\\chi(G_k)= k$, such that $\\Pi_{G_k}$ is not playable and $\\Pi_{G_k} \\in \\mathsf{RLOCAL}(O(\\log\\log n ))$. In particular, $\\Pi_{G_k}\\not\\in \\mathsf{BOREL}$ and $\\Pi_{G_k}\\not\\in\\mathsf{LOCAL}(O(\\log^*(n)))$.\n\\end{theorem}\n\nInterestingly, recent results connected to counterexamples to Hedetniemi's conjecture yield the same statement asymptotically, as $\\Delta \\to \\infty$ (see Remark \\ref{r:hedet}).\n\n\\begin{remark}\nIt can be shown that for $\\Delta=3$ both the Chv\\' atal and Gr\\\" otsch graphs are suitable examples for $k=4$.\n\\end{remark}\n\n\\begin{remark}\nAs another application of his technique Marks showed in \\cite{DetMarks} that there is a Borel $\\Delta$-regular forest that does not admit Borel perfect matching.\nThis holds even when we assume that the forest is Borel bipartite, i.e., it has Borel chromatic number $2$.\nIn order to show this result Marks works with free joins of colored hyperedges, that is, Cayley graphs of free products of cyclic groups.\nOne should think of two types of triangles ($3$-hyperedges) that are joined in such a way that every vertex is contained in both types and there are no cycles.\nWe remark that the playability condition can be formulated in this setting.\nSimilarly, one can derive a version of \\cref{thm:MainBorel}.\nHowever, we do not have any application of this generalization.\n\\end{remark}\n\n\n\n\n\n\n\n\n\\subsection{Applications of playability to homomorphism LCLs}\\label{subsec:homlcls}\n\nIn this section we find the graph examples from \\cref{th:homomorphism}.\nFirst we introduce a condition $\\Delta$-(*) that is a weaker condition than having chromatic number at most $\\Delta$, but still implies that the homomorphism LCL is not playable. Then, we will show that--similarly to the way the complete graph on $\\Delta$-many vertices, $K_\\Delta$, is maximal among graphs of chromatic number $\\leq \\Delta$--there exists a maximal graph (under homorphisms) with property $\\Delta$-(*). Recall that we assume $\\Delta >2$.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=.6\\textwidth]{fig\/h3.pdf}\n \\caption{The maximal graph with the property $\\Delta$-(*) for $\\Delta=3$.\n \\label{fig:h3}\n\\end{figure}\n\n\n\n\n\\begin{definition}[Property $\\Delta$-(*)]\nLet $\\Delta>2$ and $G=(V,E)$ be a finite graph.\nWe say that $G$ satisfies property $\\Delta$-(*) if there are sets $S_0,S_1\\subseteq V$ such that $G$ restricted to $V \\setminus S_i$ has chromatic number at most $(\\Delta-1)$ for $i\\in \\{0,1\\}$, and there is no edge between $S_0$ and $S_1$.\n\\end{definition}\n\nNote that $\\chi(G) \\leq \\Delta$ implies $\\Delta$-(*): indeed, if $A_1,\\dots,A_\\Delta$ are independent sets that cover $V(G)$, we can set $S_0=S_1=A_1$.\n\nOn the other hand, we claim that if $G$ satisfies $\\Delta$-(*) then $\\chi(G)\\le 2\\Delta-2$. In order to see this, take $S_0,S_1 \\subseteq V(G)$ witnessing $\\Delta$-(*). Then, as there is no edge between $S_0$ and $S_1$, so in particular, between $S_0 \\setminus S_1$ and $S_0 \\cap S_1$, it follows that the chromatic number of $G$'s restriction to $S_0$ is $\\leq \\Delta-1$. But then we can construct proper $\\Delta-1$-colorings of $S_0$ and $V(G) \\setminus S_0$, which shows our claim.\n\n\n\\begin{proposition}\\label{pr:CONdition*}\nLet $G$ be a graph satisfying $\\Delta$-(*).\nThen $\\Pi_G$ is not playable.\n\\end{proposition}\n\\begin{proof}\nFix $S_0,S_1$ as in the definition of $\\Delta$-(*) and assume for a contradiction that colorings $\\Lambda_1, \\dots, \\Lambda_\\Delta$ as described in Definition~\\ref{def:Playability} exist.\nBy Property $\\Delta$-(*), there exist independent sets $A_1, \\dots, A_{\\Delta-1}$ such that $S_0$ together with the $A_i$ covers $G$, i.e., such that $S_0 \\cup A_1 \\cup \\dots \\cup A_{\\Delta-1} = V(G)$.\nFor each $\\alpha \\in [\\Delta-1]$, we must have $\\Lambda_\\alpha(A_{\\alpha})=\\operatorname{Alice}$, since otherwise condition (B) in Definition~\\ref{def:Playability} is violated.\nConsequently $\\Lambda_{\\Delta}(S_0)=\\operatorname{Bob}$, otherwise condition (A) is violated.\nSimilarly $\\Lambda_\\Delta(S_1)=\\operatorname{Bob}$.\nThis shows that $\\Lambda_\\Delta$ does not satisfy condition (B) with $S = S_0$ and $T = S_1$.\n\\end{proof}\n\nNext we describe maximal examples of graphs that satisfy the condition $\\Delta$-(*). That is, we define a graph $H_\\Delta$ that satisfies $\\Delta$-(*), its chromatic number is $2\\Delta-2$ and every other graph that satisfies $\\Delta$-(*) admits a homomorphism in $H_\\Delta$.\nThe graph $H_{3}$ is depicted in \\cref{fig:h3}.\n\nRecall that the (categorical) product $G \\times H$ of graphs $G,H$ is the graph on $V(G) \\times V(H)$, such that $((g,h),(g',h')) \\in E(G \\times H)$ iff $(g,g') \\in E(G)$ and $(h,h') \\in E(H)$.\n \n Write $P$ for the product $K_{\\Delta-1}\\times K_{\\Delta-1}$. Let $V_0$ and $V_1$ be vertex disjoint copies of $K_{\\Delta-1}$.\nWe think of vertices in $V_i$ and $P$ as having labels from $[\\Delta-1]$ and $[\\Delta-1] \\times [\\Delta-1]$, respectively.\nThe graph $H_\\Delta$ is the disjoint union of $V_0$, $V_1$, $P$ and an extra vertex $\\dagger$ that is connected by an edge to every vertex in $P$, and additionally, if $v$ is a vertex in $V_0$ with label $i\\in [\\Delta-1]$, then we connect it by an edge with $(i',j)\\in P$ for every $i'\\not=i$ and $j\\in [\\Delta-1]$, and if $v$ is a vertex in $V_1$ with label $j\\in \\Delta-1$, then we connect it by an edge with $(i,j')\\in P$ for every $j'\\not=j$ and $i \\in [\\Delta-1]$.\n\n\n\\begin{proposition}\\label{pr:MinimalExample}\n\\begin{enumerate}\n \\item \\label{prop:hdelta}$H_\\Delta$ satisfies $\\Delta$-(*).\n \\item \\label{prop:hdeltachrom} $\\chi(H_\\Delta)=2\\Delta-2$.\n \\item \\label{prop:hdeltamax} A graph $G$ satisfies $\\Delta$-(*) if and only if it admits a homomorphism to $H_\\Delta$.\n\\end{enumerate}\n\n\\end{proposition}\n\\begin{proof}\n\\eqref{prop:hdelta} Set $S_0=V(V_0)\\cup \\{\\dagger\\}$ and $S_1=V(V_1)\\cup \\{\\dagger\\}$.\nBy the definition there are no edges between $S_0$ and $S_1$.\nConsider now, e.g., $V(H_\\Delta)\\setminus S_0$.\nLet $A_j$ consist of all elements in $P$ that have second coordinate equal to $j$ together with the vertex in $V_1$ that has the label $j$.\nBy the definition, the set $A_i$ is independent and $\\bigcup_{i\\in [\\Delta-1]}A_i$ covers $H_\\Delta\\setminus S_0$, and similarly for $S_1$.\n\n\\eqref{prop:hdeltachrom} By \\eqref{prop:hdelta} and the claim after the definition of $\\Delta$-(*), it is enough to show that $\\chi(H_\\Delta)\\geq 2\\Delta-2$. Towards a contradiction, assume that $c$ is a proper coloring of $H_\\Delta$ with $<2\\Delta-2$-many colors. Note the vertex $\\dagger$ guarantees that $|c(V(P))|\\leq 2\\Delta-4$, and also $\\Delta-1 \\leq |c(V(P))|$.\n\nFirst we claim that there are no indices $i,j\\in [\\Delta-1]$ (even with $i=j$) such that $c(i,r)\\not=c(i,s)$ and $c(r,j)\\not=c(s,j)$ for every $s\\not=r$: indeed, otherwise, by the definition of $P$ we had $c(i,r)\\not=c(s,j)$ for every $r,s$ unless $(i,r)=(s,j)$, which would the upper bound on the size of $c(V(P))$.\n\nTherefore, without loss of generality, we may assume that for every $i\\in [\\Delta-1]$ there is a color $\\alpha_i$ and two indices $j_i \\neq j'_i$ such that $c(i,j_i)=c(i,j'_i)=\\alpha_i$. It follows form the definition of $P$ and $j_i \\neq j'_i$ that\n $\\alpha_i\\not=\\alpha_{i'}$ whenever $i\\not= i'$. \n \n Moreover, note that any vertex in $V_1$ is connected to at least one of the vertices $(i,j_i)$ and $(i,j'_i)$, hence none of the colors $\\{\\alpha_i\\}_{i \\in [\\Delta-1]}$ can appear on $V_1$. Consequently, since $V_1$ is isomorphic to $K_{\\Delta-1}$ we need to use at least $\\Delta-1$ additional colors, a contradiction.\n\n\\eqref{prop:hdeltamax} First note that if $G$ admits a homomorphism into $H_\\Delta$, then the pullbacks of the sets witnessing $\\Delta$-(*) will witness that $G$ has $\\Delta$-(*).\n\nConversely, let $G$ be a graph that satisfies $\\Delta$-(*).\nFix the corresponding sets $S_0,S_1$ together with $(\\Delta-1)$-colorings $c_0,c_1$ of their complements.\nWe construct a homomorphism $\\Theta$ from $G$ to $H_\\Delta$. Let\n\\[\\Theta(v)=\n\\begin{cases} \\dagger &\\mbox{if } v \\in S_0 \\cap S_1, \\\\\nc_0(v) & \\mbox{if } v \\in S_1 \\setminus S_0, \\\\\nc_1(v) & \\mbox{if } v \\in S_0 \\setminus S_1,\n\\\\\n(c_0(v),c_1(v)) & \\mbox{if } v \\not \\in S_0 \\cup S_1.\n\\end{cases} \\]\n\n\nObserve that $S=S_0\\cap S_1$ is an independent set such that there is no edge between $S$ and $S_0 \\cup S_1$.\nUsing this observation, one easily checks case-by-case that $\\Theta$ is indeed a homomorphism.\n\n \n \n \n \n \n \n\\end{proof}\n\nNow, combining what we have so far, we can easily prove Theorem \\ref{th:homomorphism}.\n\n\n\\begin{proof}[Proof of Theorem \\ref{th:homomorphism}]\nIt follows that $\\Pi_{H_\\Delta}$ is not playable from \\cref{pr:MinimalExample}, \\cref{pr:CONdition*}. It is easy to see that if for a graph $G$ the LCL $\\Pi_G$ is not playable then $\\Pi_{G'}$ is not playable for every subgraph $G'$ of $G$. Since erasing a vertex decreases the chromatic number with at most one, for each $k\\leq 2\\Delta-2$ there is a subgraph $G_k$ of $H_\\Delta$ with $\\chi(G_k)=k$, such that $\\Pi_{G_k}$ is not playable.\n\nIt follows from Theorems \\ref{th:MainLOCAL} and \\ref{thm:MainBorel} that there is a $\\Delta$-regular Borel forest that admits no Borel homomorphism to any graph of $G_k$ and that $\\Pi_{G_k} \\not \\in \\mathsf{LOCAL}(O(\\log^*(n)))$.\n\nFinally, note that if $k \\geq \\Delta$ then $G_k$ can be chosen so that it contains $K_\\Delta$, yielding $\\Pi_{G_k} \\in \\mathsf{RLOCAL}(O(\\log\\log(n))$.\n\\end{proof}\n\n\\begin{remark}\n\\label{r:hedet}Recall that Hedetniemi's conjecture is the statement that if $G,H$ are finite graphs then $\\chi(G \\times H)=\\min\\{\\chi(G),\\chi(H)\\}$. This conjecture has been recently disproven by Shitov \\cite{shitov}, and strong counterexamples have been constructed later (see, \\cite{tardif2019note,zhu2021note}). We claim that these imply for $\\varepsilon>0$ the existence of finite graphs $H$ with $\\chi(H) \\geq (2-\\varepsilon)\\Delta$ to which $\\Delta$-regular Borel forests cannot have a homomorphism in BOREL, for every large enough $\\Delta$. Indeed, if a $\\Delta$-regular Borel forest admitted a Borel homomorphism to each finite graph of chromatic number at least $(2-\\varepsilon)\\Delta$, it would have such a homomorphism to their product as well. Thus, we would obtain that the chromatic number of the product of any graphs of chromatic number $(2-\\varepsilon)\\Delta$ is at least $\\Delta+1$. This contradicts Zhu's result \\cite{zhu2021note}, which states that the chromatic number of the product of graphs with chromatic number $n$ can drop to $\\approx \\frac{n}{2}$.\n\n\\end{remark}\n\n\\begin{remark}\nA natural attempt to construct graphs with large girth and not playable homomorphisms problem would be to consider random $d$-regular graphs of size $n$ for a large enough $n$. However, it is not hard to see that setting $\\Lambda(A)=\\operatorname{Alice}$ if and only if $|A|<\\frac{n}{d}$ shows that this approach cannot work.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{th:MainLOCAL}}\\label{sec:marksproof}\n\nIn this section we prove Theorem \\ref{th:MainLOCAL}, by applying Marks' game technique in the LOCAL setting. In order to define our games, we will need certain auxiliary graphs, the so-called \\emph{ID graphs}. \nThe purpose of these graphs is to define a ``playground'' for the games that we consider.\nNamely, vertices in the game are labeled by vertices from the ID graph in such a way that the at the end we obtain a homomorphism from our underlying graph to the ID graph. \n\n\n\n\n\\begin{definition}\nA pair $\\idgraphcolored{n}{t}{r}=(\\idgraph{n}{t}{r},c)$ is called an \\emph{ID graph}, if\n\\begin{enumerate}\n \\item \\label{p:idgirth} $\\idgraph{n}{t}{r}$ is graph with girth at least $2t+2$,\n \\item \\label{p:idsize} $|V(\\idgraph{n}{t}{r})|\\le n$,\n \\item \\label{p:idlabels} $c$ is a $\\Delta$-edge-coloring of $\\idgraph{n}{t}{r}$, such that every vertex is adjacent to at least one edge of each color,\n \\item \\label{p:idratio} for each $\\alpha \\in [\\Delta]$ the ratio of a maximal independent set of $\\idgraph{n}{t}{r}^{\\alpha}=(V(\\idgraph{n}{t}{r}), E(\\idgraph{n}{t}{r}) \\cap c^{-1}(\\alpha))$ is at most $r$ (i.e., $\\idgraph{n}{t}{r}^\\alpha$ is the graph formed by $\\alpha$-colored edges).\n\\end{enumerate}\n\\end{definition}\n\nBefore we define the game we show that ID graphs exist.\n\n\\begin{proposition}\\label{pr:IDgraph}\nLet $t_n\\in o(\\log(n))$, $r>0$, $\\Delta\\geq 2$.\nThen there is an ID graph $\\idgraphcolored{n}{t_n}{r}$ for every $n\\in \\mathbb{N}$ sufficiently large.\n\\end{proposition}\n\\begin{proof}\nWe use the \\emph{configuration model} for regular random graphs, see \\cite{wormald1999models}.\nThis model is defined as follows.\nLet $n$ be even.\nThen a $d$-regular random sample on $n$-vertices is just a union of $d$-many independent uniform random perfect matchings.\nNote that in this model we allow parallel edges.\n\nIt was proved by Bollob\\'as \\cite{bollobas} that the independence ratio of a random $d$-regular graph is at most $\\frac{2\\log(d)}{d}$ a.a.s.\nMoreover, this quantity is concentrated by an easy application of McDiarmid's result \\cite{mcdiarmid2002concentration}, i.e., \n\\begin{equation}\n \\P\\left(\\left|X-\\textrm{\\textbf{E}}(X)\\right|\\ge \\sqrt{n}\\right)<2\\exp\\left(-\\frac{n}{d}\\right),\n\\end{equation}\nwhere $X$ is the random variable that counts the size of a maximal independent set.\nTherefore for fixed $n$ large enough we have that the independence ratio of a random sample is at most $\\frac{3\\log(d)}{d}$ with probability at least $\\left(1-2\\exp\\left(-\\frac{n}{d}\\right)\\right)$.\n\n\nPick a $d$ large enough such that $\\frac{3\\log(d)}{d} & 1-\\left(1-2\\exp\\left(-\\frac{n}{d}\\right)\\right)^\\Delta\n\\end{split}\n\\end{equation}\nas $n\\to \\infty$.\nThis shows that there exists such a graph $\\idgraphcolored{t_n}{n}{r}$ with non-zero probability.\n\\end{proof}\n\n\n\n\nNext, we define the games. As mentioned before, the games are going to depend on the following parameters: an algorithm $\\mathcal{A}_n$ of local complexity $t\\in o(\\log(n))$, an ID graph $\\idgraphcolored{n}{t}{r}$, $\\alpha\\in \\Delta$, $\\sigma\\in V(\\idgraph{n}{t}{r})$ and $S \\subseteq \\Sigma$.\n(We will view $\\mathcal{A}_n$, $\\idgraphcolored{n}{t}{r}$ as fixed, and the rest of the parameters as variables).\n\nThe game\n$$\\mathbb{G}(\\mathcal{A}_n,n,t,\\idgraphcolored{n}{t}{r})[\\alpha,\\sigma,S]$$\nis defined as follows: two players, $\\operatorname{Alice}$ and $\\operatorname{Bob}$ assign labels to the vertices of a rooted $\\Delta$-regular tree of diameter $t$. The labels are vertices of $\\idgraph{n}{t}{r}$ and the root is labeled $\\sigma$. In the $k$-th round, where $00$. $\\idgraphingcolored{r}=(\\idgraphing{r},c)$ is an \\emph{ID graphing}, if\n\n\\begin{enumerate}\n \\item $\\idgraphing{r}$ is an acyclic locally finite Borel graphing on a standard probability measure space $(X,\\mu)$,\n \\item \\label{p:idgraphinglabels} $c$ is a Borel $\\Delta$-edge-coloring of $\\idgraphing{r}$, such that every vertex is adjacent to at least one edge of each color,\n \\item \\label{p:idgraphingratio} for each $\\alpha \\in [\\Delta]$ the $\\mu$-measure of a maximal independent set of $\\idgraphing{r}^{\\alpha}=(V(\\idgraphing{r}), E(\\idgraphing{r}) \\cap c^{-1}(\\alpha))$ is at most $r$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{proposition}\n\\label{p:idgraphing}\nFor each $r>0$ there exists an ID graphing $\\idgraphingcolored{r}$.\n\\end{proposition}\n\\begin{proof}\nLet $\\idgraphing{r}$ be a \\emph{local-global limit} of the random graphs constructed in \\cref{pr:IDgraph} (see, e.g., \\cite{hatamilovaszszegedy} for the basic results about local-global convergence).\nIt is not hard to check that this limit satisfies the required properties.\n\\end{proof}\n\n\nNow we are ready to prove the theorem. The proof will closely follow the argument given in the proof of the LOCAL version, i.e., Theorem \\ref{th:MainLOCAL}, but can be understood without reading the latter.\n\n\\begin{proof}[Proof of \\cref{thm:MainBorel}]\nLet $\\Pi$ be an LCL that is not playable, and $N\\in \\mathbb{N}$ be the number of all possible colorings of vertices of $(\\mathbb{P}_{\\alpha})_{\\alpha\\in \\Delta}$ with two colors, that is, $N=2^{\\sum_\\alpha |V(\\mathbb{P}_\\alpha)|}$.\nSet $r:=\\frac{1}{N+1}$.\n\nWe define a Borel acyclic $\\Delta$-regular graph $\\mathcal{G}$, with edges properly colored by $\\Delta$, that does not admit a Borel solution of $\\Pi$.\nVertices of $\\mathcal{G}$ are pairs $(x,A)$, where $x\\in X$ is a vertex of $\\idgraphing{r}$ and $A$ is a countable subgraph of $\\idgraphing{r}$ that is a $\\Delta$-regular tree that contains $x$ and the edge coloring of $\\idgraphingcolored{r}$ induces a proper edge coloring of $A$.\nWe say that $(x,A)$ and $(y,B)$ are connected by an $\\alpha$-edge in $\\mathcal{G}$ if $A=B$, $x,y$ are adjacent in $A$ and the edge that connects them has color $\\alpha\\in \\Delta$.\n\nSuppose for a contradiction that $\\mathcal{A}$ is a Borel function that solves $\\Pi$ on $\\mathcal{G}$.\n\nNext, we define a family of games parametrized by $\\alpha \\in [\\Delta]$, $x \\in V(\\idgraphing{r})$ and $S \\subseteq \\Sigma$. For the reader familiar with Marks' construction, let us point out that for a fixed $x$, the games are analogues to the ones he defines, with the following differences: allowed moves are vertices of the ID graphing $\\idgraphing{r}$ and restricted by its edge relation, and the winning condition is defined by a set of labels, not just merely one label.\n\nSo, the game\n$$\\mathbb{G}(\\mathcal{A},\\idgraphingcolored{r})[\\alpha,x,S]$$ \nis defined as follows: $\\operatorname{Alice}$ and $\\operatorname{Bob}$ alternatingly label vertices of a $\\Delta$-regular rooted tree. The root is labelled by $x$, and the labels come from $V(\\idgraphing{r})$. In the $k$-th round, first $\\operatorname{Alice}$ labels vertices of distance $k$ from the root on the side of the $\\alpha$ edge.\nAfter that, $\\operatorname{Bob}$ labels all remaining vertices of distance $k$, etc (see \\cref{fig:game}). We also require the assignment of labels to give rise to an edge-color preserving homomorphism to $\\idgraphingcolored{r}$. \n\nIt follows from the acyclicity of $\\idgraphing{r}$ that a play of a game determines a $\\Delta$-regular rooted subtree of $\\idgraphing{r}$ to which the restriction of the edge-coloring is proper. That is, it determines a vertex $(x,A)$ of $\\mathcal{G}$. Let $\\operatorname{Alice}$ win iff the output of $\\mathcal{A}$ on the half-edge determined by $(x,A)$ and the color $\\alpha$ is \\textbf{not} in $S$.\n\nDefine the function $\\Lambda^x_{\\alpha}:\\mathbb{P}_\\alpha\\to \\{\\operatorname{Alice},\\operatorname{Bob}\\}$ assigning the player to some $S \\in V(\\mathbb{P}_\\alpha)$ who has a winning strategy in $\\mathbb{G}(\\mathcal{A},\\idgraphingcolored{r})[\\alpha,x,S]$.\nNote that, since $\\idgraphing{r}$ is locally finite, each player has only finitely many choices at each position. Thus, it follows from Borel Determinacy Theorem that $\\Lambda^x_{\\alpha}$ is well defined.\n\nNow we show the analogue of Proposition \\ref{pr:(A)satisfied}.\n\n\\begin{proposition}\\label{pr:Borel(A)satisfied}\n$(\\Lambda^{x}_{\\alpha})_{\\alpha\\in \\Delta}$ satisfies (A) in \\cref{def:Playability}.\n\\end{proposition}\n\\begin{proof}\nAssume that $(S_\\alpha)_{\\alpha\\in \\Delta}$ is such that $\\Lambda^x_\\alpha(S_\\alpha)=\\operatorname{Alice}$.\nThis means that $\\operatorname{Alice}$ has winning strategy in all the games corresponding to $S_\\alpha$. Letting these strategies play against each other in the obvious way, produces a vertex $(x,A)$ of $\\mathcal{G}$.\nSince $\\mathcal{A}$ solves $\\Pi$ it has to output labeling of half edges $(\\mathtt{a}_{\\alpha})_{\\alpha\\in \\Delta}\\in \\mathcal{N}$, where $\\mathtt{a}_\\alpha$ is a label on the half edge that start at $(x,A)$ and has color $\\alpha$.\nNote that we must have $\\mathtt{a}_\\alpha\\not\\in S_\\alpha$ by the definition of winning strategy for $\\operatorname{Alice}$.\nThis shows that (A) of Definition \\ref{def:Playability} holds.\n\\end{proof}\n\n Let $f$ be \n defined by $$x\\mapsto (\\Lambda^x_{\\alpha})_{\\alpha\\in\\Delta}.$$ Note that $f$ has a finite range. Using the fact that the allowed moves for each player can be determined in a Borel way, uniformly in $x$, it is not hard to see that for each element $s$ in the range, $f^{-1}(s)$ is in the algebra generated by sets that can be obtained by applying game quantifiers to Borel sets (for the definition see \\cite[Section 20.D]{kechrisclassical}). It has been shown by Solovay \\cite{Solovay} and independently by Fenstad-Norman \\cite{fenstadnorman} that such sets are provably $\\Delta^1_2$, and consequently, measurable. Therefore, $f$ is a measurable map.\n \n As the number of sequences of functions $(\\Lambda^x_{\\alpha})_{\\alpha\\in\\Delta}$ is $\\leq N$, by the choice of $r$, there exists a Borel set $Y$ with $\\mu(Y) > r$, such that $f$ is constant on $Y$.\n \nSince $\\Pi$ is not playable Proposition \\ref{pr:Borel(A)satisfied} gives that there is an $\\alpha\\in \\Delta$ and $S,T\\in \\mathbb{P}_\\alpha$ that violate condition (B).\nRecall that the measure of an independent Borel set of $\\idgraphing{r}^\\alpha$ is at most $r$.\nThat means that there are $x,y\\in Y$ such that $(x,y)$ is an $\\alpha$-edge in $\\idgraphing{r}$.\nWe let the winning strategies of $\\operatorname{Bob}$ in the games\n$$\\mathbb{G}(\\mathcal{A},\\idgraphingcolored{r})[\\alpha,x,S], \\ \\mathbb{G}(\\mathcal{A},\\idgraphingcolored{r})[\\alpha,y,T],$$\nplay against each other, where we start with an $\\alpha$ edge with endpoints labeled by $x,y$.\nThis produces a labeling of a $\\Delta$-regular tree $A$ with labels from $\\idgraphing{r}$ such that $(x,A)$ and $(y,A)$ span an $\\alpha$-edge in $\\idgraphing{r}$.\nApplying $\\mathcal{A}$ on the half-edge determined by $(x,A)$ and the color $\\alpha$, we obtain $\\mathtt{a}_0\\in S$.\nSimilarly, we produce $\\mathtt{a}_1\\in T$ for $(y,A)$.\nHowever, $(\\mathtt{a}_0,\\mathtt{a}_1)\\not\\in {\\bf E}_\\alpha$ by the definition of an edge in $\\mathbb{P}_{\\alpha}$.\nThis shows that $\\mathcal{A}$ does not produce a Borel solution to $\\Pi$.\n\\end{proof}\n\n\\begin{remark}\nOne can give an alternative proof of the existence of a Borel $\\Delta$-regular forest that does not admit a Borel homomorphism to $G_k$ that avoids using the graphing $\\idgraphing{r}$ and uses the original example described by Marks \\cite{DetMarks} instead.\nThe reason is that the example graphs $G_k$ contain $K_\\Delta$, as discussed in the proof of \\cref{th:homomorphism}.\nThis takes care of the ``non-free'' as in the argument of Marks.\nThen it is enough to use the pigeonhole principle on $\\mathbb{N}$ instead of the independence ratio reasoning.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Separation Of Various Complexity Classes}\n\\label{sec:separating_examples}\n\nIn this section we provide examples of several problems that separate some classes from \\cref{fig:big_picture_trees}.\nThe examples show two things. First, we have $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon) \\not= \\mathsf{OLOCAL}(O(\\log\\log 1\/\\varepsilon))$. \nThis shows that there are problems such that their worst case complexity is at least $\\Theta(\\log n)$ on finite $\\Delta$-regular trees, but their average local complexity is constant. \nSecond, we show that there are problems in the class $\\mathsf{BOREL}$ that are not in the class $\\mathsf{RLOCAL}(O(\\log\\log n)) = \\mathsf{RLOCAL}(o(\\log n))$, on $\\Delta$-regular trees. This shows that one cannot in general hope that results from (Borel) measurable combinatorics can be turned into very efficient (sublogarithmic) distributed algorithms. \n\n\\subsection{Preliminaries}\n\\label{subsec:preliminaries_separating}\n\nWe first show that there is a strong connection between randomized local complexities and uniform local complexities. Afterwards, we introduce a generic construction that turns any LCL into a new LCL. We later use this generic transformation to construct an LCL that is contained in the set $\\mathsf{BOREL} \\setminus \\mathsf{RLOCAL}(O(\\log \\log n))$.\n\n\\paragraph{Uniform vs Local Randomized Complexity} \n\nWe will now discuss the connections between uniform and randomized complexities. \nNote that the easy part of the connection between the two concepts is turning uniform local algorithms into randomized ones, as formalized in the following proposition. \n\n\\begin{proposition}\n\\label{prop:bee_gees}\nWe have $\\mathsf{OLOCAL}(t(\\varepsilon)) \\subseteq \\mathsf{RLOCAL}(t(1\/n^{O(1)}))$. \n\\end{proposition}\n\\begin{proof}\nWe claim that an uniform local algorithm $\\mathcal{A}$ with an uniform local complexity of $t(\\varepsilon)$ can be turned into a local randomized algorithm $\\mathcal{A}'$ with a local complexity of $t(1\/n^{O(1)})$. \n\nThe algorithm $\\mathcal{A}'$ simulates $\\mathcal{A}$ on an infinite $\\Delta$-regular tree -- each vertex $u$ of degree less than $\\Delta$ in the original tree pretends that the tree continues past its virtual half-edges and the random bits of $u$ are used to simulate the random bits in this virtual subtree. \nChoosing $\\varepsilon = 1\/n^{O(1)}$, one gets that the probability of $\\mathcal{A}'$ needing to look further than $t(\\varepsilon)$ for any vertex is bounded by $1\/n^{O(1)}$, as needed in the definition of the randomized local complexity. \n\\end{proof}\n\n\nOn the other hand, we will use the following proposition from \\cite{grebik_rozhon2021toasts_and_tails}. It informally states that the existence of \\emph{any} uniform local algorithm together with the existence of a sufficiently fast randomized local algorithm for a given LCL $\\Pi$ directly implies an upper bound on the uniform complexity of $\\Pi$.\n\\begin{proposition}[\\cite{grebik_rozhon2021toasts_and_tails}]\n\\label{prop:local->uniform}\nLet $\\mathcal{A}$ be an uniform local algorithm solving an LCL problem $\\Pi$ such that its \\emph{randomized} local complexity on finite $\\Delta$-regular trees is $t(n)$ for $t(n) = o(\\log n)$. \nThen, the uniform local complexity of $\\mathcal{A}$ on infinite $\\Delta$-regular trees is $O(t(1\/\\varepsilon))$.\n\\end{proposition}\n\n\\cref{prop:local->uniform} makes our life simpler, since we can take a known randomized distributed local algorithm with local complexity $g(n) = o(\\log n)$ and check if it works without the knowledge of $n$. If yes, this automatically implies that the uniform local complexity of the algorithm is $h(\\varepsilon) = O(g(1\/\\varepsilon))$. \nIn particular, combining \\cref{prop:local->uniform} with the work of \\cite{chang_kopelowitz_pettie2019exp_separation,chang_pettie2019time_hierarchy_trees_rand_speedup} and verifying that the algorithms of \nFischer and Ghaffari~\\cite[Section 3.1.1]{fischer-ghaffari-lll} and Chang~et~al.~\\cite[Section 5.1]{ChangHLPU20} work without the knowledge of $n$, we obtain the following result. \n\n\n\\begin{theorem}\n\\label{thm:classification_local_uniform_trees}\nWe have:\n\\begin{itemize}\n \\item $\\mathsf{LOCAL}(O(1)) = \\mathsf{OLOCAL}(O(1))$\n \\item $\\mathsf{RLOCAL}(O(\\log^* n)) = \\mathsf{OLOCAL}(O(\\log^* 1\/\\varepsilon))$\n \\item $\\mathsf{RLOCAL}(O(\\log\\log n)) = \\mathsf{OLOCAL}(O(\\log\\log 1\/\\varepsilon))$\n\\end{itemize}\nMoreover, there are no other possible uniform local complexities for $t(\\varepsilon) = o(\\log 1\/\\varepsilon)$. \n\\end{theorem}\nWe note that the first two items are proven in \\cite{grebik_rozhon2021toasts_and_tails}.\nFor the third item it suffices by known reductions to find an uniform algorithm\nsolving a version of the distributed Lov\\'asz Local Lemma (LLL) on so-called tree-structured dependency graphs considered in~\\cite{ChangHLPU20}.\n\n\\begin{proof}[Proof sketch]\nThe proof follows from \\cref{prop:bee_gees} and the following ideas. \nThe first item follows from the fact that any local algorithm with local complexity $O(1)$ can simply be made uniform. More specifically, there exists a constant $n_0$ --- depending only on the problem $\\Pi$ --- such that the algorithm, being told the size of the graph is $n_0$, is correct on \\emph{any} graph of size $n \\ge n_0$. \n\nFor the second item, by the work of \\cite{chang_kopelowitz_pettie2019exp_separation,chang_pettie2019time_hierarchy_trees_rand_speedup}, it suffices to check that there is an uniform distributed $(\\Delta + 1)$-coloring algorithm with uniform local complexity $O(\\log^* 1\/\\varepsilon)$. Such an algorithm was given in \\cite{HolroydSchrammWilson2017FinitaryColoring}, or follows from the work in \\cite{Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring} and \\cref{prop:local->uniform}. \n\nSimilarly, for the third item, by the work of \\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} it suffices to check that there is an uniform distributed algorithm for a specific LLL problem on trees\nwith uniform local complexity $O(\\log\\log 1\/\\varepsilon)$.\nSuch an algorithm can be obtained by combining the randomized \\emph{pre-shattering} algorithm of Fischer and Ghaffari~\\cite[Section 3.1.1]{fischer-ghaffari-lll} and the deterministic \\emph{post-shattering} algorithm of Chang~et~al.~\\cite[Section 5.1]{ChangHLPU20} in a \\emph{graph shattering} framework~\\cite{barenboim2016locality}, which solves the LLL problem \nwith local complexity $O(\\log\\log n)$.\nBy \\cref{prop:local->uniform}, it suffices to check that this algorithm can be made to work even if it does not know the size of the graph $n$. We defer the details to \\cref{sec:LLL}.\nThis finishes the proof of \\cref{thm:classification_local_uniform_trees}. \n\\end{proof}\n\n\n\n\n\n\n\n\\paragraph{Adding Paths}\nBefore we proceed to show some separation results, we define a certain construction that turns any LCL problem $\\Pi$ into a new LCL problem $\\overline{\\Pi}$ with the following property. If the original problem $\\Pi$ cannot be solved by a fast local algorithm, then the same holds for $\\overline{\\Pi}$. However, $\\overline{\\Pi}$ might be strictly easier to solve than $\\Pi$ for $\\mathsf{BOREL}$ constructions. \n\n\\begin{definition}\nLet $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ be an LCL. \nWe define an LCL $\\overline{\\Pi}=(\\Sigma',\\mathcal{V}',\\mathcal{E}')$ as follows.\nLet $\\Sigma'$ be $\\Sigma$ together with one new label.\nLet $\\mathcal{V}'$ be the union of $\\mathcal{V}$ together with any cardinality-$\\Delta$ multiset that contains the new label exactly two times.\nLet $\\mathcal{E}'$ be the union of $\\mathcal{E}$ together with the cardinality-$2$ multiset that contains the new label twice.\n\\end{definition}\n\nIn other words, the new label determines doubly infinite lines in infinite $\\Delta$-regular trees, or lines that start and end in virtual half-edges in finite $\\Delta$-regular trees.\nMoreover, a vertex that is on such a line does not have to satisfy any other vertex constraint.\nWe call these vertices \\emph{line-vertices} and each edge on a line a \\emph{line-edge}.\n\n\n\\begin{proposition}\n\\label{pr:AddingLineLLL}\nLet $\\Pi$ be an LCL problem such that $\\overline{\\Pi}\\in \\mathsf{RLOCAL}(t(n))$ for $t(n) = o(\\log(n))$. Then also $\\Pi \\in \\mathsf{RLOCAL}(O(t(n)))$.\n\\end{proposition}\n\\begin{proof}\nLet $\\overline{\\mathcal{A}}$ be a randomized $\\mathsf{LOCAL}$ algorithm that solves $\\overline{\\Pi}$ in $t(n) = o(\\log n)$ rounds with probability at least $1 - 1\/n^C$ for some sufficiently large constant $C$. We will construct an algorithm $\\mathcal{A}$ for $\\Pi$ with complexity $t(n)$ that is correct with probability $1 - \\frac{4}{n^{C\/3 - 2}}$. The success probability of $\\mathcal{A}$ can then be boosted by ``lying to it'' that the number of vertices is $n^{O(1)}$ instead of $n$; this increases the running time by at most a constant factor and boosts the success probability back to $1 - 1\/n^C$. \n\nConsider a $\\Delta$-regular rooted finite tree $T$ of depth $10t(n)$ and let $u$ be its root vertex. \nNote that $|T| \\le \\Delta^{10t(n)+1} < n$, for $n$ large enough. \n\n\nWe start by proving that when running $\\overline{\\mathcal{A}}$ on the tree $T$, then $u$ is most likely not a line-vertex. This observation then allows us to turn $\\overline{\\mathcal{A}}$ into an algorithm $\\mathcal{A}$ that solves $\\Pi$ on any $\\Delta$-regular input tree.\nLet $X$ be the indicator of $\\mathcal{A}$ marking $u$ as a line-vertex. Moreover, for $i \\in [\\Delta]$, let $Y_i$ be the indicator variable for the following event. The number of line edges in the $i$-th subtree of $u$ with one endpoint at depth $5t(n)$ and the other endpoint at depth $5t(n) + 1$ is odd.\n\nBy a simple parity argument we can relate the value of $X$ with the values of $Y_1,Y_2,\\ldots,Y_\\Delta$ as follows. If $X = 0$, that is, $u$ is not a line-vertex, then all of the $Y_i$'s have to be $0$, as each path in the tree $T$ is completely contained in one of the $\\Delta$ subtrees of $u$. On the other hand, if $u$ is a line-vertex, then there exists exactly one path, the one containing $u$, that is not completely contained in one of the $\\Delta$ subtrees of $u$. This in turn implies that exactly two of the $\\Delta$ variables $Y_1,Y_2,\\ldots,Y_\\Delta$ are equal to $1$.\n\nThe random variables $Y_1,Y_2,\\ldots,Y_\\Delta$ are identically distributed and mutually independent. Hence, if $\\P(Y_i = 1) > \\frac{1}{n^{C\/3}}$, then the probability that there are at least $3$ $Y_i$'s equal to $1$ is strictly greater than $\\left(\\frac{1}{n^{C\/3}}\\right)^3 = \\frac{1}{n^C}$. This is a contradiction, as in that case $\\overline{\\mathcal{A}}$ does not produce a valid output, which according to our assumption happens with probability at most $\\frac{1}{n^C}$. \n\nThus, we can conclude that $\\P(Y_i = 1) \\leq \\frac{1}{n^{C\/3}}$. By a union bound, this implies that all of the $Y_i$'s are zero with probability at least $\\frac{1}{n^{C\/3 - 1}}$, and in that case $u$ is not a line-vertex. \n\nFinally, the algorithm $\\mathcal{A}$ simulates $\\overline{\\mathcal{A}}$ as if the neighborhood of each vertex was a $\\Delta$-regular branching tree up to depth at least $t(n)$. \n\nIt remains to analyze the probability that $\\mathcal{A}$ produces a valid solution for the LCL problem $\\Pi$. To that end, let $v$ denote an arbitrary vertex of the input tree. The probability that the output of $v$'s half edges satisfy the vertex constraint of the LCL problem $\\overline{\\Pi}$ is at least $1 - 1\/n^C$. Moreover, in the case that $v$ is not a line-vertex, which happens with probability at least $1 - \\frac{1}{n^{C\/3 - 1}}$, the output of $v$'s half edges even satisfy the vertex constraint of the LCL problem $\\Pi$. Hence, by a union bound the vertex constraint of the LCL problem $\\Pi$ around $v$ is satisfied with probability at least $1 - \\frac{2}{n^{C\/3 - 1}}$. With exactly the same reasoning, one can argue that the probability that the output of $\\mathcal{A}$ satisfies the edge constraint of the LCL problem $\\Pi$ at a given edge is also at least $1 - \\frac{2}{n^{C\/3 - 1}}$. Finally, doing a union bound over the $n$ vertex constraints and $n-1$ edge constraints it follows that $\\mathcal{A}$ produces a valid solution for $\\Pi$ with probability at least $1 - \\frac{4}{n^{C\/3 - 2}}$.\n\\end{proof}\n\n\\begin{remark}\nThe ultimate way how to solve LCLs of the form $\\overline{\\Pi}$ is to find a spanning forest of lines.\nThis is because once we have a spanning forest of lines, then we might declare every vertex to be a line-vertex and label every half-edge that is contained on a line with the new symbol in $\\overline{\\Pi}$.\nThe remaining half-edges can be labeled arbitrarily in such a way that the edge constraints are satisfied.\n\nPut otherwise, once we are able to construct a spanning forest of lines and $\\mathcal{E}\\not=\\emptyset$, where $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$, then $\\overline{\\Pi}$ can be solved.\nMoreover, in that case the complexity of $\\overline{\\Pi}$ is upper bounded by the complexity of finding the spanning forest of lines.\n\nLyons and Nazarov \\cite{lyons2011perfect} showed that $\\Pi_{\\text{pm}}\\in \\mathsf{fiid}$ and it is discussed in \\cite{LyonsTrees} that the construction can be iterated $(\\Delta-2)$-many times. After removing each edge that is contained in one of the $\\Delta-2$ perfect matchings each vertex has a degree of two.\nHence, it is possible to construct a spanning forest of lines as fiid.\nConsequently, $\\overline{\\Pi}\\in \\mathsf{fiid}$ for every $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ with $\\mathcal{E}\\not=\\emptyset$.\n\\end{remark}\n\n\n\\subsection{$\\mathsf{LOCAL}(O(\\log^* n)) \\not= \\mathsf{BOREL}$}\n\nWe now formalize the proof that $\\mathsf{LOCAL}(O(\\log^* n)) \\not= \\mathsf{BOREL}$ from \\cref{sec:introseparation}. \nRecall that $\\Pi_{\\chi,\\Delta}$ is the proper vertex $\\Delta$-coloring problem.\nThe following claim directly follows by \\cref{pr:AddingLineLLL} together with the fact that $\\Pi_{\\chi,\\Delta}\\not\\in \\mathsf{LOCAL}(O(\\log^* n))$.\n\n\\begin{claim}\nWe have $\\overline{\\Pi_{\\chi,\\Delta}}\\not \\in \\mathsf{LOCAL}(O(\\log^* n))$.\n\\end{claim}\n\nOn the other hand we show that adding lines helps to find a Borel solution.\nThis already provides a simple example of a problem in the set $\\mathsf{BOREL} \\setminus \\mathsf{LOCAL}(O(\\log^* n))$.\n\n\\begin{proposition}\nWe have $\\overline{\\Pi_{\\chi,\\Delta}} \\in \\mathsf{BOREL}$. \n\\end{proposition}\n\\begin{proof}\nBy \\cite{KST}, every Borel graph of finite maximum degree admits a Borel maximal independent set.\nLet $\\mathcal{G}$ be a Borel $\\Delta$-regular acyclic graph on a standard Borel space $X$.\nBy an iterative application of the fact above we find Borel sets $A_1,\\dots,A_{\\Delta-2}$ such that $A_1$ is a maximal independent set in $\\mathcal{G}$ and $A_i$ is a maximal independent set in $\\mathcal{G}\\setminus \\bigcup_{j1$.\nWe think of $A_1,\\dots,A_{\\Delta-2}$ as color classes for the first $\\Delta-2$ colors.\nLet $B=X\\setminus \\bigcup_{i\\in [\\Delta-2]}A_i$ and let $\\mathcal{H}$ be the subgraph of $\\mathcal{G}$ determined by $B$.\nIt is easy to see that the maximum degree of $\\mathcal{H}$ is $2$.\nIn particular, $\\mathcal{H}$ consists of finite paths, one-ended infinite paths or doubly infinite paths.\nWe use the extra label in $\\overline{\\Pi_{\\chi,\\Delta}}$ to mark the doubly infinite paths.\nIt remains to use the remaining $2$ colors in $[\\Delta]$ to define a proper vertex $2$-coloring of the finite and one-ended paths.\nIt is a standard argument that this can be done in a Borel way and it is easy to verify that, altogether, we found a Borel $\\overline{\\Pi_{\\chi,\\Delta}}$-coloring of $\\mathcal{G}$.\n\\end{proof}\n\n\n\n\\subsection{Examples and Lower Bound}\n\nIn this subsection we define two LCLs and show that they are not in the class $\\mathsf{RLOCAL}(o(\\log n))$.\nIn the next subsection we show that one of them is in the class $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon)$ and the other in the class $\\mathsf{BOREL}$.\nBoth examples that we present are based on the following relaxation of the perfect matching problem.\n\n\\begin{definition}[Perfect matching in power-$2$ graph]\nLet $\\Pi^2_{\\text{pm}}$ be the perfect matching problem in the power-$2$ graph, i.e., in the graph that we obtain from the input graph by adding an edge between any two vertices that have distance at most $2$ in the input graph\n\\end{definition}\n\nWe show that $\\Pi^2_{\\text{pm}}$ is not contained in $\\mathsf{RLOCAL}(o(\\log n))$. \\cref{pr:AddingLineLLL} then directly implies that $\\overline{\\Pi^2_{\\text{pm}}}$ is not contained in $\\mathsf{RLOCAL}(o(\\log n))$ as well.\nOn the other hand, we later use a one-ended spanning forest decomposition to show that $\\Pi^2_{\\text{pm}}$ is in $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon)$ and a one or two-ended spanning forest decomposition to show that $\\overline{\\Pi^2_{\\text{pm}}}$ is in $\\mathsf{BOREL}$.\n\n\nWe first show the lower bound result.\nThe proof is based on a simple parity argument as in the proof of \\cref{pr:AddingLineLLL}.\n\n\\begin{theorem}\nThe problems $\\Pi^2_{\\text{pm}}$ and $\\overline{\\Pi^2_{\\text{pm}}}$ are not in the class $\\mathsf{RLOCAL}(o(\\log n))$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose that the theorem statement does not hold for $\\Pi^2_{\\text{pm}}$.\nThen, by \\cref{thm:basicLOCAL}, there is a distributed randomized algorithm solving $\\Pi^2_{\\text{pm}}$ with probability at least $1 - 1\/n$. By telling the algorithm that the size of the input graph is $n^{2\\Delta}$ instead of $n$, we can further boost the success probability to $1 - 1\/n^{2\\Delta}$. The resulting round complexity is at most a constant factor larger, as $\\Delta = O(1)$.\nWe can furthermore assume that the resulting algorithm $\\mathcal{A}$ will always output for each vertex $v$ a vertex $M(v) \\neq v$ in $v$'s $2$-hop neighborhood, even if $\\mathcal{A}$ fails to produce a valid solution. The vertex $M(v)$ is the vertex $v$ decides to be matched to, and if $\\mathcal{A}$ produces a valid solution it holds that $M(M(v)) = v$.\n\nConsider a fully branching $\\Delta$-regular tree $T$ of depth $10t(n)$. Note that $|T| < n$ for $n$ large enough. Let $u$ be the root of $T$ and $T_i$ denote the $i$-th subtree of $u$ (so that $u \\not\\in T_i$).\nLet $S_i$ denote the set of vertices $v$ in $T_i$ such that both $v$ and $M(v)$ have distance at most $2t(n)$ from $u$.\nNote that $M(v)$ is not necessarily a vertex of $T_i$.\nLet $Y_i$ be the indicator of whether $S_i$ is even.\n\nObserve that, if $\\mathcal{A}$ does not fail on the given input graph, then the definition of the $S_i$ implies that every vertex in $\\{ u \\} \\cup \\bigcup_{i \\in [\\Delta]} S_i$ is matched to a vertex in $\\{ u \\} \\cup \\bigcup_{i \\in [\\Delta]} S_i$.\nHence, the number of vertices in $\\{ u \\} \\cup \\bigcup_{i \\in [\\Delta]} S_i$ is even, which implies that we cannot have $Y_1 = Y_2 = \\ldots = Y_\\Delta = 1$ unless $\\mathcal{A}$ fails.\nNote that $Y_i$ depends only on the output of $\\mathcal{A}$ at the vertices in $T_i$ that have a distance of precisely $2t(n) - 1$ or $2t(n)$ to $u$ (as all other vertices in $T_i$ are guaranteed to be in $S_i$). \nHence, the events $Y_i = 1$, $i \\in [\\Delta]$, are independent, and, since $\\P(Y_1 = 1) = \\ldots = \\P(Y_{\\Delta} = 1)$ (as all vertices in all $T_i$ see the same topology in their $t(n)$-hop view), we obtain $(\\P(Y_1 = 1))^{\\Delta} \\le 1\/n^{2\\Delta}$, which implies $\\P(Y_i = 1) \\le 1\/n^2$, for any $i \\in [\\Delta]$.\n\n\nHence, with probability at least $1 - \\Delta \/n^2$ we have $Y_1=Y_2=\\ldots = Y_\\Delta = 0$, by a union bound.\nLet $v_1, v_2, \\dots, v_\\Delta$ denote the neighbors of $u$ with $v_i$ being the root of subtree $T_i$.\nNote that if $\\mathcal{A}$ does not fail and $Y_1=Y_2=\\ldots = Y_\\Delta = 0$, then it has to be the case that $u$ is matched to a vertex in some subtree $T_i$ (not necessarily $v_i$), while any vertex from $N_{-i}(u) := \\{v_1, v_2, \\dots, v_{i-1}, v_{i+1}, \\dots, v_\\Delta\\}$ is matched with another vertex from $N_{-i}(u)$ (hence, we already get that $\\Delta$ needs to be odd).\nThis is because each subtree needs to have at least one vertex matched to a different subtree (since $|S_i|$ is odd for each $i \\in [\\Delta]$).\nIf $M(u)$ is a vertex in $T_i$ and, for any $v \\in N_{-i}(u)$, we have $M(v) \\in N_{-i}(u)$, we say that $u$ orients the edge going to $v_i$ outwards.\n\nConsider a path $(u_0 = u, u_1, \\dots, u_k)$, where $k = 2t(n) + 3$.\nBy the argument provided above, the probability that $u$ orients an edge outwards is at least $1 - \\Delta\/n^2 - 1\/n^{2\\Delta}$, and, if $u$ orients an edge outwards, the probability that this edge is not $uu_1$ is $(\\Delta - 1)\/\\Delta$, by symmetry.\nHence, we obtain (for sufficiently large $n$) that $\\P(\\mathcal{E}_1) \\geq 1\/2$, where $\\mathcal{E}_1$ denotes the event that vertex $u$ orients an edge different from $uu_1$ outwards.\nWith an analogous argument, we obtain that $\\P(\\mathcal{E}_2) \\geq 1\/2$, where $\\mathcal{E}_2$ denotes the event that vertex $u_k$ orients an edge different from $u_ku_{k-1}$ outwards.\nSince $\\mathcal{E}_1$ and $\\mathcal{E}_2$ are independent (due to the fact that the distance between any neighbor of $u$ and any neighbor of $u_k$ is at least $2t(n) + 1$), we obtain $\\P(\\mathcal{E}_1 \\cap \\mathcal{E}_2) \\geq 1\/4$.\nMoreover, the probability that all vertices in $\\{ u_1, \\dots, u_{k-1} \\}$ orient an edge outwards is at least $1 - (2t(n)+2)(\\Delta\/n^2 + 1\/n^{2\\Delta})$, which is at least $4\/5$ for sufficiently large $n$.\nThus, by a union bound, there is a probability of at least $1 - 3\/4 - 1\/5 - 1\/n^{2\\Delta}$ that $\\mathcal{A}$ does not fail, all vertices in $(u, u_1, \\dots, u_k)$ orient an edge outwards, and the outward oriented edges chosen by $u$ and $u_k$ are not $uu_1$ and $u_ku_{k-1}$, respectively.\nFor sufficiently large $n$, the indicated probability is strictly larger than $0$.\nWe will obtain a contradiction and conclude the proof for $\\Pi^2_{\\text{pm}}$ by showing that there is no correct solution with the described properties.\n\nAssume in the following that the solution provided by $\\mathcal{A}$ is correct and $u, u_1, \\dots, u_k$ satisfy the mentioned properties.\nSince $u$ orients an edge different from $uu_1$ outwards, $u_1$ must be matched to a neighbor of $u$, which implies that $u_1$ orients edge $u_1u$ outwards.\nUsing an analogous argumentation, we obtain inductively that $u_j$ orients edge $u_ju_{j-1}$ outwards, for all $2 \\leq j \\leq k$.\nHowever, this yields a contradiction to the fact that the edge that $u_k$ orients outwards is different from $u_ku_{k-1}$, concluding the proof (for $\\Pi^2_{\\text{pm}}$).\n\nThe theorem statement for $\\overline{\\Pi^2_{\\text{pm}}}$ follows by applying \\cref{pr:AddingLineLLL}.\n\n\n\\end{proof}\n\n\n\n\n\n\\subsection{Upper Bounds Using Forest Decompositions}\n\nWe prove an upper bound for the problems $\\Pi^2_{\\text{pm}}$ and $\\overline{\\Pi^2_{\\text{pm}}}$ defined in the previous subsection.\nWe use a technique of decomposing the input graph in a spanning forest with some additional properties, i.e., one or two ended. This technique was used in \\cite{BrooksMeas} to prove Brooks' theorem in a measurable context.\nNamely, they proved that if $\\mathcal{G}$ is a Borel $\\Delta$-regular acyclic graph and $\\mu$ is a Borel probability measure, then it is possible to erase some edges of $\\mathcal{G}$ in such a way that the remaining graph is a one-ended spanning forest on a $\\mu$-conull set.\nIt is not hard to see that one can solve $\\Pi^2_{\\text{pm}}$ on one-ended trees in an inductive manner, starting with the leaf vertices. Consequently we have $\\Pi^2_{\\text{pm}}\\in \\mathsf{MEASURE}$.\nWe provide a quantitative version of this result and thereby show that the problem of constructing a one-ended forest is contained in $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon)$.\nA variation of the construction shows that it is possible to find a one or two-ended spanning forest in a Borel way.\nThis allows us to show that $\\overline{\\Pi^2_{\\text{pm}}}\\in \\mathsf{BOREL}$.\nAs an application of the decomposition technique we show that Vizing's theorem, i.e., proper $\\Delta+1$ edge coloring $\\Pi_{\\chi',\\Delta+1}$, for $\\Delta=3$ is in the class $\\mathsf{OLOCAL}(\\operatorname{poly} \\log 1\/\\varepsilon)$.\n\n\n\\subsubsection{Uniform Complexity of the One-Forest Decomposition }\nWe start with the definition of a one-ended spanning forest.\nIt can be viewed as a variation of the edge grabbing problem $\\Pi_{\\text{edgegrab}}$.\nNamely, if a vertex $v$ grabs an incident edge $e$, then we think of $e$ as being oriented away from $v$ and $v$ selecting the other endpoint of $e$ as its parent.\nSuppose that $\\mathcal{T}$ is a solution of $\\Pi_{\\text{edgegrab}}$ on $T_\\Delta$.\nWe denote with $\\mathcal{T}^{\\leftarrow}(v)$ the subtree with root $v$.\n\n\n\\begin{definition}[One-ended spanning forest]\nWe say that a solution $\\mathcal{T}$ of $\\Pi_{\\text{edgegrab}}$ is a \\emph{one-ended spanning forest} if $\\mathcal{T}^{\\leftarrow}(v)$ is finite for every $v\\in T_\\Delta$ (see \\cref{fig:one_ended}). \n\n\\end{definition}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=.4\\textwidth]{fig\/one_ended.pdf}\n \\caption{An example of a one-ended forest decomposition of an infinite 3-regular tree. }\n \\label{fig:one_ended}\n\\end{figure}\n\n\nNote that on an infinite $\\Delta$-regular tree every connected component of a one-ended spanning forest must be infinite.\nFurthermore, recall that we discussed above that it is possible to construct a one-ended spanning forest in $\\mathsf{MEASURE}$ by \\cite{BrooksMeas}.\nNext we formulate our quantitative version.\n\n\n\n\n\n\\begin{restatable}{theorem}{oneendedtreetail}\n\\label{thm:one_ended_forest}\nThe one-ended spanning forest can be constructed by an uniform local algorithm with an uniform local complexity of $O(\\operatorname{poly}\\log1\/\\varepsilon)$. \nMore precisely, there is an uniform distributed algorithm $\\mathcal{A}$ that computes a one-ended spanning forest and if we define $R(v)$ to be the smallest coding radius that allows $v$ to compute $\\mathcal{T}^{\\leftarrow}(v)$, then\n\\[\nP(R(v) > O(\\operatorname{poly} \\log 1\/\\varepsilon)) \\le \\varepsilon. \n\\]\n\\end{restatable}\n\nThe formal proof of \\cref{thm:one_ended_forest} can be found in \\cref{sec:formalproof}.\nWe first show the main applications of the result.\nNamely, we show that $\\Pi^2_{\\text{pm}}$ can be solved inductively starting from the leaves of the one-ended trees. This proves that $\\Pi^2_{\\text{pm}}$ is in $\\mathsf{OLOCAL}(\\operatorname{poly} \\log 1\/\\varepsilon)$ and hence $\\mathsf{OLOCAL}(\\operatorname{poly} \\log 1\/\\varepsilon) \\setminus \\mathsf{RLOCAL}(o(\\log n))$ is non-empty.\n\n\\begin{theorem}\\label{thm:twomatchFFIID}\nThe problem $\\Pi^2_{\\text{pm}}$ is in the class $\\mathsf{MEASURE}$ and $\\mathsf{OLOCAL}(O(\\operatorname{poly} \\log 1\/\\varepsilon))$.\n\\end{theorem}\n\\begin{proof}\nFirst we show that every infinite one-ended directed tree $T$ admits an inductive solution from the leaves to the direction of infinity. More precisely, we define a power-$2$ perfect matching on the infinite one-ended directed tree $T$ such that the following holds. Let $v$ be an arbitrary vertex and $T^{\\leftarrow}(v)$ the finite subtree rooted at $v$. Then, all vertices in $T^{\\leftarrow}(v)$, with the possible exception of $v$, are matched to a vertex in $T^{\\leftarrow}(v)$. Moreover, for each vertex in $T^{\\leftarrow}(v) \\setminus \\{v\\}$, it is possible to determine to which vertex it is matched by only considering the subtree $T^{\\leftarrow}(v)$. \nWe show that it is possible to define a perfect matching in that way by induction on the height of the subtree $T^{\\leftarrow}(v)$. \n\nWe first start with the leaves and declare them as unmatched in the first step of the induction.\n\nNow, consider an arbitrary vertex $v$ with $R \\geq 1$ children that we denote by $v_1, v_2, \\ldots, v_R$.\nBy the induction hypothesis, all vertices in $T^{\\leftarrow}(v_i) \\setminus \\{v_i\\}$ are matched with a vertex in $T^{\\leftarrow}(v_i)$ for $i \\in [R]$ and it is possible to compute that matching given $T^{\\leftarrow}(v)$. However, some of the children of $v$ are possibly still unmatched. If the number of unmatched children of $v$ is even, then we simply pair them up in an arbitrary but fixed way and we leave $v$ unmatched. Otherwise, if the number of unmatched children of $v$ is odd, we pair up $v$ with one of its unmatched children and we pair up all the remaining unmatched children of $v$ in an arbitrary but fixed way. This construction clearly satisfies the criteria stated above. In particular, the resulting matching is a perfect matching, as every vertex gets matched eventually.\n\n\nBy \\cref{thm:one_ended_forest} there is an uniform distributed algorithm of complexity $O(\\operatorname{poly} \\log 1\/\\varepsilon)$ that computes a one-ended spanning forest $\\mathcal{T}$ on $T_\\Delta$.\nMoreover, every vertex $v\\in T_\\Delta$ can compute where it is matched with an uniform local complexity of $O(\\operatorname{poly} \\log 1\/\\varepsilon)$.\nThis follows from the discussion above, as a vertex $v$ can determine where it is matched once it has computed $\\mathcal{T}^{\\leftarrow}(w)$ for each $w$ in its neighborhood.\nHence, $\\Pi^2_{\\text{pm}}$ is in the class $\\mathsf{OLOCAL}(O(\\operatorname{poly} \\log 1\/\\varepsilon))$.\n\nSimilarly, the one-ended spanning forest $\\mathcal{T}\\subseteq \\mathcal{G}$ can be constructed in the class $\\mathsf{MEASURE}$ by \\cite{BrooksMeas}.\nBy the discussion above, this immediately implies that $\\Pi^2_{\\text{pm}}$ is in the class $\\mathsf{MEASURE}$. \n\\end{proof}\n\n\n\n\n\\subsubsection{Decomposition in $\\mathsf{BOREL}$}\nNext, we show that a similar, but weaker, decomposition can be done in a Borel way.\nNamely, we say that a subset of edges of an acyclic infinite graph $G$, denoted by $\\mathcal{T}$, is a \\emph{one or two-ended spanning forest} if every vertex $v\\in G$ is contained in an infinite connected component of $\\mathcal{T}$ and each infinite connected component of $\\mathcal{T}$ has exactly one or two directions to infinity.\nLet $S$ be such a connected component.\nNote that if $S$ has one end, then we can solve $\\Pi^2_{\\text{pm}}$ on $S$ as above.\nIf $S$ has two ends, then $S$ contains a doubly infinite path $P$ with the property that erasing $P$ splits $S\\setminus P$ into finite connected components.\nIn order to solve $\\overline{\\Pi^2_{\\text{pm}}}$ we simply solve $\\Pi^2_{\\text{pm}}$ on the finite connected components of $S\\setminus P$ (with some of the vertices in $S\\setminus P$ being potentially matched with vertices on the path $P$) and declare vertices on $P$ to be line vertices.\n\nThe high-level idea to find a one or two-ended spanning forest is to do the same construction as in the measure case and understand what happens with edges that do not disappear after countably many steps.\nA slight modification in the construction guarantees that what remains are doubly infinite paths.\n\n\\begin{restatable}{theorem}{oneendedforestborel}\n\\label{thm:OneTwoEndBorel}\nLet $\\mathcal{G}$ be a Borel $\\Delta$-regular forest. \nThen there is a Borel one or two-ended spanning forest $\\mathcal{T}\\subseteq \\mathcal{G}$.\n\\end{restatable}\n\nThe proof of \\cref{thm:OneTwoEndBorel} can be found in \\cref{sec:oneortwo}.\nWe remark that the notation used in the proof is close to the notation in the proof that gives a measurable construction of a one-ended spanning forest in \\cite{BrooksMeas}.\nNext, we show more formally how \\cref{thm:OneTwoEndBorel} implies that $\\overline{\\Pi^2_{\\text{pm}}}\\in \\mathsf{BOREL}$.\n\n\\begin{theorem}\nThe problem $\\overline{\\Pi^2_{\\text{pm}}}$ is in the class $\\mathsf{BOREL}$.\n\\end{theorem}\n\\begin{proof}\n\nLet $\\mathcal{T}$ be the one or two-ended spanning forest given by \\cref{thm:OneTwoEndBorel} and $S$ be a connected component of $\\mathcal{T}$.\nIf $S$ is one-ended, then we use the first part of the proof of \\cref{thm:twomatchFFIID} to find a solution of $\\Pi^2_{\\text{pm}}$ on $S$.\nSince deciding that $S$ is one-ended as well as computing an orientation towards infinity can be done in a Borel way, this yields a Borel labeling.\n\nSuppose that $S$ is two-ended.\nThen, there exists a doubly infinite path $P$ in $S$ with the property that the connected components of $S\\setminus P$ are finite.\nMoreover, it is possible to detect $P$ in a Borel way.\nThat is, declaring the vertices on $P$ to be line vertices and using the special symbol in $\\overline{\\Pi^2_{\\text{pm}}}$ for the half-edges on $P$ yields a Borel measurable labeling.\nLet $C\\not=\\emptyset$ be one of the finite components in $S\\setminus P$ and $v\\in C$ be the vertex of distance $1$ from $P$.\nOrient the edges in $C$ towards $v$, this can be done in a Borel way since $v$ is uniquely determined for $C$.\nThen the first part of the proof of \\cref{thm:twomatchFFIID} shows that one can inductively find a solution to $\\Pi^2_{\\text{pm}}$ on $C$ in such a way that all vertices, possibly up to $v$, are matched.\nIf $v$ is matched, then we are done.\nIf $v$ is not matched, then we add the edge that connects $v$ with $P$ to the matching. Note that it is possible that multiple vertices are matched with the same path vertex, but this is not a problem according to the definition of $\\overline{\\Pi^2_{\\text{pm}}}$. \nIt follows that this defines a Borel function on $H(\\mathcal{G})$ that solves $\\overline{\\Pi^2_{\\text{pm}}}$.\n\\end{proof}\n\n\n\n\n\n\\subsubsection{Vizing's Theorem for $\\Delta=3$}\nFinding a measurable or local version of Vizing's Theorem, i.e., proper edge $(\\Delta+1)$-coloring $\\Pi_{\\chi',\\Delta+1}$, was studied recently in \\cite{grebik2020measurable,weilacher2021borel,BernshteynVizing}.\nIt is however not known, even on trees, whether $\\Pi_{\\chi',\\Delta+1}$ is in $\\mathsf{RLOCAL}(O(\\log\\log n))$.\nHere we use the one-ended forest construction to show that $\\Pi_{\\chi',\\Delta+1}$ is in the class $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon)$ for $\\Delta=3$.\n\n\n\n\\iffalse\n\\paragraph{Composition of Uniform Local Algorithms}\nWhen designing algorithms, one often composes several simple algorithms to obtain a more sophisticated algorithm. For example, an algorithm $\\mathcal{A}$ might first invoke algorithm $\\mathcal{A}_1$ and after $\\mathcal{A}_1$ finished, $\\mathcal{A}$ invokes algorithm $\\mathcal{A}_2$ with the input of $\\mathcal{A}_2$ being equal to the output of $\\mathcal{A}_1$. Then, the final output of $\\mathcal{A}$ is simply the output of $\\mathcal{A}_2$. We call such a composition a sequential composition. In the context of the classic deterministic local model, the round complexity of $\\mathcal{A}$ is simply the sum of the round complexities of the two algorithms $\\mathcal{A}_1$ and $\\mathcal{A}_2$ that are used as a subroutine. Here, we prove two such composition results in the context of uniform local algorithms. These general results will help us in \\cref{app:one_ended} to analyze the uniform complexity of an uniform algorithm that computes a so-called one-ended forest in a principled manner.\n\n\n\n\\begin{lemma}[Parallel Composition]\n\\label{lem:parallel_composition}\nLet $A_1$ and $A_2$ be two distributed uniform algorithms with an uniform local complexity of $t_{A_1}(\\varepsilon)$ and $t_{A_2}(\\varepsilon)$, respectively. \nLet $A$ be the parallel composition of $A_1$ and $A_2$. That is, the coding radius of algorithm $\\mathcal{A}$ at a vertex $u$ is the smallest radius such that $A$ knows the local answer of both $A_1$ and $A_2$ at $u$. \nThen, $t_A(\\varepsilon) \\leq \\max(t_{A_1}(\\varepsilon\/2), t_{A_2}(\\varepsilon\/2))$.\n\\end{lemma}\n\\begin{proof}\nThe probability that the coding radius of algorithm $\\mathcal{A}_1$ at vertex $u$ is larger than $\\max(t_{A_1}(\\varepsilon\/2), t_{A_2}(\\varepsilon\/2))$ is at most $\\varepsilon\/2$ and the same holds for the coding radius of algorithm $\\mathcal{A}_2$ at vertex $u$. Hence, by a simple union bound, the probability that the coding radius of algorithm $\\mathcal{A}$ at a vertex $u$ is larger than $\\max(t_{A_1}(\\varepsilon\/2), t_{A_2}(\\varepsilon\/2))$ is at most $\\varepsilon\/2 + \\varepsilon\/2 = \\varepsilon$. \\\\ Hence, $t_A(\\varepsilon) \\leq \\max(t_{A_1}(\\varepsilon\/2), t_{A_2}(\\varepsilon\/2))$, as desired.\n\\end{proof}\n\n\\fi\n\n\n\n\\begin{proposition}\nLet $\\Delta=3$. We have $\\Pi_{\\chi',\\Delta+1} \\in\\mathsf{OLOCAL}(\\operatorname{poly} \\log 1\/\\varepsilon)$.\n\\end{proposition}\n\\begin{proof}[Proof sketch.]\nBy \\cref{thm:one_ended_forest}, we can compute a one-ended forest decomposition $\\mathcal{T}$ with uniform local complexity $O(\\operatorname{poly} \\log 1\/\\varepsilon)$. \nNote that every vertex has at least one edge in $\\mathcal{T}$, hence the edges in $G \\setminus \\mathcal{T}$ form paths. \nThese paths can be $3$-edge colored by using the uniform version of Linial's coloring algorithm. This algorithm has an uniform local complexity of $O(\\log^* 1\/\\varepsilon)$. By \\cref{lem:sequential_composition}, the overall complexity is $O(\\operatorname{poly} \\log (\\Delta^{\\log^* 1\/\\varepsilon} \/ \\varepsilon) + \\log^* 1\/\\varepsilon)) = O(\\operatorname{poly} \\log 1\/\\varepsilon)$. \n\nFinally, we color the edges of $T$. We start from the leaves and color the edges inductively. In particular, whenever we consider a vertex $v$ we color the at most two edges in $T^{\\leftarrow}(v)$ below $v$. \nNote that there always is at least one uncolored edge going from $v$ in the direction of $T$. Hence, we can color the at most two edges below $v$ in $T^{\\leftarrow}(v)$ greedily -- each one neighbors with at most $3$ colored edges at any time. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{thm:one_ended_forest}} \n\\label{sec:formalproof}\n\n\nIn this section we formally prove \\cref{thm:one_ended_forest}. \n\n\n\\oneendedtreetail*\n\nWe follow the construction of the one-ended spanning forest in \\cite{BrooksMeas}.\nThe construction proceeds in rounds. Before each round, some subset of the vertices have already decided which incident edge to grab, and we refer to these vertices as settled vertices. All the remaining vertices are called unsettled. The goal in each round is to make a large fraction of the unsettled vertices settled. To achieve this goal, our construction relies on certain properties that the graph induced by all the unsettled vertices satisfies. \n\nOne important such property is that the graph induced by all the unsettled vertices is expanding. That is, the number of vertices contained in the neighborhood around a given vertex grows exponentially with the radius. The intuitive reason why this is a desirable property is the following. Our algorithm will select a subset of the unsettled vertices and clusters each unsettled vertex to the closest selected vertex. As the graph is expanding and any two vertices in the selected subset are sufficiently far away, there will be a lot of edges leaving a given cluster. For most of these inter-cluster edges, both of the endpoints will become settled. This in turn allows one to give a lower bound on the fraction of vertices that become settled in each cluster.\n\nTo ensure that the graph induced by all the unsettled vertices is expanding, a first condition we impose on the graph induced by all the unsettled vertices is that it has a minimum degree of at least $2$. While this condition is a first step in the right direction, it does not completely suffice to ensure the desired expansion, as an infinite path has a minimum degree of $2$ but does not expand sufficiently. Hence, our algorithm will keep track of a special subset of the unsettled vertices, the so-called hub vertices. Each hub vertex has a degree of $\\Delta$ in the graph induced by all the unsettled vertices. That is, all of the neighbors of a hub vertex are unsettled as well. Moreover, each unsettled vertex has a bounded distance to the closest hub vertex, where the specific upper bound on the distance to the closest hub vertex increases with each round. As we assume $\\Delta > 2$, the conditions stated above suffice to show that the graph induced by all the unsettled vertices expands.\n\nNext, we explain in more detail how a vertex decides which edge to grab. Concretely, if a vertex becomes settled, it grabs an incident edge such that the other endpoint of that edge is strictly closer to the closest unsettled vertex as the vertex itself. For example, if a vertex becomes settled and there is exactly one neighbor that is still unsettled, then the vertex will grab the edge that the vertex shares with its unsettled neighbor. Grabbing edges in that way, one can show two things. First, for a settled vertex $v$, the set $T^{\\leftarrow}(v)$ of vertices behind $v$ does not change once $v$ becomes settled. The intuitive reason for this is that the directed edges point towards the unsettled vertices and therefore the unsettled vertices lie before $v$ and not after $v$. Moreover, one can also show that $T^{\\leftarrow}(v)$ only contains finitely many vertices. The reason for this is that at the moment a vertex $v$ becomes settled, there exists an unsettled vertex that is sufficiently close to $v$, where the exact upper bound on that distance again depends on the specific round in which $v$ becomes settled. \n\nWhat remains to be discussed is at which moment a vertex decides to become settled. As written above, in each round the algorithm considers a subset of the unsettled vertices and each unsettled vertex is clustered to the closest vertex in that subset. This subset only contains hub vertices and it corresponds to an MIS on the graph with the vertex set being equal to the set of hub vertices and where two hub vertices are connected by an edge if they are sufficiently close in the graph induced by all the unsettled vertices. Now, each cluster center decides to connect to exactly $\\Delta$ different neighboring clusters, one in each of its $\\Delta$ subtrees. Remember that as each cluster center is a hub vertex, all of its neighbors are unsettled as well. Now, each unsettled vertex becomes settled except for those that lie on the unique path between two cluster centers such that one of the cluster centers decided to connect to the other one.\n\n\n\n\n\n\n\\paragraph{Construction}\nThe algorithm proceeds in rounds. After each round, a vertex is either settled or unsettled and a settled vertex remains settled in subsequent rounds. Moreover, some unsettled vertices are so-called hub vertices. We denote the set of unsettled, settled and hub vertices after the $i$-th round with $U_i$, $S_i$ and $H_i$, respectively. We set $U_0 = H_0 = V$ prior to the first round and we always set $S_i = V \\setminus U_i$. \\cref{fig:constr1} illustrates the situation after round $i$. Moreover, we denote with $O_i$ a partial orientation of the edges of the infinite $\\Delta$-regular input tree $T$. In the beginning, $O_0$ corresponds to the partial orientation with no oriented edges. Each vertex is incident to at most $1$ outwards oriented edge in $O_i$. If a vertex is incident to an outwards oriented edge in $O_i$, then the other endpoint of the edge will be its parent in the one-ended forest decomposition. For each vertex $v$ in $S_i$, we denote with $T_{v,i}$ the smallest set that satisfies the following conditions. First, $v \\in T_{v,i}$. Moreover, if $u \\in T_{v,i}$ and $\\{w,u\\}$ is an edge that according to $O_i$ is oriented from $w$ to $u$, then $w \\in T_{v,i}$. We later show that $T_{v,i}$ contains exactly those vertices that are contained in the subtree $\\mathcal{T}^\\leftarrow(v)$.\n\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics{fig\/one_ended_construction.pdf}\n \\caption{The picture illustrates the situation after some round $i$ of the construction. The vertices on the paths are all contained in $U_i$, while the vertices corresponding to the big black dots are additionally contained in $H_i$. Note that the length of each path is at least $d_i$. However, the distance between $u$ and $v$ in the original graph can be much smaller.}\n \\label{fig:constr1}\n\\end{figure}\n\n\nAfter the $i$-th round, the construction satisfies the following invariants.\n\n\n\n\\begin{enumerate}\n \n \\item For each $v \\in S_{i-1}$, we have $T_{v,i} = T_{v,i-1}$.\n \n \\item Let $v$ be an arbitrary vertex in $S_i$. Then, $T_{v,i}$ contains finitely many vertices and furthermore $T_{v,i} \\subseteq S_i$.\n \n \\item The minimum degree of the graph $T[U_i]$ is at least $2$ and each vertex in $H_i$ has a degree of $\\Delta$ in $T[U_i]$.\n \n \\item Each vertex in $U_i$ has a distance of at most $\\sum_{j=0}^i d_j$ to the closest vertex in $H_i$ in the graph $T[U_i]$.\n\\end{enumerate}\n\nWe now describe how to compute $U_i, S_i$, $H_i$ and $O_i$. The construction during the $i$-th round is illustrated in $\\cref{fig:constr2}$. Note that we can assume that the invariants stated above are satisfied after the $(i-1)$-th round. In the $i$-th round we have a parameter $d_i$ that we later set to $2^{2^i}$.\n\n\\begin{enumerate}\n \\item $H_i$ is a subset of $H_{i-1}$ that satisfies the following property. No two vertices in $H_i$ have a distance of at most $d_i$ in $T[U_{i-1}]$. Moreover, each vertex in $H_{i-1} \\setminus H_i$ has a distance of at most $d_i$ to the closest vertex in $H_i$ in the graph $T[U_{i-1}]$. Note that we can compute $H_i$ by computing an MIS in the graph with vertex set $H_{i-1}$ and where two vertices are connected iff they have a distance of at most $d_i$ in the graph $T[U_{i-1}]$. Hence, we can use Ghaffari's MIS algorithm \\cite{ghaffari2016MIS} to compute the set $H_i$.\n \\item Next, we describe how to compute $U_i$. \n We assign each vertex $u \\in U_{i-1}$ to the closest vertex in the graph $T[U_{i-1}]$ that is contained in $H_i$, with ties being broken arbitrarily. We note that there exists a node in $H_i$ with a distance of at most $(\\sum_{j=0}^{i-1} d_j) + d_i$ to $u$. To see why, note that Invariant $(4)$ from round $i - 1$ implies that there exists a vertex $w$ in $H_{i-1}$ with a distance of at most $\\sum_{j=0}^{i-1} d_j$ to $u$. We are done if $w$ is also contained in $H_i$. If not, then it follows from the way we compute $H_i$ that there exists a vertex in $H_i$ with a distance of at most $d_i$ to $w$, but then the triangle inequality implies that the distance from $u$ to that vertex is at most $(\\sum_{j=0}^{i-1} d_j) + d_i$, as desired. For each vertex $v \\in H_i$, we denote with $C(v)$ the set of all vertices in $U_{i-1}$ that got assigned to $v$. Now, let $E_v$ denote the set of edges that have exactly one endpoint in $C(v)$. We can partition $E_v$ into $E_{v,1} \\sqcup E_{v,2} \\sqcup \\ldots \\sqcup E_{v,\\Delta}$, where for $\\ell \\in [\\Delta]$, $E_{v,\\ell}$ contains all the edges in $E_v$ that are contained in the $\\ell$-th subtree of $v$. \n Invariants $(3)$ and $(4)$ after the $(i-1)$-th round imply that $E_{v,\\ell} \\neq \\emptyset$. Moreover, $E_{v,\\ell}$ contains only finitely many edges, as we have shown above that the cluster radius is upper bounded by $\\sum_{j=0}^{i} d_j$.\n \n Now, for $\\ell \\in [\\Delta]$, we choose an edge $e_\\ell$ uniformly at random from the set $E_{v,\\ell}$. Let $u_\\ell \\neq v$ be the unique vertex in $H_i$ such that one endpoint of $e_\\ell$ is contained in $C(u_\\ell)$. We denote with $P_{v,\\ell}$ the set of vertices that are contained in the unique path between $v$ and $u_\\ell$. Finally, we set $U_i = \\bigcup_{v \\in H_i, \\ell \\in [\\Delta]} P_{v,\\ell}$.\n \\item It remains to describe how to compute the partial orientation $O_i$. All edges that are oriented in $O_{i-1}$ will be oriented in the same direction in $O_i$. Additionally, we orient for each vertex $u$ that got settled in the $i$-th round, i.e., $u \\in S_i \\cap U_{i-1}$, exactly one incident edge away from $u$. Let $w$ be the closest vertex to $u$ in $T[U_{i-1}]$ that is contained in $U_i$, with ties being broken arbitrarily. We note that it follows from the discussions above that the distance between $u$ and $w$ in the graph $T[U_{i-1}]$ is at most $\\sum_{j=0}^{i} d_j$. We now orient the edge incident to $u$ that is on the unique path between $u$ and $w$ outwards. We note that this orientation is well-defined in the sense that we only orient edges that were not oriented before and that we don't have an edge such that both endpoints of that edge want to orient the edge outwards.\n\\end{enumerate}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics{fig\/one_ended_construction2.pdf}\n \\caption{The picture illustrates the situation during some round $i$ of the construction. The vertices on all paths are contained in $U_{i-1}$ and the vertices corresponding to the big black dots are contained in $H_{i-1}$. The vertices $v$, $u_1$, $u_2$ and $u_3$ are the only vertices in the picture that are contained in $H_i$. All the vertices on the red paths remain unsettled during round $i$ while all the vertices on the black paths become settled. Each vertex that becomes settled during the $i$-th round orients one incident edge outwards, illustrated by the green edges. Each green edge is oriented towards the closest vertex that is still unsettled after the $i$-th round. We note that some of the edges in the picture are oriented away from $v$.}\n \\label{fig:constr2}\n\\end{figure}\n\n\nWe now prove by induction that our procedure satisfies all the invariants.\nPrior to the first round, all invariants are trivially satisfied. Hence, it remains to show that the invariants hold after the $i$-th round, given that the invariants hold after the $(i-1)$-th round. \n\n\\begin{enumerate}\n\n \\item Let $v \\in S_{i-1}$ be arbitrary. As $O_i$ is an extension of $O_{i-1}$, it follows from the definition of $T_{v,i-1}$ and $T_{v,i}$ that $T_{v,i-1} \\subseteq T_{v,i}$. Now assume that $T_{v,i} \\not \\subseteq T_{v,i-1}$. This would imply the existence of an edge $\\{u,w\\}$ such that $u \\in T_{v,i-1}$ and the edge was oriented during the $i$-th round from $w$ to $u$. As $u \\in T_{v,i-1}$, Invariant $(2)$ from the $(i-1)$-th round implies $u \\in S_{i-1}$. This is a contradiction as during the $i$-th round only edges with both endpoints in $U_{i-1}$ can be oriented. Therefore it holds that $T_{v,i} = T_{v,i-1}$, as desired.\n\n \\item Let $v \\in S_i$ be arbitrary. If it also holds that $v \\in S_{i-1}$, then the invariant from round $i-1$ implies that $T_{v,i-1} = T_{v,i}$ contains finitely many vertices and $T_{v,i-1} \\subseteq S_{i-1} \\subseteq S_i$. Thus, it suffices to consider the case that $v \\in S_i \\cap U_{i-1}$. Note that $T_{v,i} \\cap U_i = \\emptyset$. Otherwise there would exist an edge that is oriented away from a vertex in $U_i$ according to $O_i$, but this cannot happen according to the algorithm description. Hence, $T_{v,i} \\subseteq S_i$. Now, for the sake of contradiction, assume that $T_{v,i}$ contains infinitely many vertices. From the definition of $T_{v,i}$, this implies that there exists a sequence of vertices $(v_k)_{k \\geq 1}$ with $v = v_1$ such that for each $k \\geq 1$, $\\{v_{k+1},v_k\\}$ is an edge in $T$ that is oriented from $v_{k+1}$ to $v_k$ according to $O_i$. For each $k \\geq 1$ we have $v_k \\in S_i$. We furthermore know that $v_k \\in U_{i-1}$, as otherwise $T_{v_k,i} = T_{v_k,i-1}$ would contain infinitely many vertices, a contradiction. From the way we orient the edges, a simple induction proof implies that the closest vertex of $v_k$ in the graph $T[U_{i-1}]$ that is contained in $U_i$ has a distance of at least $k$. However, we previously discussed that the distance between $v_k$ and the closest vertex in $U_i$ in the graph $T[U_{i-1}]$ is at most $\\sum_{j=0}^{i} d_j$. This is a contradiction. Hence, $T_{v,i}$ contains finitely many vertices.\n \n \\item It follows directly from the description of how to compute $U_i$ and $H_i$ that the minimum degree of the graph $T[U_i]$ is at least $2$ and that each vertex in $H_i$ has a degree of $\\Delta$ in $T[U_i]$.\n \n \\item Let $u \\in U_i$ be arbitrary. We need to show that $u$ has a distance of at most $\\sum_{j=0}^i d_j$ to the closest vertex in $H_i$ in the graph $T[U_i]$. Let $v$ be the vertex with $u \\in C(v)$. We know that there is no vertex in $H_i$ that is closer to $u$ in $T[U_{i-1}]$ than $v$. Hence, the distance between $u$ and $v$ is at most $\\sum_{j=0}^i d_j$ in the graph $T[U_{i-1}]$. Moreover, from the description of how we compute $U_i$, it follows that all the vertices on the unique path between $u$ and $v$ are contained in $U_i$. Hence, each vertex in $U_i$ has a distance of at most $\\sum_{j=0}^i d_j$ to the closest vertex in $H_i$ in the graph $T[U_i]$, as desired.\n \\end{enumerate}\n\nWe derive an upper bound on the coding radius of the algorithm in three steps. First, for each $i \\in \\mathbb{N}$ and $\\varepsilon > 0$, we derive an upper bound on the coding radius for computing with probability at least $1 - \\varepsilon$ for a given vertex in which of the sets $S_i, U_i$ and $H_i$ it is contained in and for each incident edge whether and how it is oriented in $O_i$, given that each vertex receives as additional input in which of the sets $S_{i-1}, U_{i-1}$ and $H_{i-1}$ it is contained in and for each incident edge whether and how it is oriented in $O_{i-1}$. Given this upper bound on the coding radius, we can use the sequential composition lemma (\\cref{lem:sequential_composition}) and a simple induction proof to give an upper bound on the coding radius for computing with probability at least $1 - \\varepsilon$ for a given vertex in which of the sets $S_i, U_i$ and $H_i$ it is contained in and for each incident edge whether and how it is oriented in $O_i$, this time without providing any additional input. \n\nSecond, we analyze after how many rounds a given vertex is settled with probability at least $1- \\varepsilon$ for a given $\\varepsilon > 0$. \n\nFinally, we combine these two upper bounds to prove \\cref{thm:one_ended_forest}.\n \n \n\\begin{lemma}\n\\label{lem:tail1}\nFor each $i \\in \\mathbb{N}$ and $\\varepsilon \\in (0,0.01]$, let $g_i(\\varepsilon)$ denote the smallest coding radius such that with probability at least $1 - \\varepsilon$ one knows for a given vertex in which of the sets $S_i, U_i$ and $H_i$ it is contained in and for each incident edge whether and how it is oriented in $O_i$, given that each vertex receives as additional input in which of the sets $S_{i-1}, U_{i-1}$ and $H_{i-1}$ it is contained in and for each incident edge whether and how it is oriented in $O_{i-1}$. It holds that $g_i(\\varepsilon) = O(d^2_i \\log (\\Delta\/\\varepsilon))$.\n\\end{lemma}\n\n\\begin{proof}\n When running Ghaffari's uniform MIS algorithm on some graph $G'$ with maximum degree $\\Delta'$, each vertex knows with probability at least $1-\\varepsilon$ whether it is contained in the MIS or not after $O(\\log (\\Delta') + \\log (1\/\\varepsilon))$ communication rounds in $G'$ (\\cite{ghaffari2016MIS}, Theorem 1.1). For the MIS computed during the $i$-th round, $\\Delta' =\\Delta^{O(d_i)}$ and each communication round in $G'$ can be simulated with $O(d_i)$ communication rounds in the tree $T$. Hence, to know for a given vertex with probability at least $1-\\varepsilon$ whether it is contained in the MIS or not, it suffices consider the $O(d_i^2\\log(\\Delta) + d_i\\log(1\/\\varepsilon))$-hop neighborhood around that vertex. Now, let $u$ be an arbitrary vertex. In order to compute in which of the sets $S_i, U_i$ and $H_i$ $u$ is contained in and for each incident edge of $u$ whether and how it is oriented in $O_i$, we not only need to know whether $u$ is in the MIS or not. However, it suffices if we know for all the vertices in the $O(d_i)$-hop neighborhood of $u$ whether they are contained in the MIS or not (on top of knowing for each vertex in the neighborhood in which of the sets $S_{i-1}, U_{i-1}$ and $H_{i-1}$ it is contained in and for each incident edge whether and how it is oriented in $O_{i-1}$). Hence, by a simple union bound over the at most $\\Delta^{O(d_i)}$ vertices in the $O(d_i)$-hop neighborhood around $u$, we obtain $g_i(\\varepsilon) = O(d_i) + O(d_i^2\\log(\\Delta) + d_i\\log(\\Delta^{O(d_i)}\/\\varepsilon)) = O(d_i^2 \\log(\\Delta\/\\varepsilon))$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:tail2}\nFor each $i \\in \\mathbb{N}$ and $\\varepsilon \\in (0,0.01]$, let $h_i(\\varepsilon)$ denote the smallest coding radius such that with probability at least $1 - \\varepsilon$ one knows for a given vertex in which of the sets $S_i, U_i$ and $H_i$ it is contained in and for each incident edge whether and how it is oriented in $O_i$. Then, there exists a constant $c$ independent of $\\Delta$ such that $h_i(\\varepsilon) \\leq 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta\/\\varepsilon)$.\n\\end{lemma}\n\n\\begin{proof}\nBy prove the statement by induction on $i$. For a large enough constant $c$, it holds that $h_1(\\varepsilon) \\leq 2^{2^{3}} \\cdot(c\\log(\\Delta)) \\log(\\Delta\/\\varepsilon)$. Now, consider some arbitrary $i$ and assume that $h_i(\\varepsilon) \\leq 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta\/\\varepsilon)$ for some large enough constant $c$. We show that this implies $h_{i+1}(\\varepsilon) \\leq 2^{2^{(i+1)+2}} \\cdot(c\\log(\\Delta))^{i+1} \\log(\\Delta\/\\varepsilon)$.\nBy the sequential composition lemma (\\cref{lem:sequential_composition}) and assuming that $c$ is large enough, we have\n\n\\begin{align*}\n h_{i+1}(\\varepsilon) &\\leq h_{i}((\\varepsilon\/2)\/\\Delta^{g_{i+1}(\\varepsilon\/2) + 1}) + g_{i+1}(\\varepsilon\/2) \\\\\n &\\leq 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta \\cdot \\Delta^{g_{i+1}(\\varepsilon\/2) + 1}\/(\\varepsilon\/2)) + g_{i+1}(\\varepsilon\/2) \\\\\n &\\leq 2 \\cdot 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta \\cdot \\Delta^{g_{i+1}(\\varepsilon\/2) + 1}\/(\\varepsilon\/2)) \\\\\n &\\leq 2 \\cdot 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta^{(c\/10) \\cdot 2^{2^{i+2}}\\log(\\Delta\/\\varepsilon)}\/(\\varepsilon\/2)) \\\\\n &\\leq 4 \\cdot 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta^{(c\/10) \\cdot 2^{2^{i+2}}\\log(\\Delta\/\\varepsilon)}) \\\\\n &\\leq (4\/10) \\cdot 2^{2^{i+2}} \\cdot 2^{2^{i+2}} (c \\log(\\Delta))^{i+1} \\log(\\Delta\/\\varepsilon) \\\\\n &\\leq 2^{2^{(i+1)+2}} \\cdot(c\\log(\\Delta))^{i+1} \\log(\\Delta\/\\varepsilon),\n\\end{align*}\n\nas desired.\n\n\\end{proof}\n\n\n \n \n\n\\begin{lemma}\\label{lm:tail0}\nLet $u$ be an arbitrary vertex. For each $\\varepsilon \\in (0,0.01]$, let $f(\\varepsilon)$ denote the smallest $i \\in \\mathbb{N}$ such that $u$ is settled after the $i$-th round with probability at least $1 - \\varepsilon$. There exists a fixed $c \\in \\mathbb{R}$ independent of $\\Delta$ such that $f(\\varepsilon) \\leq \\lceil 1 + \\log \\log \\frac{1}{c} \\log_\\Delta(1\/\\varepsilon)\\rceil$.\n\\end{lemma}\n\n\\begin{proof}\n\nLet $i \\in \\mathbb{N}$ be arbitrary. We show that a given vertex is settled after the $i$-th round with probability at least $1 - O(1\/\\Delta^{\\Omega(d_i\/d_{i-1})})$. For the sake of analysis, we run the algorithm on a finite $\\Delta$-regular high-girth graph instead of an infinite $\\Delta$-regular tree. For now, we additionally assume that no vertex realizes that we don't run the algorithm on an infinite $\\Delta$-regular tree. That is, we assume that for each vertex the coding radius to compute all its local information after the $i$-th round is much smaller than the girth of the graph. \n\nWith this assumption, we give a deterministic upper bound on the fraction of vertices that are not settled after the $i$-th round. On the one hand, the number of vertices that are not settled after the $i$-th round can be upper bounded by $|H_i| \\cdot \\Delta \\cdot O(d_i)$. On the other hand, we will show that the fraction of vertices that are contained in $H_i$ is upper bounded by $1\/\\Delta^{\\Omega(d_i\/d_{i-1})}$. We do this by showing that $C(v)$ contains $\\Delta^{\\Omega(d_i\/d_{i-1})}$ many vertices for a given $v \\in H_i$. Combining these two bounds directly implies that the fraction of unsettled vertices is smaller than \n\n\\[\\frac{\\Delta \\cdot O(d_i)}{\\Delta^{\\Omega(d_i\/d_{i-1})}} = 1\/\\Delta^{\\Omega(d_i\/d_{i-1})}.\\]\n\n\nLet $v \\in H_i$ be arbitrary. We show that $|C(v)| = \\Delta^{\\Omega(d_i\/d_{i-1})}$. Using Invariants (3) and (4) together with a simple induction argument, one can show that there are at least $(\\Delta-1)^{\\lfloor D \/ ((2\\sum_{j=0}^{i-1}d_{j}) + 1)\\rfloor}$ vertices contained in $H_{i-1}$ and whose distance to $v$ is at most $D$ in the graph induced by the vertices in $U_{i-1}$. Furthermore, from the way we defined the clustering $C(v)$ and the fact that two vertices in $H_i$ have a distance of at least $d_i$ in the graph $T[U_{i-1}]$, it follows that all vertices in $U_{i-1}$ having a distance of at most $d_i\/2 -1$ to $v$ are contained in the cluster $C(v)$. Hence, the total number of vertices in $C(v)$ is at least $(\\Delta-1)^{\\lfloor (d_i\/2 - 1) \/ ((2\\sum_{j=0}^{i-1}d_{j}) + 1)\\rfloor} = \\Delta^{\\Omega(d_i\/d_{i-1})}$, as promised.\n\nNow we remove the assumption that no vertex realizes that we don't run the algorithm on an infinite $\\Delta$-regular tree. If a vertex does realize that we don't run the algorithm on an infinite $\\Delta$-regular tree, then we consider the vertex as being unsettled after the $i$-th round. By considering graphs with increasing girth, we can make the expected fraction of vertices that realize that we are not on an infinite $\\Delta$-regular tree arbitrarily small. Combining this observation with the previous discussion, this implies that the expected fraction of vertices that are not settled after the $i$-th round is at most $1\/\\Delta^{\\Omega(d_i\/d_{i-1})}$. By symmetry, each vertex has the same probability of being settled after the $i$-th round. Hence, the probability that a given vertex is settled after the $i$-th round when run on a graph with sufficiently large girth is $1 - 1\/\\Delta^{\\Omega(d_i\/d_{i-1})}$, and the same holds for each vertex when we run the algorithm on an infinite $\\Delta$-regular tree. Thus, there exists a constant $c$ such that the probability that a given vertex is unsettled after the $i$-th round is at most $1\/\\Delta^{c \\cdot d_i\/d_{i-1}} = 1\/\\Delta^{c \\cdot 2^{2^{i-1}}}$. Setting $i = \\lceil 1 + \\log \\log \\frac{1}{c} \\log_\\Delta(1\/\\varepsilon)\\rceil$ finishes the proof.\n\\end{proof}\n\n\n\n\n\nWe are now finally ready to finish the proof of \\cref{thm:one_ended_forest}. Let $u$ be an arbitrary vertex and $\\varepsilon \\in (0,0.01]$. We need to compute an upper bound on the coding radius that is necessary for $u$ to know all the vertices in $\\mathcal{T}^{\\leftarrow}(u)$ with probability at least $1-\\varepsilon$. To compute such an upper bound for the required coding radius it suffices to find an $i^*$ and an $R^*$ such that the following holds. First, $u$ is settled after $i^*$ rounds with probability at least $1 - \\varepsilon\/2$. Second, $u$ knows for each edge in its $O(d_{i^*})$-hop neighborhood whether it is oriented in the partial orientation $O_{i^*}$, and if yes, in which direction, by only considering its $R^*$-hop neighborhood with probability at least $1 - \\varepsilon\/2$. By a union bound, both of these events occur with probability at least $1-\\varepsilon$. Moreover, if both events occur $u$ knows all the vertices in the set $\\mathcal{T}^{\\leftarrow}(u)$. The reason is as follows. If $u$ is settled after round $i^*$, then it follows from the previous analysis that only vertices in the $O(d_{i^*})$-hop neighborhood of $u$ can be contained in $\\mathcal{T}^{\\leftarrow}(u)$. Moreover, for each vertex in its $O(d_{i^*})$-hop neighborhood, $u$ can determine if the vertex is contained in $\\mathcal{T}^{\\leftarrow}(u)$ if it knows all the edge orientations on the unique path between itself and that vertex after the $i^*$-th round. Hence, it remains to find concrete values for $i^*$ and $R^*$. According to \\cref{lm:tail0}, we can choose $i^* = \\lceil 1 + \\log \\log \\frac{1}{c} \\log_\\Delta (2\/\\varepsilon)\\rceil$ for some large enough constant $c$. Moreover, it follows from a union bound that all vertices in the $O(d_{i^*})$-hop neighborhood around $u$ know with probability at least $1-\\varepsilon\/2$ the orientation of all its incident edges according to $O_{i^*}$ by only considering their $h_{i^*}((\\varepsilon\/2)\/\\Delta^{O(d_{i^*})})$-hop neighborhood. Hence, we can set \n\n\\begin{align*}\nR^* &= O(d_{i^*}) + h_{i^*}((\\varepsilon\/2)\/\\Delta^{O(d_{i^*})}) \\\\\n &\\leq O(d_{i^*}) + 2^{2^{i^* + 2}} \\cdot (c \\log(\\Delta))^{i^*}\\log(\\Delta^{O(d_{i^*})}\/\\varepsilon) \\\\\n &= O(2^{2^{i^*}}\\cdot 2^{2^{i^* + 2}} (c \\log(\\Delta))^{i^*}\\log(\\Delta\/\\varepsilon)) \\\\\n &= O(2^{2^{i^* + 3}} (c \\log(\\Delta))^{i^*}\\log(\\Delta\/\\varepsilon)) \\\\\n &= O(2^{2^{\\log \\log \\frac{1}{c} \\log (2\/\\varepsilon) + 5}} (c \\log(\\Delta))^{\\log \\log \\frac{1}{c} \\log (2\/\\varepsilon) + 2}\\log(\\Delta\/\\varepsilon)) \\\\\n &= \\log(1\/\\varepsilon)^{32} \\cdot \\log(\\Delta\/\\varepsilon) \\cdot \\log\\log(1\/\\varepsilon)^{O(\\log\\log(\\Delta))}.\n\\end{align*}\n\nAs $\\Delta = O(1)$, it therefore holds that\n\n\\[P(R(v)) = \\operatorname{poly}(\\log(1\/\\varepsilon)) \\geq 1 - \\varepsilon, \\]\n\nas desired.\n\n\n\n\n\n\n\\section{$\\mathsf{BAIRE} = \\mathsf{LOCAL}(O(\\log n))$}\n\\label{sec:baire}\n\nIn this section, we show that on $\\Delta$-regular trees the classes $\\mathsf{BAIRE}$ and $\\mathsf{LOCAL}(O(\\log(n)))$ are the same.\nAt first glance, this result looks rather counter-intuitive.\nThis is because in finite $\\Delta$-regular trees every vertex can see a leaf of distance $O(\\log(n))$, while there are no leaves at all in an infinite $\\Delta$-regular tree.\nHowever, there is an intuitive reasons why these classes are the same: in both setups there is a technique to decompose an input graph into a hierarchy of subsets. Furthermore, the existence of a solution that is defined inductively with respect to these decompositions can be characterized by the same combinatorial condition of Bernshteyn \\cite{Bernshteyn_work_in_progress}.\nWe start with a high-level overview of the decomposition techniques used in both contexts.\n\n\n\\paragraph{{\\sf Rake}\\ and {\\sf Compress}}\nThe hierarchical decomposition in the context of distributed computing is based on a variant of a decomposition algorithm of\nMiller and Reif~\\cite{MillerR89}.\nTheir original decomposition algorithm works as follows.\nStart with a tree $T$, and repeatedly apply the following two operations alternately: {\\sf Rake}\\ (remove all degree-1 vertices) and {\\sf Compress}\\ (remove all degree-2 vertices).\nThen $O(\\log n)$ iterations suffice to remove all vertices in $T$~\\cite{MillerR89}. To view it another way, this produces a decomposition of the vertex set $V$ into $2L - 1$ layers \\[V = \\VR{1} \\cup \\VC{1} \\cup \\VR{2} \\cup \\VC{2} \\cup \\VR{3} \\cup \\VC{3} \\cup \\cdots \\cup \\VR{L},\\] \nwith $L= O(\\log n)$, where $\\VR{i}$ is the set of vertices removed during the $i$-th {\\sf Rake}\\ operation and $\\VC{i}$ is the set of vertices removed during the $i$-th {\\sf Compress}\\ operation. We will use a variant~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} of this decomposition in the proof of \\cref{thm:ellfull_to_logn}.\n\n\n\n\nVariants of this decomposition turned out to be useful in designing $\\mathsf{LOCAL}$ algorithms~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup,ChangHLPU20,chang2020n1k_speedups}. \nIn our context, we assume that the given LCL satisfies a certain combinatorial condition and then find a solution inductively, in the reversed order of the construction of the decomposition.\nNamely, in the {\\sf Rake}\\ step we want to be able to existentially extend the inductive partial solution to all relative degree $1$-vertices (each $v \\in \\VR{i}$ has degree at most 1 in the subgraph induced by $\\VR{i} \\cup \\cdots \\cup \\VR{L}$) and in the {\\sf Compress}\\ step we want to extend the inductive partial solution to paths with endpoints labeled from the induction (the vertices in $\\VC{i}$ form degree-2 paths in the subgraph induced by $\\VC{i} \\cup \\cdots \\cup \\VR{L}$).\n\n\n\\paragraph{$\\TOAST$}\nFinding a hierarchical decomposition in the context of descriptive combinatorics is tightly connected with the notion of \\emph{Borel hyperfiniteness}.\nUnderstanding what Borel graphs are Borel hyperfinite is a major theme in descriptive set theory \\cite{doughertyjacksonkechris,gaojackson,conley2020borel}.\nIt is known that grids, and generally polynomial growth graphs are hyperfinite, while, e.g., acyclic graphs are not in general hyperfinite \\cite{jackson2002countable}.\nA strengthening of hyperfiniteness that is of interest to us is called a \\emph{toast} \\cite{gao2015forcing,conleymillerbound}.\nA $q$-toast, where $q\\in \\mathbb{N}$, of a graph $G$ is a collection $\\mathcal{D}$ of fintie subsets of $G$ with the property that (i) every pair of vertices is covered by an element of $\\mathcal{D}$ and (ii) the boundaries of every $D\\not=E\\in \\mathcal{D}$ are at least $q$ apart.\nThe idea to use a toast structure to solve LCLs appears in \\cite{conleymillerbound} and has many applications since then \\cite{gao2015forcing,marksunger}.\nThis approach has been formalized in \\cite{grebik_rozhon2021toasts_and_tails}, where the authors introduce $\\TOAST$ algorithms.\nRoughly speaking, an LCL $\\Pi$ admits a $\\TOAST$ algorithm if there is $q\\in \\mathbb{N}$ and a partial extending function (the function is given a finite subset of a tree that is partially colored and outputs an extension of this coloring on the whole finite subset) that has the property that whenever it is applied inductively to a $q$-toast, then it produces a $\\Pi$-coloring.\nAn advantage of this approach is that once we know that a given Borel graph admits, e.g., a Borel toast structure and a given LCL $\\Pi$ admits a $\\TOAST$ algorithm, then we may conclude that $\\Pi$ is in the class $\\mathsf{BOREL}$.\nSimilarly for $\\mathsf{MEASURE}$, $\\mathsf{BAIRE}$ or $\\mathsf{OLOCAL}$, we refer the reader to \\cite{grebik_rozhon2021toasts_and_tails} for more details and results concerning grids.\n\nIn the case of trees there is no way of constructing a Borel toast in general, however, it is a result of Hjorth and Kechris \\cite{hjorth1996borel} that every Borel graph is hyperfinite on a comeager set for every compatible Polish topology.\nA direct consequence of \\cite[Lemma~3.1]{marks2016baire} together with a standard construction of toast via Voronoi cells gives the following strengthening to toast.\nWe include a sketch of the proof for completeness.\n\n\\begin{restatable}{proposition}{BaireToast}\n\\label{pr:BaireToast}\nLet $\\mathcal{G}$ be a Borel graph on a Polish space $(X,\\tau)$ with degree bounded by $\\Delta\\in \\mathbb{N}$.\nThen for every $q>0$ there is a Borel $\\mathcal{G}$-invariant $\\tau$-comeager set $C$ on which $\\mathcal{G}$ admits a Borel $q$-toast.\n\\end{restatable}\n\\begin{proof}[Proof sketch]\nLet $\\{A_n\\}_{n\\in \\mathbb{N}}$ be a sequence of MIS with parameter $f(n)$ as in \\cite[Lemma~3.1]{marks2016baire} for a sufficiently fast growing function $f(n)$, e.g., $f(n)=(2q)^{n^2}$.\nThen $C=\\bigcup_{n\\in \\mathbb{N}}A_n$ is a Borel $\\tau$-comeager set that is $\\mathcal{G}$-invariant.\nWe produce a toast structure in a standard way, e.g., see~\\cite[Appendix~A]{grebik_rozhon2021toasts_and_tails}.\n\nLet $B_n(x)$ denote the ball of radius $f(n)\/3$ and $R_n(x)$ the ball of radius $f(n)\/4$ around $x\\in A_n$.\nIteratively, define cells $\\mathcal{D}_n=\\{C_{n}(x)\\}_{x\\in A_n}$ as follows.\nSet $C_1(x)=R_1(x)$.\nSuppose that $\\mathcal{D}_n$ has been defined and set\n\\begin{itemize}\n\t\\item $H^{n+1}(x,n+1):=R_{n+1}(x)$ for every $x\\in A_{n+1}$,\n\t\\item if $1\\le i\\le n$ and $\\left\\{H^{n+1}(x,i+1)\\right\\}_{x\\in A_{k+1}}$ has been defined, then we put\n\t$$H^{n+1}(x,i)=\\bigcup \\left\\{B_i(y):H^{n+1}(x,i+1)\\cap B_i(y)\\not=\\emptyset\\right\\}$$\n\tfor every $x\\in A_{n+1}$,\n\t\\item set $C_{n+1}(x):=H^{n+1}(x,1)$ for every $x\\in A_{n+1}$, this defines $\\mathcal{D}_{n+1}$.\n\\end{itemize}\nThe fact that $\\mathcal{D}=\\bigcup_{n\\in\\mathbb{N}} \\mathcal{D}_n$ is a $q$-toast on $C$ follows from the fact that $C=\\bigcup_{n\\in \\mathbb{N}}A_n$ together with the fact that the boundaries are $q$ separated which can be shown as in \\cite[Appendix~A]{grebik_rozhon2021toasts_and_tails}.\n\\end{proof}\n\n\n\nTherefore to understand LCLs in the class $\\mathsf{BAIRE}$ we need understand what LCLs on trees admit $\\TOAST$ algorithm.\nIt turns out that these notions are equivalent, again by using the combinatorial characterization of Bernshteyn \\cite{Bernshteyn_work_in_progress} that we now discuss.\n\n\n\\paragraph{Combinatorial Condition -- $\\ell$-full set}\nIn both decompositions, described above, we need to extend a partial coloring along paths that have their endpoints colored from the inductive step.\nThe precise formulation of the combinatorial condition that captures this demand was extracted by Bernshteyn \\cite{Bernshteyn_work_in_progress}.\nHe proved that it characterizes the class $\\mathsf{BAIRE}$ for Cayley graphs of virtually free groups.\nNote that this class contains, e.g., $\\Delta$-regular trees with a proper edge $\\Delta$-coloring.\n\n\\begin{definition}[Combinatorial condition -- an $\\ell$-full set]\n\\label{def:ellfull}\nLet $\\Pi = (\\Sigma, \\mathcal{V}, \\mathcal{E})$ be an LCL and $\\ell\\geq 2$.\nA set $\\mathcal{V}' \\subseteq \\mathcal{V}$ is \\emph{$\\ell$-full} whenever the following is satisfied.\nTake a path with at least $\\ell$ vertices, and add half-edges to it so that each vertex has degree $\\Delta$.\nTake any $c_1, c_2 \\in \\mathcal{V}'$ and label arbitrarily the half-edges around the endpoints with $c_1$ and $c_2$, respectively.\nThen there is a way to label the half-edges around the remaining $\\ell-2$ vertices with configurations from $\\mathcal{V}'$ such that all the $\\ell-1$ edges on the path have valid edge configuration on them. \n\\end{definition}\n\nNow we are ready to formulate the result that combines Bernshteyn's result \\cite{Bernshteyn_work_in_progress} (equivalence between (1.) and (2.), and the moreover part) with the main results of this section.\nThis also shows the remaining implications in \\cref{fig:big_picture_trees}.\n\n\\begin{theorem}\\label{thm:MainBaireLog}\nLet $\\Pi$ be an LCL on regular trees. \nThen the following are equivalent:\n\\begin{enumerate}\n \\item $\\Pi\\in \\mathsf{BAIRE}$,\n \\item $\\Pi$ admits a $\\TOAST$ algorithm,\n \\item $\\Pi$ admits an $\\ell$-full set,\n \\item $\\Pi\\in \\mathsf{LOCAL}(O(\\log(n)))$.\n\\end{enumerate}\nMoreover, any of the equivalent conditions is necessary for $\\Pi\\in \\mathsf{fiid}$.\n\\end{theorem}\n\nNext we discuss the proof of \\cref{thm:MainBaireLog}.\nWe refer the reader to Bernshteyn's paper \\cite{Bernshteyn_work_in_progress} for full proofs in the case of $\\mathsf{BAIRE}$ and $\\mathsf{fiid}$, here we only sketch the argument for completeness.\nWe also note that instead of using the toast construction, he used a path decomposition of acyclic graphs of Conley, Marks and Unger \\cite{conley2020measurable}.\n\n\n\\subsection{Sufficiency}\n\nWe start by showing that the combinatorial condition is sufficient for $\\mathsf{BAIRE}$ and $\\mathsf{LOCAL}(O(\\log(n)))$.\nNamely, it follows from the next results together with \\cref{pr:BaireToast} that (2.) implies all the other conditions in \\cref{thm:MainBaireLog}.\nAs discussed above the main idea is to color inductively along the decompositions.\n\n\\begin{proposition}\\label{pr:toastable}\nLet $\\Pi=(\\Sigma, \\mathcal{V}, \\mathcal{E})$ be an LCL that admits $\\ell$-full set $\\mathcal{V}'\\subseteq \\mathcal{V}$ for some $\\ell>0$.\nThen $\\Pi$ admits a $\\TOAST$ algorithm that produces a $\\Pi$-coloring for every $(2\\ell+2)$-toast $\\mathcal{D}$.\n\\end{proposition}\n\\begin{proof}[Proof sketch]\nOur aim is to build a partial extending function.\nSet $q:=2\\ell+2$.\nLet $E$ be a piece in a $q$-toast $\\mathcal{D}$ and suppose that $D_1,\\dots,D_k\\in \\mathcal{D}$ are subsets of $E$ such that the boundaries are separated.\nSuppose, moreover, that we have defined inductively a coloring of half-edges of vertices in $D=\\bigcup D_i$ using only vertex configurations from $\\mathcal{V}'$ such that every edge configuration $\\mathcal{E}$ is satisfied for every edge in $D$.\n\nWe handle each connected component of $E\\setminus D$ separately.\nLet $A$ be one of them.\nLet $u\\in A$ be a boundary vertex of $E$.\nSuch an vertex exists since every vertex in $E$ has degree $\\Delta$.\nThe distance of $u$ and any $D_i$ is at least $2\\ell+2$ for every $i\\in [k]$.\nWe orient all the edges from $A$ towards $u$.\nMoreover if $v_i\\in A$ is a boundary vertex of some $D_i$ we assign to $v_i$ a path $V_i$ of length $\\ell$ towards $u$.\nNote that $V_i$ and $V_j$ have distance at least $1$, in particular, are disjoint for $i\\not=j\\in [k]$ .\nNow, until you encounter some path $V_i$, color any in manner half-edges of vertices in $A$ inductively starting at $u$ in such a way that edge configurations $\\mathcal{E}$ are satisfied on every edge and only vertex configurations from $\\mathcal{V}'$ are used.\nUse the definition of $\\ell$-full set to find a coloring of any such $V_i$ and continue in a similar manner until the whole $A$ is colored.\n\\end{proof}\n\n\n\n\n\\begin{proposition}[$\\ell$-full $\\Rightarrow$ $\\mathsf{LOCAL}(O(\\log n))$]\\label{thm:ellfull_to_logn}\nLet $\\Pi = (\\Sigma, \\mathcal{V}, \\mathcal{E})$ be an LCL with an $\\ell$-full set $\\mathcal{V}' \\subseteq \\mathcal{V}$. Then $\\Pi$ can be solved in $O(\\log n)$ rounds in $\\mathsf{LOCAL}$.\n\\end{proposition}\n\\begin{proof}\nThe proof uses a variant of the rake-and-compress decomposition considered in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}.\n\n\\paragraph{The Decomposition}\nThe decomposition is parameterized an integer $\\ell' \\geq 1$, and it decomposes the vertices of $T$ into $2L - 1$ layers \\[V = \\VR{1} \\cup \\VC{1} \\cup \\VR{2} \\cup \\VC{2} \\cup \\VR{3} \\cup \\VC{3} \\cup \\cdots \\cup \\VR{L},\\] \nwith $L= O(\\log n)$. We write $\\GC{i}$ to denote the subtree induced by the vertices $\\left(\\bigcup_{j=i+1}^{L} \\VR{j}\\right) \\cup \\left( \\bigcup_{j=i}^{L-1} \\VC{j}\\right)$. Similarly, $\\GR{i}$ is the subtree induced by the vertices \n$\\left(\\bigcup_{j=i}^L \\VR{j}\\right) \\cup \\left( \\bigcup_{j=i}^{L-1} \\VC{j}\\right)$.\nThe sets $\\VR{i}$ and $\\VC{i}$ are required to satisfy the following requirements. \n\\begin{itemize}\n \\item Each $v \\in \\VR{i}$ has degree at most one in the graph $\\GR{i}$.\n \\item Each $v \\in \\VC{i}$ has degree exactly two in the graph $\\GC{i}$. Moreover, the $\\VC{i}$-vertices in $\\GC{i}$ form paths with $s$ vertices, with $\\ell' \\leq s \\leq 2 \\ell'$.\n\\end{itemize}\nFor any given constant $\\ell' \\geq 1$, it was shown in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} that such a decomposition of a tree $T$ can be computed in $O(\\log n)$ rounds. See \\cref{fig:rake_and_compress} for an example of such a decomposition with $\\ell' = 4$.\n\n\\paragraph{The Algorithm}\nGiven such a decomposition with $\\ell' = \\max\\{1, \\ell - 2\\}$, $\\Pi$ can be solved in $O(\\log n)$ rounds by labeling the vertices in this order: $\\VR{L}$, $\\VC{L-1}$, $\\VR{L-1}$, $\\ldots$, $\\VR{1}$, as follows. The algorithm only uses the vertex configurations in the $\\ell$-full set $\\mathcal{V}'$.\n\n\\paragraph{Labeling $\\VR{i}$}\nSuppose all vertices in $\\VR{L}, \\VC{L-1}, \\VR{L-1}, \\ldots, \\VC{i}$ have been labeled using $\\mathcal{V}'$. Recall that each $v \\in \\VR{i}$ has degree at most one in the graph $\\GR{i}$.\nIf $v \\in \\VR{i}$ has no neighbor in $\\VR{L} \\cup \\VC{L-1} \\cup \\VR{L-1} \\cup \\cdots \\cup \\VC{i}$, then we can label the half edges surrounding $v$ by any $c \\in \\mathcal{V}'$.\nOtherwise, $v \\in \\VR{i}$ has exactly one neighbor $u$ in $\\VR{L} \\cup \\VC{L-1} \\cup \\VR{L-1} \\cup \\cdots \\cup \\VC{i}$. Suppose the vertex configuration of $u$ is $c$, where the half-edge label on $\\{u,v\\}$ is $\\mathtt{a} \\in c$. A simple observation from the definition of $\\ell$-full sets is that for any $c \\in \\mathcal{V}'$ and any $\\mathtt{a} \\in c$, there exist $c' \\in \\mathcal{V}'$ and $\\mathtt{a}' \\in c'$ in such a way that $\\{\\mathtt{a}, \\mathtt{a}'\\} \\in \\mathcal{E}$.\nHence we can label the half edges surrounding $v$ by $c' \\in \\mathcal{V}'$ where the half-edge label on $\\{u,v\\}$ is $\\mathtt{a}' \\in c'$. \n\n\n\\paragraph{Labeling $\\VC{i}$} Suppose all vertices in $\\VR{L}, \\VC{L-1}, \\VR{L-1}, \\ldots, \\VR{i+1}$ have been labeled using $\\mathcal{V}'$. Recall that the $\\VC{i}$-vertices in $\\GC{i}$ form degree-2 paths $P=(v_1, v_2, \\ldots, v_s)$, with $\\ell' \\leq s \\leq 2 \\ell'$.\nLet $P'= (x, v_1, v_2, \\ldots, v_s, y)$ be the path resulting from appending to $P$ the neighbors of the two end-points of $P$ in $\\GC{i}$. The two vertices $x$ and $y$ are in $\\VR{L} \\cup \\VC{L-1} \\cup \\VR{L-1} \\cup \\cdots \\cup \\VR{i+1}$, so they have been assigned half-edge labels using $\\mathcal{V}'$. Since $P'$ contains at least $\\ell'+2 \\geq \\ell$ vertices, the definition of $\\ell$-full sets ensures that we can label $v_1, v_2, \\ldots, v_s$ using vertex configurations in $\\mathcal{V}'$ in such a way that the half-edge labels on $\\{x, v_1\\}, \\{v_1, v_2\\}, \\ldots, \\{v_s, y\\}$ are all in $\\mathcal{E}$.\n\\end{proof}\n\n\n\n\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width= .8\\textwidth]{fig\/y1.pdf}\n \\caption{The variant of the rake-and-compress decomposition used in the proof of \\cref{thm:ellfull_to_logn}. }\n \\label{fig:rake_and_compress}\n\\end{figure}\n\n\n\n\\subsection{Necessity}\n\nWe start by sketching that (2.) in \\cref{thm:MainBaireLog} is necessary for $\\mathsf{BAIRE}$ and $\\mathsf{fiid}$.\n\n\n\\begin{theorem}[Bernshteyn \\cite{Bernshteyn_work_in_progress}]\nLet $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ be an LCL and suppose that $\\Pi\\in \\mathsf{BAIRE}$ or $\\Pi\\in \\mathsf{fiid}$.\nThen $\\Pi$ admits an $\\ell$-full set $\\mathcal{V}'\\subseteq\\mathcal{V}$ for some $\\ell>0$.\n\\end{theorem}\n\\begin{proof}[Proof Sketch]\n\n\nWe start with $\\mathsf{BAIRE}$.\nSuppose that every Borel acyclic $\\Delta$-regular graph admits a Borel solution on a $\\tau$-comeager set for every compatible Polish topology $\\tau$.\nIn particular, this holds for the Borel graph induced by the standard generators of the free product of $\\Delta$-copies of $\\mathbb{Z}_2$ on the free part of the shift action on the alphabet $\\{0,1\\}$ endowed with the product topology.\nLet $F$ be such a solution.\nWrite $\\mathcal{V}'\\subseteq \\mathcal{V}$ for the configurations of half-edge labels around vertices that $F$ outputs on a non-meager set.\nLet $C$ be a comeager set on which $F$ is continuous. Then, every element of $\\mathcal{V}'$ is encoded by some finite window in the shift on $C$, that is, for each element there are a $k \\in \\mathbb{N}$ and function $s:B(1,k) \\to \\{0,1\\}$ such that $F$ is constant on the set $N_s \\cap C$ (where $N_s$ is the basic open neighbourhood determined by $s$, and $B(1,k)$ is the $k$-neighbourhood of the identity in the Cayley graph of the group).\nSince $\\mathcal{V}'$ is finite, we can take $t>0$ to be the maximum of such $k$'s.\nIt follows by standard arguments that $\\mathcal{V}'$ is $\\ell$-full for $\\ell>2t+1$.\n\nA similar argument works for the $\\mathsf{fiid}$, however, for the sake of brevity, we sketch a shorter argument that uses the fact that there must be a correlation decay for factors of iid's. Let $\\Pi\\in \\mathsf{fiid}$.\nThat is, there is an $\\operatorname{Aut}(T)$-equivariant measurable function from iid's on $T$ (without colored edges this time) into the space of $\\Pi$-colorings.\nLet $\\mathcal{V}'$ be the set of half-edges configurations around vertices that have non-zero probability to appear.\nLet $u,v\\in T$ be vertices of distance $k_0\\in \\mathbb{N}$.\nBy \\cite{BGHV} the correlation between the configurations around $u$ and $v$ tends to $0$ as $k_0\\to \\infty$.\nThis means that if the distance is big enough, then all possible pairs of $\\mathcal{V}'$ configurations need to appear.\n\\end{proof}\n\nTo finish the proof of \\cref{thm:MainBaireLog} we need to demonstrate the following theorem. Note that $\\mathsf{LOCAL}(n^{o(1)}) = \\mathsf{LOCAL}(O(\\log n))$ according to the $\\omega(\\log n)$ -- $n^{o(1)}$ complexity gap~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}.\n\n\\begin{restatable}{theorem}{logcomb}\\label{thm:logn_to_ellfull}\nLet $\\Pi = (\\Sigma, \\mathcal{V}, \\mathcal{E})$ be an LCL solvable in $\\mathsf{LOCAL}(n^{o(1)})$ rounds.\nThen there exists an $\\ell$-full set $\\mathcal{V}' \\subseteq \\mathcal{V}$ for some $\\ell \\geq 2$.\n\\end{restatable}\n\n\nThe rest of the section is devoted to the proof of \\cref{thm:logn_to_ellfull}. We start with the high-level idea of the proof.\nA natural attempt for showing $\\mathsf{LOCAL}(n^{o(1)})$ $\\Rightarrow$ $\\ell$-full is to simply take any $\\mathsf{LOCAL}(n^{o(1)})$ algorithm $\\mathcal{A}$ solving $\\Pi$, and then take $\\mathcal{V}'$ to be all vertex configurations that can possibly occur in an output of $\\mathcal{A}$. It is not hard to see that this approach does not work in general, because the algorithm might use a special strategy to label vertices with degree smaller than $\\Delta$. Specifically, there might be some vertex configuration $c$ used by $\\mathcal{A}$ so that some $\\mathtt{a} \\in c$ will only be used to label \\emph{virtual} half edges. It will be problematic to include $c$ in $\\mathcal{V}'$.\n\nTo cope with this issue, we do not deal with general bounded-degree trees. Instead, we construct recursively a sequence $(W^\\ast_1, W^\\ast_2, \\ldots, W^\\ast_L)$ of sets of rooted, layered, and \\emph{partially labeled} tree in a special manner. A tree $T$ is included in $W^\\ast_i$ if it can be constructed by gluing a multiset of rooted trees in $W^\\ast_{i-1}$ and a new root vertex $r$ in a certain fixed manner. A vertex is said to be in layer $i$ if it is introduced during the $i$-th step of the construction, i.e., it is introduced as the root $r$ during the construction of $W^\\ast_i$ from $W^\\ast_{i-1}$. All remaining vertices are said to be in layer 0.\n\n\nWe show that each $T \\in W^\\ast_L$ admits a correct labeling that extends the given partial labeling, as these partial labelings are computed by a simulation of $\\mathcal{A}$.\nMoreover, in these correct labelings, the variety of possible configurations of half-edge labels around vertices in different non-zero layers is the same for each layer.\nThis includes vertices of non-zero layer whose half-edges are labeled by the given partial labeling.\nWe simply pick $\\mathcal{V}'$ to be the set of all configurations of half-edge labels around vertices that can appear in a non-zero layer in a correct labeling of a tree $T \\in W^\\ast_L$.\nOur construction ensures that each $c \\in \\mathcal{V}'$ appears as the labeling of some degree-$\\Delta$ vertex in some tree that we consider.\n\n\nThe proof that $\\mathcal{V}'$ is an $\\ell$-full set is based on finding paths using vertices of non-zero layers connecting two vertices with any two vertex configurations in $\\mathcal{V}'$ in different lengths. These paths exist because the way rooted trees in $W^\\ast_{i-1}$ are glued together in the construction of $W^\\ast_{i}$ is sufficiently flexible.\nThe reason that we need $\\mathcal{A}$ to have complexity $\\mathsf{LOCAL}(n^{o(1)})$ is that the construction of the trees can be parameterized by a number $w$ so that all the trees have size polynomial in $w$ and the vertices needed to be assigned labeling are at least distance $w$ apart from each other. Since the number of rounds of $\\mathcal{A}$ executed on trees of size $w^{O(1)}$ is much less than $w$, each labeling assignment can be calculated locally and independently. \nThe construction of the trees as well as the analysis are based on a machinery developed in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}.\n Specifically, we will consider the equivalence relation $\\overset{\\star}{\\sim}$ defined in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} and prove some of its properties, including a pumping lemma for bipolar trees. The exact definition of $\\overset{\\star}{\\sim}$ in this paper is different from the one in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} because the mathematical formalism describing LCL problems in this paper is different from the one in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}.\nAfter that, we will consider a procedure for gluing trees parameterized by a labeling function $f$ similar to the one used in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}. \nWe will apply this procedure iteratively to generate a set of trees. We will show that the desired $\\ell$-full set $\\mathcal{V}' \\subseteq \\mathcal{V}$ can be constructed by considering the set of all possible correct labeling of these trees. \n \n\n\\paragraph{The Equivalence Relation $\\overset{\\star}{\\sim}$} \nWe consider trees with a list of designated vertices $v_1, v_2, \\ldots, v_k$ called \\emph{poles}. A \\emph{rooted tree} is a tree with one pole $r$, and a \\emph{bipolar tree} is a tree with two poles $s$ and $t$. \nFor a tree $T$ with its poles $S=(v_1, v_2, \\ldots, v_k)$ with $\\deg(v_i) = d_i < \\Delta$, we denote by $h_{(T,S)}$ the function that maps each choice of the \\emph{virtual} half-edge labeling surrounding the poles of $T$ to $\\textsf{YES}$ or $\\textsf{NO}$, indicating whether such a partial labeling can be completed into a correct complete labeling of $T$. \nMore specifically, consider \n\\[X=(\\mathcal{I}_1, \\mathcal{I}_2, \\ldots, \\mathcal{I}_k),\\]\nwhere $\\mathcal{I}_i$ is a size-$(\\Delta - d_i)$ multiset of labels in $\\Sigma$, for each $1 \\leq i \\leq k$. Then $h_{(T,S)}(X) = \\textsf{YES}$ if there is a correct labeling of $T$ such that the $\\Delta - d_i$ virtual half-edge labels surrounding $v_i$ are labeled by $\\mathcal{I}_i$, for each $1 \\leq i \\leq k$. This definition can be generalized to the case $T$ is already partially labeled in the sense that some of the half-edge labels have been fixed. In this case, $h_{(T,S)}(X) = \\textsf{NO}$ whenever $X$ is incompatible with the given partial labeling.\n\nLet $T$ be a tree with poles $S = (v_1, v_2, \\ldots, v_k)$ and let $T'$ be another tree with poles $S' = (v_1', v_2', \\ldots, v_k')$ such that $\\deg(v_i) = \\deg(v_i') = d_i$ for each $1 \\leq i \\leq k$.\nThen we write $T_1 \\overset{\\star}{\\sim} T_2$ if $h_{(T,S)} = h_{(T',S')}$.\n\nGiven an LCL problem $\\Pi$, it is clear that the number of equivalence classes of rooted trees and bipolar trees w.r.t.~$\\overset{\\star}{\\sim}$ is finite.\nFor a rooted tree $T$, denote by $\\textsf{Class}_1(T)$ the equivalence class of $T$. For a bipolar tree $H$, denote by $\\texttt{type}(H)$ the equivalence class of $H$.\n\n\n\n\n\\paragraph{Subtree Replacement}\nThe following lemma provides a sufficient condition that the equivalence class of a tree $T$ is invariant of the equivalence class of its subtree $T'$.\nWe note that a real half edge in $T$ might become virtual in its subtree $T'$.\nConsider a vertex $v$ in $T'$ and its neighbor $u$ that is in $T$ but not in $T'$. Then the half edge $(v, \\{u,v\\})$ is real in $T$ and virtual in $T'$.\n\n\n\\begin{lemma}[Replacing subtrees]\\label{lem:replace}\nLet $T$ be a tree with poles $S$.\nLet $T'$ be a connected subtree of $T$ induced by $U \\subseteq V$, where $V$ is the set of vertices in $T$.\nWe identify a list of designated vertices $S' = (v_1', v_2', \\ldots, v_k')$ in $U$ satisfying the following two conditions to be the poles of $T'$.\n\\begin{itemize}\n \\item $S \\cap U \\subseteq S'$.\n \\item Each edge $e=\\{u,v\\}$ connecting $u \\in U$ and $ v \\in V \\setminus U$ must satisfy $u \\in S'$.\n\\end{itemize}\nLet $T''$ be another tree with poles $S'' = (v_1'', v_2'', \\ldots, v_k'')$ that is in the same equivalence class as $T'$.\nLet $T^\\ast$ be the result of replacing $T'$ by $T''$ in $T$ by identifying $v_i' = v_i''$ for each $1 \\leq i \\leq k$.\nThen $T^\\ast$ is in the same equivalence class as $T$. \n\\end{lemma}\n\\begin{proof}\nTo prove the lemma, by symmetry, it suffices to show that starting from any correct labeling $\\mathcal{L}$ of $T$, it is possible to find a correct labeling $\\mathcal{L}^\\ast$ of $T^\\ast$ in such a way that the multiset of the\nvirtual half-edge labels surrounding each pole in $S$ remain the same.\n\nSuch a correct labeling $\\mathcal{L}^\\ast$ of $T^\\ast$ is constructed as follows. If $v \\in V \\setminus U$, then we simply adopt the given labeling $\\mathcal{L}$ of $v$ in $T$.\nNext, consider the vertices in $U$.\nSet $X=(\\mathcal{I}_1, \\mathcal{I}_2, \\ldots, \\mathcal{I}_k)$ to be the one compatible with the labeling $\\mathcal{L}$ of $T$ restricted to the subtree $T'$ in the sense that $\\mathcal{I}_i$ is the multiset of the virtual half-edge labels surrounding the pole $v_i'$ of $T'$, for each $1 \\leq i \\leq k$. We must have $h_{T',S'}(X) = \\textsf{YES}$.\nSince $T'$ and $T''$ are in the same equivalence class, we have $h_{T'',S''}(X) = \\textsf{YES}$, and so we can find a correct labeling $\\mathcal{L}''$ of $T''$ that is also compatible with $X$. Combining this labeling $\\mathcal{L}''$ of the vertices $U$ in $T''$ with the labeling $\\mathcal{L}$ of the vertices $V \\setminus U$ in $T$, we obtain a desired labeling $\\mathcal{L}^\\ast$ of $T^\\ast$.\n\nWe verify that $\\mathcal{L}^\\ast$ gives a correct labeling of $T^\\ast$. Clearly, the size-$\\Delta$ multiset that labels each vertex $v \\in V$ is in $\\mathcal{V}$, by the correctness of $\\mathcal{L}$ and $\\mathcal{L}''$. Consider any edge $e=\\{u,v\\}$ in $T^\\ast$. Similarly, if $\\{u,v\\} \\subseteq V \\setminus U$ or $\\{u,v\\} \\cap (V \\setminus U) = \\emptyset$, then the size-$2$ multiset that labels $e$ is in $\\mathcal{E}$, by the correctness of $\\mathcal{L}$ and $\\mathcal{L}''$. For the case that $e=\\{u,v\\}$ connects a vertex $u \\notin V$ and a vertex $v \\in V \\setminus U$, we must have $u \\in S'$ by the lemma statement. Therefore, the label of the half edge $(u,e)$ is the same in both $\\mathcal{L}$ and $\\mathcal{L}^\\ast$ by our choice of $\\mathcal{L}''$. Thus, the size-$2$ multiset that labels $e$ is in $\\mathcal{E}$, by the correctness of $\\mathcal{L}$.\n\nWe verify that the \nvirtual half-edge labels surrounding each pole in $S$ are the same in both $\\mathcal{L}$ and $\\mathcal{L}^\\ast$. Consider a pole $v \\in S$. If $v \\in V \\setminus U$, then the labeling of $v$ is clearly the same in both $\\mathcal{L}$ and $\\mathcal{L}^\\ast$.\nIf $v \\notin V\\setminus U$, then the condition $S \\cap U \\subseteq S'$ in the statement implies that $v \\in S'$. In this case, the way we pick $\\mathcal{L}''$ ensures that the \nvirtual half-edge labels surrounding $v \\in S'$ are the same in both $\\mathcal{L}$ and $\\mathcal{L}^\\ast$.\n\\end{proof}\n\nIn view of the proof of \\cref{lem:replace}, as long as the conditions in \\cref{lem:replace} are met, we are able to abstract out a subtree by its equivalence class when reasoning about correct labelings of a tree. This observation will be applied repeatedly in the subsequent discussion.\n\n\n\\paragraph{A Pumping Lemma} We will prove a pumping lemma of bipolar trees using \\cref{lem:replace}. \nSuppose $T_i$ is a tree with a root $r_i$ for each $1 \\leq i \\leq k$, then $H=(T_1, T_2, \\ldots, T_k)$ denotes the bipolar tree resulting from concatenating the roots $r_1, r_2, \\ldots, r_k$ into a path $(r_1, r_2, \\ldots, r_k)$ and setting the two poles of $H$ by $s = r_1$ and $t = r_k$.\nA simple consequence of \\cref{lem:replace} is that $\\texttt{type}(H)$ is determined by $\\textsf{Class}_1(T_1), \\textsf{Class}_1(T_2), \\ldots, \\textsf{Class}_1(T_k)$.\nWe have the following pumping lemma. \n\n\\begin{lemma}[Pumping lemma]\\label{lem:pump}\nThere exists a finite number $ {\\ell_{\\operatorname{pump}}} > 0$ such that as long as $k \\geq {\\ell_{\\operatorname{pump}}} $, any bipolar tree $H=(T_1, T_2, \\ldots, T_k)$ can be decomposed into $H = X \\circ Y \\circ Z$ with $0 < |Y| < k$ so that $X \\circ Y^i \\circ Z$ is in the same equivalence class as $H$ for each $i \\geq 0$. \n\\end{lemma}\n\\begin{proof}\nSet $ {\\ell_{\\operatorname{pump}}} $ to be the number of equivalence classes for bipolar trees plus one. By the pigeon hole principle, there exist $1 \\leq a < b \\leq k$ such that $(T_1, T_2, \\ldots, T_a)$ and $(T_1, T_2, \\ldots, T_b)$ are in the same equivalence class. Set $X = (T_1, T_2, \\ldots, T_a)$, $Y = (T_{a+1}, T_{a+2}, \\ldots, T_b)$, and $Z = (T_{b+1}, T_{b+2}, \\ldots, T_k)$. As we already know that $\\texttt{type}(X) = \\texttt{type}(X\\circ Y)$, \\cref{lem:replace} implies that \n$\\texttt{type}(X\\circ Y) = \\texttt{type}(X^2\\circ Y)$ by replacing $X$ by $X\\circ Y$ in the bipolar tree $X\\circ Y$.\nSimilarly, $\\texttt{type}(X \\circ Y^i)$ is the same for for each $i \\geq 0$. Applying \\cref{lem:replace} again to replace $X \\circ Y$ by $X \\circ Y^i$ in $H = X \\circ Y \\circ Z$, we conclude that $X \\circ Y^i \\circ Z$ is in the same equivalence class as $H$ for each $i \\geq 0$. \n\\end{proof}\n\n\n\n\\paragraph{A Procedure for Gluing Trees} Suppose that we have a set of rooted trees $W$. We devise a procedure that generates a new set of rooted trees by gluing the rooted trees in $W$ together. \nThis procedure is parameterized by a labeling function $f$. Consider a bipolar tree \\[H = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)\\] \nwhere $T^m$ is formed by attaching the roots $r_1, r_2, \\ldots, r_{\\Delta - 2}$ of the rooted trees $T_1^m, T_2^m, \\ldots, T_{\\Delta -2}^m$ to the root $r^m$ of $T^m$.\n\n\nThe labeling function $f$ assigns the half-edge labels surrounding $r^m$ based on \n\\[\\texttt{type}(H^l), \\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m), \\texttt{type}(H^r)\\]\nwhere $H^l = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l)$ and $H^r = (T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)$.\n\nWe write $T^{m}_\\ast$ to denote the result of applying $f$ to label the root $r^m$ of $T^{m}$ in the bipolar tree $H$, and we write \\[H_\\ast = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T_\\ast^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)\\] to denote the result of applying $f$ to $H$. We make the following observation.\n\n\\begin{lemma}[Property of $f$]\\label{lem:labeling_function}\nThe two equivalence classes\n$\\textsf{Class}_1(T_\\ast^m)$ and $\\texttt{type}(H_\\ast)$ are determined by \\[\\texttt{type}(H^l), \\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m), \\texttt{type}(H^r),\\] and the labeling function $f$.\n\\end{lemma}\n\\begin{proof}\nBy \\cref{lem:replace}, once the half-edge labelings of the root $r^m$ of $T^{m}$ is fixed, $\\textsf{Class}_1(T_\\ast^m)$ is determined by $\\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m)$. \nTherefore, indeed $\\textsf{Class}_1(T_\\ast^m)$ is determined by\n\\[\\texttt{type}(H^l), \\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m), \\texttt{type}(H^r),\\] and the labeling function $f$.\nSimilarly, applying \\cref{lem:replace} to the decomposition of $H_\\ast$ into $H^l$, $T_\\ast^m$, and $H^r$, we infer that $\\texttt{type}(H_\\ast)$ depends only on $\\texttt{type}(H^l)$, $\\textsf{Class}_1(T_\\ast^m)$, and $\\texttt{type}(H^r)$.\n\\end{proof}\n\nThe three sets of trees $X_f(W)$, $Y_f(W)$, and $Z_f(W)$ are constructed as follows.\n\n\\begin{itemize}\n \\item $X_f(W)$ is the set of all rooted trees resulting from appending $\\Delta-2$ arbitrary rooted trees $T_1, T_2, \\ldots, T_{\\Delta-2}$ in $W$ to a new root vertex $r$.\n \\item $Y_f(W)$ is the set of bipolar trees constructed as follows. For each choice of $2 {\\ell_{\\operatorname{pump}}} +1$ rooted trees $T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r$ from $X_f(W)$, concatenate them into a bipolar tree \n \\[H = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r),\\] let $H_\\ast$ be the result of applying the labeling function $f$ to $H$, and then add $H_\\ast$ to $Y_f(W)$.\n \\item $Z_f(W)$ is the set of rooted trees constructed as follows. For each \\[H_\\ast = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T_\\ast^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r) \\in Y_f(W),\\] add $(T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T_\\ast^m)$ to $Z_f(W)$, where we set the root of $T_1^l$ as the root, and add $(T_\\ast^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)$ to $Z_f(W)$, where we set the root of ${T_{ {\\ell_{\\operatorname{pump}}} }^r}$ as the root. \n\\end{itemize}\n\nWe write $\\textsf{Class}_1(S) = \\bigcup_{T \\in S} \\{\\textsf{Class}_1(T)\\}$ and $\\texttt{type}(S) = \\bigcup_{H \\in S} \\{\\texttt{type}(H)\\}$, and \nwe make the following observation.\n\n\\begin{lemma}[Property of $X_f(W)$, $Y_f(W)$, and $Z_f(W)$]\\label{lem:sets_of_trees}\nThe three sets of equivalence classes $\\textsf{Class}_1(X_f(W))$, $\\texttt{type}(Y_f(W))$, and $\\textsf{Class}_1(Z_f(W))$ depend only on $\\textsf{Class}_1(W)$ and the labeling function $f$.\n\\end{lemma}\n\\begin{proof}\nThis is a simple consequence of \\cref{lem:replace,lem:labeling_function}.\n\\end{proof}\n\n\\paragraph{A Fixed Point $W^\\ast$}\nGiven a fixed labeling function $f$,\nwe want to find a set of rooted trees $W^\\ast$ that is a \\emph{fixed point} for the procedure $Z_f$ in the sense that \\[\\textsf{Class}_1(Z_f(W^\\ast)) = \\textsf{Class}_1(W^\\ast).\\] \nTo find such a set $W^\\ast$, we construct a two-dimensional array of rooted trees $\\{W_{i,j}\\}$, as follows.\n\n\\begin{itemize}\n \\item For the base case, $W_{1,1}$ consists of only the one-vertex rooted tree.\n \\item Given that $W_{i,j}$ as been constructed, we define $W_{i,j+1} = Z_f(W_{i,j})$.\n \\item Given that $W_{i,j}$ for all positive integers $j$ have been constructed, $W_{i+1,1}$ is defined as follows. Pick $b_i$ as the smallest index such that $\\textsf{Class}_1(W_{i, b_i}) = \\textsf{Class}_1(W_{i, a_i})$ for some $1 \\leq a_i < b_i$. By the pigeon hole principle, the index $b_i$ exists, and it is upper bounded by $2^C$, where $C$ is the number of equivalence classes for rooted trees. We set \n \\[W_{i+1,1} = W_{i, a_i} \\cup W_{i, a_i + 1} \\cup \\cdots \\cup W_{i, b_i - 1}.\\]\n\\end{itemize}\n\nWe show that the sequence $\\textsf{Class}_1(W_{i,a_i}), \\textsf{Class}_1(W_{i,a_i+1}), \\textsf{Class}_1(W_{i,a_i+2}), \\ldots$ is periodic with a period $c_i = b_i - a_i$.\n\n\\begin{lemma}\\label{lem:W-aux-1}\nFor any $i \\geq 1$ and for any $j \\geq a_i$, we have $\\textsf{Class}_1(W_{i,j}) = \\textsf{Class}_1(W_{i,j+c_i})$, where $c_i = b_i - a_i$.\n\\end{lemma}\n\\begin{proof}\nBy \\cref{lem:sets_of_trees}, $\\textsf{Class}_1(W_{i,j})$ depends only on $\\textsf{Class}_1(W_{i,j-1})$. Hence the lemma follows from the fact that $\\textsf{Class}_1(W_{i, b_i}) = \\textsf{Class}_1(W_{i, a_i})$.\n\\end{proof}\n\nNext, we show that $\\textsf{Class}_1(W_{2,1}) \\subseteq \\textsf{Class}_1(W_{3, 1}) \\subseteq \\textsf{Class}_1(W_{4, 1}) \\subseteq \\cdots$.\n\n\\begin{lemma}\\label{lem:W-aux-2}\nFor any $i \\geq 2$, we have $\\textsf{Class}_1(W_{i,1}) \\subseteq \\textsf{Class}_1(W_{i,j})$ for each $j > 1$, and so $\\textsf{Class}_1(W_{i,1}) \\subseteq \\textsf{Class}_1(W_{i+1, 1})$.\n\\end{lemma}\n\\begin{proof}\nSince $W_{i,1} = \\bigcup_{a_{i-1} \\leq l \\leq b_{i-1} - 1} W_{i-1, l}$, we have \\[\\bigcup_{a_{i-1}+j-1 \\leq l \\leq b_{i-1} +j} W_{i-1, l} \\subseteq W_{i,j}\\] according to the procedure of constructing $W_{i,j}$. By \\cref{lem:W-aux-1}, \\[ \\textsf{Class}_1\\left(\\bigcup_{a_{i-1} \\leq l \\leq b_{i-1} - 1} W_{i-1, l}\\right) = \\textsf{Class}_1\\left(\\bigcup_{a_{i-1}+j-1 \\leq l \\leq b_{i-1} +j} W_{i-1, l}\\right)\\] for all $j \\geq 1$, and so we have $\\textsf{Class}_1(W_{i,1}) \\subseteq \\textsf{Class}_1(W_{i,j})$ for each $j > 1$. The claim $\\textsf{Class}_1(W_{i,1}) \\subseteq \\textsf{Class}_1(W_{i+1, 1})$ follows from the fact that $W_{i+1,1} = \\bigcup_{a_{i} \\leq l \\leq b_{i} - 1} W_{i, l}$.\n\\end{proof}\n\nSet $i^\\ast$ to be the smallest index $i \\geq 2$ such that $\\textsf{Class}_1(W_{i,1}) = \\textsf{Class}_1(W_{i+1, 1})$. By the pigeon hole principle and \\cref{lem:W-aux-2}, the index $i^\\ast$ exists, and it is upper bounded by $C$, the number of equivalence classes for rooted trees. We set \\[W^\\ast = W_{i^\\ast, 1}.\\] \n\nThe following lemma shows that $\\textsf{Class}_1(Z_f(W^\\ast)) = \\textsf{Class}_1(W^\\ast)$, as needed.\n\n\\begin{lemma}\\label{lem:W-aux-3}\nFor any $i \\geq 2$, if $\\textsf{Class}_1(W_{i,1}) = \\textsf{Class}_1(W_{i+1, 1})$, then $\\textsf{Class}_1(W_{i,j})$ is the same for all $j \\geq 1$. \n\\end{lemma}\n\\begin{proof}\nBy \\cref{lem:W-aux-1} and the way we construct $W_{i+1, 1}$, we have \\[\\textsf{Class}_1(W_{i, j}) \\subseteq \\textsf{Class}_1(W_{i+1, 1}) \\ \\ \\text{for each} \\ \\ j \\geq a_i.\\]\nBy \\cref{lem:W-aux-2}, we have \\[\\textsf{Class}_1(W_{i, 1}) \\subseteq \\textsf{Class}_1(W_{i, j}) \\ \\ \\text{for each} \\ \\ j > 1.\\]\nTherefore, $\\textsf{Class}_1(W_{i,1}) = \\textsf{Class}_1(W_{i+1, 1})$ implies that $\\textsf{Class}_1(W_{i, j}) = \\textsf{Class}_1(W_{i,1})$ for each $j \\geq a_i$. Hence we must have $\\textsf{Class}_1(Z_f(W_{i,1})) = \\textsf{Class}_1(W_{i,1})$, and so $\\textsf{Class}_1(W_{i,j})$ is the same for all $j \\geq 1$. \n\\end{proof}\n\nWe remark that simply selecting $W^\\ast$ to be any set such that $\\textsf{Class}_1(Z_f(W^\\ast)) = \\textsf{Class}_1(W^\\ast)$ is not enough for our purpose. As we will later see, it is crucial that the set $W^\\ast$ is constructed by iteratively applying the function $Z_f$ and taking the union of previously constructed sets.\n\n\n\n\n\n\n\n\\paragraph{A Sequence of Sets of Trees}\nWe define $W_1^\\ast = W^\\ast$ and $W_i^\\ast = Z_f(W_{i-1}^\\ast)$ for each $1 < i \\leq L$, where $L$ is some sufficiently large number to be determined.\n\nThe way we choose $W^\\ast$ guarantees that $\\textsf{Class}_1(W_i^\\ast) = \\textsf{Class}_1(W^\\ast)$ for all $1 \\leq i \\leq L$. \nFor convenience, we write $X_i^\\ast = X_f(W_i^\\ast)$ and $Y_i^\\ast = Y_f(W_i^\\ast)$.\nSimilarly, $\\textsf{Class}_1(X_i^\\ast)$ is the same for all $1 \\leq i \\leq L$ and $\\texttt{type}(Y_i^\\ast)$ is the same for all $1 \\leq i \\leq L$.\n\n\nOur analysis will rely on the assumption that all rooted trees in $W^\\ast$ admit correct labelings. Whether this is true depends only on $\\textsf{Class}_1(W^\\ast)$, which depends only on the labeling function $f$. We say that $f$ is \\emph{feasible} if it leads to a set $W^\\ast$ where all the rooted trees therein admit correct labelings. The proof that a feasible labeling function $f$ exists is deferred.\n\nThe assumption that $f$ is feasible implies that all trees in $W_i^\\ast$, $X_i^\\ast$, and $Y_i^\\ast$, for all $1 \\leq i \\leq L$, admit correct labelings. All rooted trees in $X_i^\\ast$ admit correct labelings because they are subtrees of the rooted trees in $W_{i+1}^\\ast$. A correct labeling of any bipolar tree $H \\in Y_i^\\ast$ can be obtained by combining any correct labelings of the two rooted trees in $W_{i+1}^\\ast$ resulting from $H$.\n\n\n\n\n\\paragraph{Layers of Vertices}\nWe assign a layer number $\\lambda(v)$ to each vertex $v$ in a tree based on the step that $v$ is introduced in the construction \\[W_1^\\ast \\rightarrow W_2^\\ast \\rightarrow \\cdots \\rightarrow W_L^\\ast.\\]\n\nIf a vertex $v$ is introduced as the root vertex of a tree in $X_i^\\ast = X_f(W_i^\\ast)$, then we say that the layer number of $v$ is $\\lambda(v) = i \\in \\{1,2, \\ldots, L\\}$. A vertex $v$ has $\\lambda(v) = 0$ if it belongs to a tree in $W_1^\\ast$.\n\n\nFor any vertex $v$ with $\\lambda(v) = i$ in a tree $T \\in X_j^\\ast$ with $i \\leq j$, we write $T_v$ to denote the subtree of $T$ such that $T_v \\in X_i^\\ast$ where $v$ is the root of $T_v$. \n\n\n \nWe construct a sequence of sets $R_1, R_2, \\ldots, R_L$ as follows.\nWe go over all rooted trees $T \\in X_L^\\ast$, all possible correct labeling $\\mathcal{L}$ of $T$, and all vertices $v$ in $T$ with $\\lambda(v) = i \\in \\{1,2, \\ldots, L\\}$. Suppose that the $\\Delta-2$ rooted trees in the construction of $T_v \\in X_i^\\ast = X_f(W_i^\\ast)$ are $T_1$, $T_2$, $\\ldots$, $T_{\\Delta-2}$, and let $r_i$ be the root of $T_i$. Consider the following parameters.\n\\begin{itemize}\n \\item $c_i = \\textsf{Class}_1(T_i)$.\n \\item $\\mathtt{a}_i$ is the real half-edge label of $v$ in $T_v$ for the edge $\\{v, r_i\\}$, under the correct labeling $\\mathcal{L}$ of $T$ restricted to $T_v$.\n \\item $\\mathcal{I}$ is the size-$2$ multiset of virtual half-edge labels of $v$ in $T_v$, under the correct labeling $\\mathcal{L}$ of $T$ restricted to $T_v$.\n\\end{itemize}\nThen we add \n$(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I})$ to $R_i$.\n\n\n\n\n\\begin{lemma}\\label{lem:Rsets1}\nFor each $1 \\leq i < L$, $R_i$ is determined by $R_{i+1}$.\n\\end{lemma}\n\\begin{proof}\nWe consider the following alternative way of constructing $R_i$ from $R_{i+1}$.\nEach rooted tree $T'$ in $W_{i+1}^\\ast = Z_f(W_i^\\ast)$ can be described as follows.\n\\begin{itemize}\n \\item Start with a path $(r_1, r_2, \\ldots, r_{ {\\ell_{\\operatorname{pump}}} + 1})$, where $r_1$ is the root of $T'$.\n \\item For each $1 \\leq j \\leq {\\ell_{\\operatorname{pump}}} +1$, append $\\Delta-2$ rooted trees $T_{j,1}, T_{j,2}, \\ldots, T_{j, \\Delta-2} \\in W_i^\\ast$ to $r_j$.\n \\item Assign the labels to the half edges surrounding $r_{ {\\ell_{\\operatorname{pump}}} + 1}$ according to the labeling function $f$.\n\\end{itemize}\n\n Now, consider the function $\\phi$ that maps each equivalence class $c$ for rooted trees to a subset of $\\Sigma$ defined as follows:\n$\\mathtt{a} \\in \\phi(c)$ if there exist \\[(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I}) \\in R_{i+1}\\] and $1 \\leq j \\leq \\Delta -2$ such that $c = c_j$ and $\\mathtt{a} = \\mathtt{a}_j$.\n\nWe go over all possible $T' \\in W_{i+1}^\\ast = Z_f(W_i^\\ast)$.\nNote that the root $r$ of $T'$ has exactly one virtual half edge. For each $\\mathtt{b} \\in \\Sigma$ such that $\\{\\mathtt{a}, \\mathtt{b}\\} \\in \\mathcal{E}$ for some $\\mathtt{a} \\in \\phi(\\textsf{Class}_1(T'))$, we go over all possible correct labelings $\\mathcal{L}$ of $T'$ where the virtual half edge of $r$ is labeled $\\mathtt{b}$. For each $1 \\leq j \\leq {\\ell_{\\operatorname{pump}}} +1$, consider the following parameters.\n\\begin{itemize}\n \\item $c_l = \\textsf{Class}_1(T_{j,l})$.\n \\item $\\mathtt{a}_l$ is the half-edge label of $r_j$ for the edge $\\{r_j, r_{j,l}\\}$.\n \\item $\\mathcal{I}$ is the size-$2$ multiset of the remaining two half-edge labels of $v$.\n\\end{itemize}\nThen we add \n$(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I})$ to $R_i$. \n\nThis construction of $R_i$ is equivalent to the original construction of $R_i$ because a correct labeling of $T' \\in W_i^\\ast$ can be extended to a correct labeling of a tree $T \\in X_L^\\ast$ that contains $T'$ as a subtree if and only if the virtual half-edge label of the root $r$ of $T'$ is $\\mathtt{b} \\in \\Sigma$ such that $\\{\\mathtt{a}, \\mathtt{b}\\} \\in \\mathcal{E}$ for some $\\mathtt{a} \\in \\phi(\\textsf{Class}_1(T'))$.\n\n\nIt is clear that this construction of $R_i$ only depends on $\\textsf{Class}_1(W_i^\\ast)$, the labeling function $f$, and the function $\\phi$, which depends only on $R_{i+1}$. Since the labeling function $f$ is fixed and $\\textsf{Class}_1(W_i^\\ast)$ is the same for all $i$, we conclude that $R_{i}$ depends only on $R_{i+1}$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:Rsets2}\nWe have $R_1 \\subseteq R_2 \\subseteq \\cdots \\subseteq R_L$.\n\\end{lemma}\n\\begin{proof}\nFor the base case, we show that $R_{L-1} \\subseteq R_L$. In fact, our proof will show that $R_{i} \\subseteq R_L$ for each $1 \\leq i < L$. Consider any \n\\[(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I}) \\in R_i.\\] \nThen there is a rooted tree $T \\in X_i^\\ast$ that is formed by attaching $\\Delta-2$ rooted trees of equivalence classes $c_1, c_2, \\ldots, c_{\\Delta-2}$ to the root vertex so that if we label the half edges surrounding the root vertex according to $\\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}$, and $\\mathcal{I}$, then this partial labeling can be completed into a correct labeling of $T$.\n\nBecause $\\textsf{Class}_1(W_i^\\ast) = \\textsf{Class}_1(W_L^\\ast)$, there is also a rooted tree $T' \\in X_L^\\ast$ that is formed by attaching $\\Delta-2$ rooted trees of equivalence classes $c_1, c_2, \\ldots, c_{\\Delta-2}$ to the root vertex. Therefore, if we label the root vertex of $T'$ in the same way as we do for $T$, then this partial labeling can also be completed into a correct labeling of $T'$. Hence we must have \n\\[(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I}) \\in R_L.\\] \n\n\nNow, suppose that we already have $R_i \\subseteq R_{i+1}$ for some $1 < i < L$. We will show that $R_{i-1} \\subseteq R_{i}$. \nDenote by $\\phi_{i}$ and $\\phi_{i+1}$ the function $\\phi$ in \\cref{lem:Rsets1} constructed from $R_{i}$ and $R_{i+1}$. \nWe have $\\phi_i(c) \\subseteq \\phi_{i+1}(c)$ for each equivalence class $c$, because $R_i \\subseteq R_{i+1}$. Therefore, in view of the alternative construction described in the proof of \\cref{lem:Rsets1}, we have $R_{i-1} \\subseteq R_{i}$.\n\\end{proof}\n\n\n\n\\paragraph{The Set of Vertex Configurations $\\mathcal{V}'$}\nBy \\cref{lem:Rsets1,lem:Rsets2}, if we pick $L$ to be sufficient large, we can have $R_1 = R_2 = R_3$.\nMore specifically, if we pick \n\\[L \\geq C^{\\Delta-2} \\cdot \\binom{|\\Sigma|+1}{|\\Sigma|-1} + 3,\\]\nthen there exists an index $3 \\leq i \\leq L$ such that $R_i = R_{i-1}$, implying that $R_1 = R_2 = R_3$. Here $C$ is the number of equivalence classes for rooted trees and $\\binom{|\\Sigma|+1}{|\\Sigma|-1}$ is the number of size-$2$ multisets of elements from $\\Sigma$.\n\n\n\nThe set $\\mathcal{V}'$ is defined by including all size-$\\Delta$ multisets $\\{\\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}\\} \\cup \\mathcal{I}$ such that \\[(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I}) \\in R_{1}\\] for some $c_1, c_2, \\ldots, c_{\\Delta-2}$.\n\n\n\n\\paragraph{The Set $\\mathcal{V}'$ is $\\ell$-full}\nTo show that the set $\\mathcal{V}'$ is $\\ell$-full, we consider the subset $\\mathcal{V}^\\ast \\subseteq \\mathcal{V}'$ defined by the set of vertex configurations used by the labeling function $f$ in the construction of $Y_f(W_i^\\ast)$. The definition of $\\mathcal{V}^\\ast$ is invariant of $i$ as $\\textsf{Class}_1(W_i^\\ast)$ is the same for all $i$. Clearly, for each $x \\in \\mathcal{V}^\\ast$, there is a rooted tree $T \\in W_2^\\ast$ where the root of a subtree $T' \\in X_1^\\ast$ of $T$ has its size-$\\Delta$ multiset of half-edge labels fixed to be $x$ by $f$, and so $\\mathcal{V}^\\ast \\subseteq \\mathcal{V}'$.\n\nFor notational simplicity, we write $x \\overset{k}{\\leftrightarrow} x'$ if there is a correct labeling of a $k$-vertex path $(v_1, v_2, \\ldots, v_k)$ using only vertex configurations in $\\mathcal{V}'$ so that the vertex configuration of $v_1$ is $x$ and the vertex configuration of $v_k$ is $x'$. If it is further required that the half-edge label of $v_1$ for the edge $\\{v_1, v_2\\}$ is $\\mathtt{a} \\in x$, then we write $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} x'$. The notation $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{a}')$ is defined similarly.\n\n\n\\begin{lemma}\\label{lem:find_path_1}\nFor any $x \\in \\mathcal{V}' \\setminus \\mathcal{V}^\\ast$ and $\\mathtt{a} \\in x$, there exist a vertex configuration $x' \\in \\mathcal{V}^\\ast$ and a number $2 \\leq k \\leq 2 {\\ell_{\\operatorname{pump}}} + 1$ such that $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} x'$.\n\\end{lemma}\n\\begin{proof}\nLet $T \\in W_L^\\ast$ be chosen so that there is a correct labeling $\\mathcal{L}$ where $x$ is a vertex configuration of some vertex $v$ with $\\lambda(v) = 2$.\n\nTo prove the lemma, it suffices to show that for each of the $\\Delta$ neighbors $u$ of $v$, it is possible to find a path $P = (v,u, \\ldots, w)$ meeting the following conditions.\n\\begin{itemize}\n \\item $w$ is a vertex whose half-edge labels have been fixed by $f$. This ensures that the vertex configuration of $w$ is in $\\mathcal{V}^\\ast$.\n \\item All vertices in $P$ are within layers 1,2, and 3. This ensures that the vertex configuration of all vertices in $P$ are in $\\mathcal{V}'$.\n \\item The number of vertices $k$ in $P$ satisfies $2 \\leq k \\leq 2 {\\ell_{\\operatorname{pump}}} + 1$. \n\\end{itemize}\n\nWe divide the proof into three cases. Refer to \\cref{fig:ell_full_construction1} for an illustration, where squares are vertices whose half-edge labels have been fixed by $f$.\n\n\\paragraph{Case 1} Consider the subtree $T_v \\in X_2^\\ast$ of $T$ whose root is $v$. In view of the construction of the set $X_f(W_2^\\ast)$, $v$ has $\\Delta-2$ children $u_1, u_2, \\ldots, u_{\\Delta-2}$ in $T_v$, where the subtree $T_i$ rooted at $u_i$ is a rooted tree in $W_2^\\ast$. \n\nFor each $1 \\leq i \\leq \\Delta-2$, according to the structure of the trees in the set $W_2^\\ast = Z_f(W_1^\\ast)$, there is a path $(u_i = w_1, w_2, \\ldots, w_{ {\\ell_{\\operatorname{pump}}} +1})$ in $T_i$ containing only layer-1 vertices, where the half-edge labels of $w_{ {\\ell_{\\operatorname{pump}}} +1}$ have been fixed by $f$. Hence $P = (v, u_i = w_1, w_2, \\ldots, w_{ {\\ell_{\\operatorname{pump}}} +1})$ is a desired path with $k = {\\ell_{\\operatorname{pump}}} + 2$ vertices.\n \n \n \n \n \n\\paragraph{Case 2} Consider the subtree $T' \\in W_3^\\ast = Z_f(W_2^\\ast)$ that contains $v$ in $T$.\nSimilarly, according to the structure of the trees in the set $W_3^\\ast = Z_f(W_2^\\ast)$, there is a path $(r = w_1', w_2', \\ldots, w_{ {\\ell_{\\operatorname{pump}}} +1}')$ in $T'$ containing only layer-2 vertices so that $r$ is the root of $T'$, $v = w_i'$ for some $1 \\leq i' \\leq {\\ell_{\\operatorname{pump}}} +1$, and the half-edge labels of $w_{ {\\ell_{\\operatorname{pump}}} +1}'$ have been fixed by $f$.\nSince $x \\in \\mathcal{V}' \\setminus \\mathcal{V}^\\ast$, we have $v \\neq w_{ {\\ell_{\\operatorname{pump}}} +1}'$. Hence $P = (v = w_i', w_{i+1}', \\ldots, w_{ {\\ell_{\\operatorname{pump}}} +1}')$ is a desired path with $2 \\leq k \\leq {\\ell_{\\operatorname{pump}}} + 1$ vertices.\n\n \n \n \n\n\\paragraph{Case 3} There is only one remaining neighbor of $v$ to consider. In view of the construction of $X_f(W_3^\\ast)$, there is a layer-3 vertex $v'$ adjacent to the vertex $r$, the root of the tree $T' \\in W_3^\\ast$ considered in the previous case. If the half-edge labels of $v'$ have been fixed by $f$, then $P = (v = w_i', w_{i-1}', \\ldots, w_{1}'=r, v')$ is a desired path.\nOtherwise, similar to the analysis in the previous case, we can find a path $P' = (v', \\ldots, w)$ connecting $v'$ to a vertex $w$ whose half-edge labels have been fixed by $f$. All vertices in $P'$ are of layer-3, and the number of vertices in $P'$ is within $[2, {\\ell_{\\operatorname{pump}}} + 1]$. \nCombining $P'$ with the path $(v = w_i', w_{i-1}', \\ldots, w_{1}'=r, v')$, we obtain the desired path $P$ whose number of vertices satisfies $2 \\leq k \\leq 2 {\\ell_{\\operatorname{pump}}} + 1$.\n \n \n\\end{proof}\n\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width= .75\\textwidth]{fig\/y2.pdf}\n \\caption{An illustration for the proof of \\cref{lem:find_path_1}.}\n \\label{fig:ell_full_construction1}\n\\end{figure}\n\nFor \\cref{lem:find_path_2}, note that if $\\mathtt{a}$ appears more than once in the multiset $x$, then we still have $\\mathtt{a} \\in x \\setminus \\{\\mathtt{a}\\}$.\n\n\n\\begin{lemma}\\label{lem:find_path_2}\nFor any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$, $\\mathtt{a}' \\in x' \\in \\mathcal{V}^\\ast$, and $0 \\leq t \\leq {\\ell_{\\operatorname{pump}}} - 1$, we have $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ for some $\\mathtt{b} \\in x \\setminus \\{\\mathtt{a}\\}$ and $\\mathtt{b}' \\in x' \\setminus \\{\\mathtt{a}'\\}$\n with $k = 2 {\\ell_{\\operatorname{pump}}} + 4 + t$.\n\\end{lemma}\n\\begin{proof}\nFor any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$, we can find a rooted tree $T_{x, \\mathtt{a}} \\in W_2^\\ast$ such that the path $(v_1$, $v_2$, $\\ldots$, $v_{ {\\ell_{\\operatorname{pump}}} +1})$ of the layer-1 vertices in $T_{x, \\mathtt{a}}$ where $v_1 = r$ is the root of $T_{x, \\mathtt{a}}$ satisfies the property that the half-edge labels of $v_{ {\\ell_{\\operatorname{pump}}} +1}$ have been fixed to $x$ by $f$ where the half-edge label for $\\{v_{ {\\ell_{\\operatorname{pump}}} }, v_{ {\\ell_{\\operatorname{pump}}} +1}\\}$ is in $x \\setminus \\{\\mathtt{a}\\}$.\n\nNow, observe that for any choice of $(\\Delta-2)( {\\ell_{\\operatorname{pump}}} +1)$ rooted trees \\[\\{T_{i,j}\\}_{1 \\leq i \\leq {\\ell_{\\operatorname{pump}}} +1, 1 \\leq j \\leq \\Delta-2}\\] in $W_2^\\ast$, there is a rooted tree $T' \\in W_3^\\ast = Z_f(W_2^\\ast)$ formed by appending $T_{i, 1}$, $T_{i,2}$, $\\ldots$, $T_{i, \\Delta-2}$ to $u_i$ in the path $(u_1$, $u_2$, $\\ldots$, $u_{ {\\ell_{\\operatorname{pump}}} +1})$ and fixing the half-edge labeling of $u_{ {\\ell_{\\operatorname{pump}}} +1}$ by $f$. All vertices in $(u_1, u_2, \\ldots, u_{ {\\ell_{\\operatorname{pump}}} +1})$ are of layer-2.\n\nWe choose any $T' \\in W_2^\\ast$ with $T_{i,j} = T_{x, \\mathtt{a}}$ and $T_{i',j'} = T_{x', \\mathtt{a}'}$ such that $i' - i = t + 1$.\nThe possible range of $t$ is $[0, {\\ell_{\\operatorname{pump}}} - 1]$.\nConsider any $T \\in W_L^\\ast$ that contains $T'$ as its subtree, and consider any correct labeling of $T$.\nThen the path $P$ resulting from concatenating the path $(v_{ {\\ell_{\\operatorname{pump}}} +1}, v_{ {\\ell_{\\operatorname{pump}}} }, \\ldots, v_1)$ in $T_{x, \\mathtt{a}}$, the path $(u_i, u_{i+1}, \\ldots, u_{i'})$, and the path $(v_1', v_2', \\ldots, v_{ {\\ell_{\\operatorname{pump}}} +1}')$ in $T_{x', \\mathtt{a}'}$\nshows that $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ \nfor $k = ( {\\ell_{\\operatorname{pump}}} +1) + (t+2) + ( {\\ell_{\\operatorname{pump}}} +1) = 2 {\\ell_{\\operatorname{pump}}} + 4 + t$. See \\cref{fig:ell_full_construction2} for an illustration.\n\nFor the rest of the proof, we show that the desired rooted tree $T_{x, \\mathtt{a}} \\in W_2^\\ast$ exists for any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$.\nFor any $x \\in \\mathcal{V}^\\ast$, we can find a bipolar tree\n\\[H_\\ast = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T_\\ast^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r) \\in Y_1^\\ast = Y_f(W_1^\\ast)\\]\nsuch that the vertex configuration of the root $r^m$ of $T_\\ast^m$ is fixed to be $x$ by the labeling function $f$.\nThen, for any $\\mathtt{a} \\in x$, at least one of the two rooted trees in $W_2^\\ast = Z_f(W_1^\\ast)$ resulting from cutting $H_\\ast$ satisfies the desired requirement. \n\\end{proof}\n\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width= .65\\textwidth]{fig\/y3.pdf}\n \\caption{An illustration for the proof of \\cref{lem:find_path_2}.}\n \\label{fig:ell_full_construction2}\n\\end{figure} \n\nIn the subsequent discussion, the length of a path refers to the number of edges in a path. In particular, the length of a $k$-vertex path is $k-1$.\n\nThe following lemma is proved by iteratively applying \\cref{lem:find_path_2} via intermediate vertex configurations $\\tilde{x} \\in \\mathcal{V}^\\ast$. \n\n\\begin{lemma}\\label{lem:find_path_3}\nFor any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$ and $\\mathtt{a}' \\in x' \\in \\mathcal{V}^\\ast$, we have $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ for some $\\mathtt{b} \\in x \\setminus \\{\\mathtt{a}\\}$ and $\\mathtt{b}' \\in x' \\setminus \\{\\mathtt{a}'\\}$\n for all $k \\geq 6 {\\ell_{\\operatorname{pump}}} + 10$.\n\\end{lemma}\n\\begin{proof}\nBy applying \\cref{lem:find_path_2} for three times,\nwe infer that \\cref{lem:find_path_2} also works for any path length $k - 1 = (k_1 - 1) + (k_2 - 1) + (k_3 - 1)$, where $k_i - 1 = 2 {\\ell_{\\operatorname{pump}}} + 3 + t_i$ for $0 \\leq t_i \\leq {\\ell_{\\operatorname{pump}}} -1$. \n\nSpecifically, we first apply \\cref{lem:find_path_2} to find a path realizing $(x,\\mathtt{b}) \\overset{k_1}{\\leftrightarrow} \\tilde{x}$ for some $\\tilde{x} \\in \\mathcal{V}^\\ast$, and for some $\\mathtt{b} \\in x \\setminus \\{\\mathtt{a}\\}$.\nLet $\\tilde{\\mathtt{a}} \\in \\tilde{x}$ be the real half-edge label used in the last vertex of the path. \nWe extend the length of this path by $k_2 - 1$ by attaching to it the path realizing $(\\tilde{x},\\tilde{\\mathtt{b}}) \\overset{k_2}{\\leftrightarrow} \\tilde{x}$, for some $\\tilde{\\mathtt{b}} \\in \\tilde{x} \\setminus \\{\\tilde{\\mathtt{a}}\\}$.\nFinally, let $\\tilde{\\mathtt{c}} \\in \\tilde{x}$ be the real half-edge label used in the last vertex of the current path. \nWe extend the length of the current path by $k_3$ by attaching to it the path realizing $(\\tilde{x},\\tilde{\\mathtt{d}}) \\overset{k_3}{\\leftrightarrow} (x', \\mathtt{b}')$, for some $\\tilde{\\mathtt{d}} \\in \\tilde{x} \\setminus \\{\\tilde{\\mathtt{x}}\\}$, and for some $\\mathtt{b}' \\in x' \\setminus \\{\\mathtt{a}'\\}$.\nThe resulting path realizes $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ for some $\\mathtt{b} \\in x \\setminus \\{\\mathtt{a}\\}$ and $\\mathtt{b}' \\in x' \\setminus \\{\\mathtt{a}'\\}$.\n\n\nTherefore, \\cref{lem:find_path_2} also works with path length of the form $k-1 = 6 {\\ell_{\\operatorname{pump}}} + 9 + t$, for any $t \\in [0, 3 {\\ell_{\\operatorname{pump}}} - 3]$. \n\nAny $k-1 \\geq 6 {\\ell_{\\operatorname{pump}}} + 9$ can be written as $k-1 = b(2 {\\ell_{\\operatorname{pump}}} +3) + (6 {\\ell_{\\operatorname{pump}}} + 9 + t)$, for some $t \\in [0, 3 {\\ell_{\\operatorname{pump}}} - 3]$ and some integer $b \\geq 0$.\nTherefore, similar to the above, we can find a path showing $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ by first applying \\cref{lem:find_path_2} with path length\n$2 {\\ell_{\\operatorname{pump}}} +3$ for $b$ times, and then applying the above variant of \\cref{lem:find_path_2} to extend the path length by $6 {\\ell_{\\operatorname{pump}}} + 9 + t$.\n\\end{proof}\n\nWe show that \\cref{lem:find_path_1,lem:find_path_3} imply that $\\mathcal{V}'$ is $\\ell$-full for some $\\ell$. \n\n\\begin{lemma}[$\\mathcal{V}'$ is $\\ell$-full] \\label{lem:find_path_4}\nThe set $\\mathcal{V}'$ is $\\ell$-full for $\\ell = 10 {\\ell_{\\operatorname{pump}}} + 10$. \n\\end{lemma}\n\\begin{proof}\nWe show that for any target path length $k-1 \\geq 10 {\\ell_{\\operatorname{pump}}} + 9$, and for any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$ and $\\mathtt{a}' \\in x' \\in \\mathcal{V}^\\ast$, we have $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} (x',\\mathtt{a}')$.\n\n\nBy \\cref{lem:find_path_1}, there exists a vertex configuration $\\tilde{x} \\in \\mathcal{V}^\\ast$ so that we can find a path $P$ realizing $(x,\\mathtt{a}) \\overset{\\ell_1}{\\leftrightarrow} \\tilde{x}$ for some path length $1 \\leq (\\ell_1-1) \\leq 2 {\\ell_{\\operatorname{pump}}} $.\nSimilarly, there exists a vertex configuration $\\tilde{x}' \\in \\mathcal{V}^\\ast$ so that we can find a path $P'$ realizing $(x',\\mathtt{a}') \\overset{\\ell_2}{\\leftrightarrow} \\tilde{x}'$ for some path length $1 \\leq (\\ell_2-1) \\leq 2 {\\ell_{\\operatorname{pump}}} $. \n\n\nLet $\\tilde{\\mathtt{a}}$ be the real half-edge label for $\\tilde{x}$ in $P$, and let $\\tilde{\\mathtt{a}}'$ be the real half-edge label for $\\tilde{x}'$ in $P'$.\nWe apply \\cref{lem:find_path_3} to find a path $\\tilde{P}$ realizing $(x,\\mathtt{b}) \\overset{\\tilde{\\ell}}{\\leftrightarrow} (x', \\mathtt{b}')$ for some $\\mathtt{b} \\in \\tilde{x} \\setminus \\{\\tilde{\\mathtt{a}}\\}$ and $\\mathtt{b}' \\in \\tilde{x}' \\setminus \\{\\tilde{\\mathtt{a}}'\\}$ with path length \n\\[\\tilde{\\ell}-1 = (k-1) - (\\ell_1-1) - (\\ell_2-1) \\geq 6 {\\ell_{\\operatorname{pump}}} + 9.\\] \nThe path formed by concatenating $P_1$, $\\tilde{P}$, and $P_2$ shows that $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} (x',\\mathtt{a}')$.\n\\end{proof}\n\n\n\n\n\n\n \n\n\n\n\\paragraph{A Feasible Labeling Function $f$ exists} We show that a feasible labeling function $f$ exists given that $\\Pi$ can be solved in $\\mathsf{LOCAL}(n^{o(1)})$ rounds. We will construct a labeling function $f$ in such a way each equivalence class in $\\textsf{Class}_1(W^\\ast)$ contains only rooted trees that admit legal labeling. That is, for each $c \\in \\textsf{Class}_1(W^\\ast)$, the mapping $h$ associated with $c$ satisfies $h(X) = \\textsf{YES}$ for some $X$.\n\n\n\nWe will consider a series of modifications in the construction \\[W \\rightarrow X_f(W) \\rightarrow Y_f(W) \\rightarrow Z_f(W)\\] that do not alter the sets of equivalence classes in these sets, conditioning on the assumption that all trees that we have processed admit correct labelings. That is, whether $f$ is feasible is invariant of the modifications.\n\n\n\\paragraph{Applying the Pumping Lemma} Let $w > 0$ be some target length for the pumping lemma. In the construction of $Y_f(W)$, when we process a bipolar tree \n\\[H = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r),\\] we apply \\cref{lem:pump} to the two subtrees $H^l = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l)$ and $H^r = (T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)$ to obtain two new bipolar trees $H_+^l$ and $H_+^r$.\nThe $s$-$t$ path in the new trees $H_+^l$ and $H_+^r$ contains $w + x$ vertices, for some $0 \\leq x < {\\ell_{\\operatorname{pump}}} $. The equivalence classes do not change, that is, $\\texttt{type}(H^l) = \\texttt{type}(H_+^l)$ and $\\texttt{type}(H^r) = \\texttt{type}(H_+^r)$.\n\n\nWe replace $H^l$ by $H_+^l$ and replace $H^r$ by $H_+^r$ in the bipolar tree $H$.\nRecall that the outcome of applying the labeling function $f$ to the root $r$ of the rooted tree $T_{ {\\ell_{\\operatorname{pump}}} +1}$ depends only on \\[\\texttt{type}(H^l), \\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m), \\texttt{type}(H^r),\\] so applying the pumping lemma to $H^l$ and $H^r$ during the construction of $Y_f(W)$ does not alter $\\textsf{Class}_1(T_\\ast^m)$ and $\\texttt{type}(H^\\ast)$ for the resulting bipolar tree $H^\\ast$ and its middle rooted tree $T_\\ast^m$, by \\cref{lem:labeling_function}.\n\n\n\n\\paragraph{Reusing Previous Trees} During the construction of the fixed point $W^\\ast$, we remember all bipolar trees to which we have applied the feasible function $f$.\n\nDuring the construction of $Y_f(W)$. Suppose that we are about to process a bipolar tree $H$, and there is already some other bipolar tree $\\tilde{H}$ to which we have applied $f$ before so that \n$\\texttt{type}(H^l)=\\texttt{type}(\\tilde{H}^l)$, $\\textsf{Class}_1(T_{i}^m) = \\textsf{Class}_1(\\tilde{T}_{i}^m)$ for each $1 \\leq i \\leq \\Delta-2$, and $\\texttt{type}(H^r)=\\texttt{type}(\\tilde{H}^r)$.\nThen we replace $H$ by $\\tilde{H}$, and then we process $\\tilde{H}$ instead.\n\nBy \\cref{lem:labeling_function}, this modification does not alter $\\textsf{Class}_1(T_\\ast^m)$ and $\\texttt{type}(H^\\ast)$ for the resulting bipolar tree $H^\\ast$.\n\n\\paragraph{Not Cutting Bipolar Trees} We consider the following different construction of $Z_f(W)$ from $Y_f(W)$. For each $H^\\ast \\in Y_f(W)$, we simply add two copies of $H^\\ast$ to $Z_f(W)$, one of them has $r = s$ and the other one has $r = t$.\nThat is, we do not cut the bipolar tree $H^\\ast$, as in the original construction of $Z_f(W)$.\n\nIn general, this modification might alter $\\textsf{Class}_1(Z_f(W))$.\nHowever, it is not hard to see that if all trees in $Y_f(W)$ already admit correct labelings, then $\\textsf{Class}_1(Z_f(W))$ does not alter after the modification.\n\nSpecifically, let $T$ be a rooted tree in $Z_f(W)$ with the above modification. \nThen $T$ is identical to a bipolar tree $H^\\ast=(T_1, T_2, \\ldots, T_k) \\in Y_f(W)$, where there exists some $1 < i < k$ such that the root $r_i$ of $T_i$ has its half-edge labels fixed by $f$.\nSuppose that the root of $T$ is the root $r_1$ of $T_1$.\nIf we do not have the above modification, then the rooted tree $T'$ added to $Z_f(W)$ corresponding to $T$ is $(T_1, T_2, \\ldots, T_i)$, where the root of $T'$ is also $r_1$. \n\nGiven the assumption that all bipolar trees in $Y_f(W)$ admit correct labelings, it is not hard to see that $\\textsf{Class}_1(T') = \\textsf{Class}_1(T)$. For any correct labeling $\\mathcal{L}'$ of $T'$, we can extend the labeling to a correct labeling $\\mathcal{L}$ of $T$ by labeling the remaining vertices according to any arbitrary correct labeling of $H^\\ast$. This is possible because the labeling of $r_i$ has been fixed. For any correct labeling $\\mathcal{L}$ of $T$, we can obtain a correct labeling $\\mathcal{L}'$ of $T'$ by restricting $\\mathcal{L}$ to $T'$.\n\n\n\n\n\n\n\\paragraph{Simulating a $\\mathsf{LOCAL}$ Algorithm} We will show that there is a labeling function $f$ that makes all the rooted trees in $W^\\ast$ to admit correct solutions, where we apply the above three modifications in the construction of $W^\\ast$. In view of the above discussion, such a function $f$ is feasible.\n\nThe construction of $W^\\ast$ involves only a finite number $k$ of iterations of $Z_f$ applied to some previously constructed rooted trees. If we view the target length $w$ of the pumping lemma as a variable, then the size of a tree in $W^\\ast$ can be upper bounded by $O(w^{k})$. \n\n\nSuppose that we are given an arbitrary $n^{o(1)}$-round $\\mathsf{LOCAL}$ algorithm $\\mathcal{A}$ that solves $\\Pi$. It is known~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup,chang2020n1k_speedups} that randomness does not help for LCL problems on bounded-degree trees with round complexity $\\Omega(\\log n)$, so we assume that \n$\\mathcal{A}$ is deterministic.\n\nThe runtime of $\\mathcal{A}$ on a tree of size \n$O(w^{k})$ can be made to be at most $t = w\/10$ if $w$ is chosen to be sufficiently large. \n\nWe pick the labeling function $f$ as follows. \nWhenever we encounter a new bipolar tree $H$ to which we need to apply $f$, we simulate $\\mathcal{A}$ locally at the root $r^m$ of the middle rooted tree $T^m$ in $H$. Here we assume that the pumping lemma was applied before simulating $\\mathcal{A}$. Moreover, when we do the simulation, we assume that the number of vertices is $n = O(w^{k})$. \n\nTo make the simulation possible, we just locally generate arbitrary distinct identifiers in the radius-$t$ neighborhood $S$ of $r^m$. This is possible because $t = w\/10$ is so small that for any vertex $v$ in any tree $T$ constructed by recursively applying $Z_f$ for at most $k$ iterations, the radius-$(t+1)$ neighborhood of $v$ intersects at most one such $S$-set. Therefore, it is possible to complete the identifier assignment to the rest of the vertices in $T$ so that for any edge $\\{u,v\\}$, the set of identifiers seen by $u$ and $v$ are distinct in an execution of a $t$-round algorithm. \n\nThus, by the correctness of $\\mathcal{A}$, the labeling of all edges $\\{u,v\\}$ are configurations in $\\mathcal{E}$ and the labeling of all vertices $v$ are configurations in $\\mathcal{V}$.\nOur choice of $f$ implies that all trees in $W^\\ast$ admit correct labeling, and so $f$ is feasible. Combined with \\cref{lem:find_path_4}, we have proved that $\\mathsf{LOCAL}(n^{o(1)})$ $\\Rightarrow$ $\\ell$-full.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Concluding Remarks}\\label{sec:concludingRemarksTrees}\n\nHere we collect some open problems we find interesting. \n\n\n\\begin{enumerate}\n \\item What is the relation of $\\mathsf{ffiid}, \\mathsf{MEASURE}, \\mathsf{fiid}$? In particular is perfect matching in the class $\\mathsf{ffiid}$ or $\\mathsf{MEASURE}$? \n \\item Are there LCL problems whose uniform complexity is nontrivial and slower than $\\operatorname{poly}\\log 1\/\\varepsilon$?\n \n \n \n \n \n \n \\item Suppose that $\\Pi_G$ is a homomorphism LCL, is it true that $\\Pi_G\\in\\mathsf{BOREL}$ if and only if $\\Pi_G\\in O(\\log^*(n))$?\n \\item Are there finite graphs of chromatic number bigger than $2\\Delta-2$ such that the corresponding homomorphism LCL is not in $\\mathsf{BOREL}$? See also Remark \\ref{r:hedet}.\n\\end{enumerate}\n\n\\subsection*{Acknowledgements} \nWe would like to thank Anton Bernshteyn, Endre Cs\u00f3ka, Mohsen Ghaffari, Jan Hladk\u00fd, Steve Jackson, Alexander Kechris, Edward Krohne, Oleg Pikhurko, Brandon Seward, Jukka Suomela, and Yufan Zheng for insightful discussions. \nWe especially thank Yufan Zheng for suggesting potentially hard instances of homomorphism LCL problems.\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}