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+{"text":"\n\\section{Additional Definitions}\n\n\\begin{definition}[Pairwise Independence Hash Functions]\\label{def:pairwise}\nLet $\\mathcal{H}$ be a family of functions from $\\{1,\\ldots, N\\}$ to $\\{1,\\ldots, M\\}$. The family $\\mathcal{H}$ is \\emph{pairwise independent} if for every $x,y \\in \\{1,\\ldots, N\\}$ such that $x \\neq y$ and for every $a,b \\in \\{1,\\ldots, M\\}$ it holds that \n$$\\Pr_{h \\in \\mathcal{H}}[h(x)=a \\wedge h(y)=b]=1\/M^2~.$$\nThat is, if $h$ is chosen uniformly at random from $\\mathcal{H}$, then the random variable $h(x)$ and $h(y)$ are uniformly distributed and pairwise independent. \n\\end{definition}\n\n\\begin{fact}\\label{fc:pairwise}\\cite{TCS-010}\nThere is an explicit family $\\mathcal{H}$ of pairwise independent has functions from $\\{0,1\\}^n \\to \\{0,1\\}^m$ constructed using $O(\\max\\{\nm,n\\})$ bits and computable in $\\operatorname{\\text{{\\rm poly}}}(n,m)$ time.\n\\end{fact}\n\\section{Overview of the Cycle Space Sampling Technique} \\label{sec:cycle_space_overview}\n\nThe cycle space sampling technique allows to detect cuts in a graph using a connection between cuts and cycles in a graph.\nThis beautiful technique was introduced by Pritchard and Thurimella \\cite{pritchard2011fast}, that showed its applicability for distributed algorithms identifying small cuts in a graph. We next give a short overview of the technique, for full details see \\cite{pritchard2011fast}.\n\nThe \\emph{cycle space} of a graph is the family of all subsets of edges $F$ that have even degree at each vertex, any such subset $\\phi \\subseteq E$ is called a \\emph{binary circulation}. The \\emph{cut space} is the family of all induced edge cuts. It is easy to see that if we take a cycle $C$ in a graph and an induced edge cut, then the number of edges of the cycle that cross the cut is even. The cycle space technique extends this observation and shows that the cycle space and cut space are orthogonal vector spaces. Using this, they show the following (see Propositions 2.2 and 2.5 in \\cite{pritchard2011fast}).\n\n\\begin{claim} \\label{claim_cycle} \nLet $\\phi$ be a uniformly random binary circulation and $F \\subseteq E$. Then\n$$Pr[|F \\cap \\phi| \\ is \\ even] = \\left\\{\n                \\begin{array}{ll}\n                  1,\\ if\\ F\\ is\\ an\\ induced\\ edge\\ cut\\\\\n                  1\/2,\\ otherwise\n                \\end{array}\n              \\right. $$ \n\\end{claim}   \n\nHence, sampling a random binary circulation allows to detect if a subset of edges is an induced edge cut with probability $1\/2$. To reduce the failure probability to $1\/2^b$ we can choose $b$ random binary circulations. To use this technique, the authors provide an efficient way to sample a random binary circulation, we describe next. Let $T$ be a spanning tree of the graph. For any non-tree edge $e$, adding $e$ to the graph creates a cycle. These cycles are the \\emph{fundamental cycles}, and it is shown that the \\emph{fundamental cycles} are a basis for the cycle space. Based on this, they show that sampling a random binary circulation can be done by choosing each fundamental cycle with probability $1\/2$, or equivalently choosing each non-tree edge with probability $1\/2$. The binary circulation $\\phi$ sampled has all the non-tree edges sampled, and each tree edge that appears in odd number of sampled cycles. Given the sampled non-tree edges in $\\phi$, the tree edges in $\\phi$ can be identified using a simple scan of the tree, as shown in \\cite{pritchard2011fast}. Choosing $b$ random binary circulations, is equivalent to choosing a $b$-bit random string $\\phi(e)$ for each non-tree edge. For a tree edge $t$, we define $\\phi(t) = \\oplus_{e \\in C_t} \\phi(e)$, where $C_t$ are all non-tree edges $e$ such that $t$ is in the fundamental cycle of $e$. This again can be computed by a simple scan of the tree, and takes $O((n+m)b)$ time if the labels have size $b$. This gives the following.\n\n\\cycle*\n\nTo see this, let $\\phi_1,...,\\phi_b$ be the sampled binary circulations. If $F$ is an induced edge cut, then from Claim \\ref{claim_cycle}, for every sampled circulation $\\phi_i$, we have that $|F \\cap \\phi_i|$ is even, and hence for all $i$, the $i$'th bit of $\\Moplus_{e \\in F} \\phi(e)$ is equal to 0 as needed. Otherwise, for all $i$, the $i$'th bit $\\Moplus_{e \\in F} \\phi(e)$ equals $0$ with probability $1\/2$, hence the probability that the whole vector equals $0$ is $1\/2^b$, as needed.\n\\section{Fault-Tolerant Approximate Distance Labels}\\label{sec:ft-distance}\nGiven integer parameters $f,k \\geq 1$, an $(f,k)$ \\emph{FT approximate distance labeling scheme} assigns labels $\\mathsf{DistLabel}\\xspace: V \\cup E \\to \\{0,1\\}^{q}$ such that given the labels of $s,t$ and a subset $F \\subseteq E$, $|F|\\leq f$, there exists a decoding algorithm that outputs a distance estimate $\\delta_{G \\setminus F}(s,t)$ satisfying:\n$$\\mbox{\\rm dist}_{G\\setminus F}(s,t) \\leq \\delta_{G \\setminus F}(s,t) \\leq k\\cdot\\mbox{\\rm dist}_{G\\setminus F}(s,t)~.$$\n\nWe next show that there is an efficient transformation from any FT connectivity labeling scheme into an FT approximate distance labeling scheme. This transformation increases the label size by a multiplicative factor of $\\widetilde{O}(n^{1\/k})$. This technique was first introduced by \\cite{chechik2012f} in the context of distance sensitivity oracles, and it is based on the notion of tree covers.\n\n\\begin{definition}[Tree Covers]\\label{def:tree-cover}\nLet $G=(V,E)$ be an undirected graph with edge weights $\\omega$, and let $\\rho,k$ be two integers. Define $B_{\\rho}(v)=\\{ u \\in V ~\\mid~ \\mbox{\\rm dist}_G(u,v)\\leq \\rho\\}$. A tree cover $\\mathsf{TC}\\xspace(G, \\omega, \\rho,k)$ is a collection of rooted trees $\\mathcal{T}=\\{T_1,\\ldots, T_\\ell\\}$ with root $r(T)$ for every $T \\in \\mathcal{T}$ such that:\n\\begin{enumerate}[noitemsep]\n\\item For every vertex $v$ there exists a tree $T \\in \\mathcal{T}$ such that $B_{\\rho}(v) \\subseteq T$.\n\\item The radius of each tree $T$ is at most $(2k-1)\\cdot \\rho$.\n\\item Each vertex participates in $(k \\cdot n^{1\/k})$ trees.\n\\end{enumerate}\nLet $|\\mathsf{TC}\\xspace(G, \\omega, \\rho,k)|$ denote the number of trees in the tree cover $\\mathsf{TC}\\xspace(G, \\omega, \\rho,k)$.\n\\end{definition}\n\n\\begin{proposition}\\cite{Peleg:2000}\nFor any $n$-vertex graph $G=(V,E, \\omega)$, and any parameters $\\rho,k$, one can compute tree covers $\\mathsf{TC}\\xspace(G, \\omega, \\rho,k)$ in time $\\widetilde{O}(|E(G)| \\cdot n^{1\/k})$.\n\\end{proposition}\n\n\\begin{lemma}[From Connectivity Labels to Approximate Distance Labels]\\label{lem:reduction}\nLet $G=(V,E, \\omega)$ be a weighted undirected $n$-vertex graph where $\\omega(e)\\in [1,W]$, and let \n$\\mathsf{ConnLabel}\\xspace: V \\cup E \\to \\{0,1\\}^s$ be an $f$-FT connectivity labeling scheme for $G$ with decoding time $t$. Then for every integer $k\\geq 1$, there is an $(f,(8k-2)(|F|+1))$ FT approximate distance labeling scheme $\\mathsf{DistLabel}\\xspace: V \\cup E \\to \\{0,1\\}^{q}$ for $G$, where $q=O(s \\cdot k \\cdot n^{1\/k}\\cdot \\log (nW))$, and with decoding time $\\widetilde{O}(t \\log{(nW)})$.\n\\end{lemma}\n\n\\paragraph{The labeling algorithm.}\nFor every vertex $u$, the label $\\mathsf{DistLabel}\\xspace(u)$ consists of $K=\\log(nW)$ sub-labels of FT connectivity labels in distinct subgraphs of $G$ defined as follows. The $i^{th}$ sub-label addresses all distances that are at most $2^i$ in $G$. Let $H_i$ be set of heavy edges in $G$ of weight at least $2^i$, and define the $i^{th}$ tree-cover by \n\\begin{equation}\\label{eq:TC-i}\n\\mathsf{TC}\\xspace_i=\\mathsf{TC}\\xspace(G\\setminus H_i,\\omega, 2^i,k)~.\n\\end{equation}\nFor each tree $T_{i,j} \\in \\mathsf{TC}\\xspace_i$, the algorithm applies the FT connectivity scheme on the graph $G_{i,j}=G[V(T_{i,j})]$. For every vertex $u$ and $i \\in \\{1,\\ldots, K\\}$, let $i^*(u)$ be an index of a tree in $\\mathsf{TC}\\xspace_i$ that covers the $2^i$-ball of $u$. I.e., $B_{2^i}(v) \\subseteq T_{i,i^*(u)}$. \nThe label of every $u \\in V$ is then given by:\n$$\\mathsf{DistLabel}\\xspace(u)=\\{\\langle \\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(u), i,j \\rangle ~\\mid~ i \\in [1,K], j \\in \\{1,\\ldots, |\\mathsf{TC}\\xspace_i|\\}, u \\in G_{i,j}\\} \\bigcup \\{i^*(u) ~\\mid~ i \\in [1,K]\\}~.$$\n\nSimilarly, the label of each edge $e \\in G$ contains the FT connectivity label of $e$ in each of the instances $(G_{i,j}, T_{i,j})$:\n$$\\mathsf{DistLabel}\\xspace(e)=\\{\\langle \\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(e), i,j \\rangle ~\\mid~ i \\in [1,K], j \\in \\{1,\\ldots, |\\mathsf{TC}\\xspace_i|\\}, e \\in G_{i,j}\\}.$$ \n\nThe time for assigning the labels is the time for constructing the tree cover and computing the indexes $i^{*}(v)$, and the time for assigning the connectivity labels on each one of the trees. The first part requires polynomial time. The second depends on the connectivity labels. For example, using our scheme from Section \\ref{sec:ftconn-sketch} the time complexity of the second part is $\\widetilde{O}(mn^{1\/k})$, as it is linear in the total number of vertices and edges in the trees.\n\n\n\n\\paragraph{The decoding algorithm.}\nConsider the query $\\langle s,t,F\\rangle$. \nThe algorithm has $K$ phases, in each phase $i \\in [1,K]$ the decoding algorithm of the FT connectivity labels is applied on the instance $G_{i,i^*(s)}, T_{i,i^*(s)}$ where $G_{i,i^*(s)}$ contains the $2^i$ ball of $s$ in $G$. \nIf $t \\notin G_{i,i^*(s)}$, the phase $i$ ends and we continue to phase $i+1$. \nOtherwise, the algorithm decides if $s$ and $t$ are connected in $G_{i,i^*(s)} \\setminus F$ in the following manner. \nLet $F_i=F \\cap G_{i,i^*(s)}$, this subset of edges can be obtained from the labels of the $F$ edges. \nSince the labels of $s,t$ and $F_i$ contain the FT connectivity labels in the subgraph $G_{i,i^*(s)}$ and the tree $T_{i,i^*(s)}$, the algorithm can apply the decoding algorithm of the FT connectivity scheme.\nIf $s$ and $t$ are indeed connected in $G_{i,i^*(s)}\\setminus F_i$, the algorithm returns the estimate $\\delta_{G \\setminus F}(s,t)= (4k-1) \\cdot (|F|+1) \\cdot 2^{i}$. Otherwise, it proceeds to the next phase.\n\nOverall, let $i$ be the minimum index in $\\{1,\\ldots, K\\}$ for which $s$ and $t$ are connected in the subgraph $G_{i,i^*(s)} \\setminus F$. Then the decoding algorithm returns the distance estimate $\\delta_{G \\setminus F}(s,t)=(4k-1) \\cdot (|F|+1) \\cdot 2^{i}$. If no such $i$ exists, the decoding algorithm returns $\\delta_{G \\setminus F}(s,t)=\\infty$, which implies that $s$ and $t$ are not connected in $G \\setminus F$. \n\nThe decoding time is $\\widetilde{O}(t \\log{(nW)})$, where $t$ is the decoding time of the connectivity labels, as we use the decoding algorithm of the connectivity labels $K$ times on the graphs $G_{i,i^*(s)}$. To obtain this, we need to make sure that given the labels of $s,t,F$ we can easily find their connectivity label in the graph $G_{i,i^*(s)}$ if exist. This can be easily done if we store the connectivity labels in a sorted order.\n\n\\paragraph{Analysis.}\nWe now analyze the construction, and start by bounding the size of the labels. By the properties of the tree-cover in Def. \\ref{def:tree-cover}, each vertex appears in $O(K \\cdot k \\cdot n^{1\/k})$ subgraphs. Thus, $\\mathsf{DistLabel}\\xspace(u)$consists of $O(K \\cdot k n^{1\/k})$ FT connectivity labels and the label size is bounded by $O(K \\cdot k n^{1\/k} \\cdot s)$ bits, as desired. Next, we show correctness. By the correctness of the FT connectivity labeling scheme, it is sufficient to show the following. Let $P_{s,t,F}$ be an $s$-$t$ shortest path in $G \\setminus F$ of length $(2^{i-1}, 2^{i}]$. By the properties of the tree cover, there is a tree $T_{i,i^*(s)} \\in \\mathsf{TC}\\xspace_i$ that contains all the vertices of the path $P_{s,t,F}$. Therefore, we have that $s$ and $t$ are connected in $G_{i,i^*(s)}\\setminus F$. Since the labels of $s, t$ and $F_i=F \\cap G_{i,i^*(s)}$ contain the FT connectivity labels in $G_{i,i^*(s)}$, we get that the distance estimate returned by the algorithm satisfies that\n$$\\mbox{\\rm dist}_{G \\setminus F}(s,t)\\leq \\delta_{G \\setminus F}(s,t) \\leq (4k-1)(|F|+1)  \\cdot 2^i \\leq (8k-2)(|F|+1) \\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)~.$$\nTo see this, let $j \\leq i$ be the first index such that $s$ and $t$ are connected in $G_{j,j^*(s)}\\setminus F$. The algorithm returns the estimate $(4k-1)(|F|+1) \\cdot 2^j \\leq (4k-1)(|F|+1) \\cdot 2^i = (8k-2)(|F|+1) \\cdot 2^{i-1} \\leq (8k-2)(|F|+1) \\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$. To prove the left inequality, we show that if $s$ and $t$ are connected in $G_{j,j^*(s)}\\setminus F$, there is indeed a path between them in $G \\setminus F$ of length at most $\\delta_{G \\setminus F}(s,t) = (4k-1)(|F|+1) \\cdot 2^j$. First, from the tree cover properties, the radius of $T_{j,j^*(s)}$ is at most $(2k-1)2^j$, implying that any two vertices in $T_{j,j^*(s)}$ are at distance at most $(4k -2) \\cdot 2^j$ from each other. Now the graph $T_{j,j^*(s)} \\setminus F$ has at most $|F|+1$ connected components. Since $G_{j,j^*(s)}\\setminus F$ is connected, it implies that there is a path between $s$ and $t$ in $G_{j,j^*(s)}\\setminus F$. This path traverses at most $|F|+1$ different components in $T_{j,j^*(s)} \\setminus F$, and at most $|F|$ edges connecting them, each one of weight at most $2^j$. As the diameter of each component is bounded by $(4k - 2) \\cdot 2^j$, the length of the path is at most $(4k - 2) \\cdot 2^j \\cdot (|F|+1) + 2^j \\cdot |F| \\leq (4k - 1) \\cdot 2^j \\cdot (|F| + 1)$, as needed.\n\n\\remove{\nLet $\\mathcal{T}=\\bigcup_{i=1}^{K}\\mathsf{TC}\\xspace_i$ be the collection of tree covers with $K=O(\\log (nW))$ scales of distances. We call an edge $(u,v)$ a \\emph{tree edge} if it appears on at least one of the trees in $\\mathcal{T}$. \nUsing Lemma \\ref{lem:useful-recovery-edges}, we have the following decoding algorithm, which becomes useful in the context of routing schemes, as described in the next section. \\mtodo{there could be a problem with treating edges globally as tree edges or non-tree edges, I think for each edge $e \\in G_{i,j}$ we need to know its label in this graph (either tree or non-tree label), extended identifiers are also different for different trees because of the ancestry labels.}\n\\begin{lemma}\\label{lem:approx-dist-recovery}\nConsider the $(f,(8k-2)(f+1))$ approximate distance labels $\\mathsf{DistLabel}\\xspace$ obtained by using Lemma \\ref{lem:reduction} with the FT connectivity labeling scheme of Sec. \\ref{sec:ftconn-sketch}. For any triplet $s,t,F \\subseteq E$ and $|F|\\leq f$, let $F_T$ be the tree edges of $F$. Then, given the labels $\\{\\mathsf{DistLabel}\\xspace(w), w \\in \\{s,t\\} \\cup F_T\\}$ and the extended edge identifiers of $F \\setminus F_T$, the decoding algorithm can be modified to return a labeled $s$-$t$ path $\\widehat{P}$ of length $O(f)$ that provides a succinct description of an $s$-$t$ path in $G \\setminus F$, along with indices $i,j$. Each $G$-edge $e$ of $\\widehat{P}$ is augmented with port information and the extended identifier of $e$; and each non-$G$ edge $e'=(u,v)$ corresponds to a $u$-$v$ path in $T_{i,j} \\setminus F$. In addition, the length of the $s$-$t$ path encoded by $\\widehat{P}$ is bounded by $(8k-2)(|F|+1)\\cdot \\mbox{\\rm dist}_{G\\setminus F}(s,t)$. \n\\end{lemma}\n\\begin{proof}\n\n\n\n\\end{proof}\n}\n\n\\section{Introduction}\nDistributed graph representation is concerned with augmenting each vertex (and possibly also edges) with useful and low-space information in order to efficiently address various graph queries in a distributed manner. As the\nvertices and edges of the network may occasionally fail or malfunction, it is desirable\nto make these representations robust against failures. In this paper, we provide new constructions of succinct \\emph{labeled-based distributed data structures} that can handle connectivity, distance queries and routing in the presence of edge failures. \n\nConnectivity labels are short names attached to each vertex in the $n$-vertex input graph $G$, such that given the labels of a pair of vertices $s$ and $t$ (and no any other information), it is possible to deduce if $s$ and $t$ are connected in $G$. The primary complexity measure of the  labeling scheme is the label length (maximum length of a label). In general, labels can be viewed as the \\emph{logical} names of the vertices \\cite{kannan1992implicat,peleg2005informative}, as they are considerably more informative than the physical names that usually correspond to arbitrary $O(\\log n)$-bit identifiers. For example, in routing applications the label of the vertex is treated as its ``address\". It is quite immediate to provide connectivity labeling schemes of logarithmic length. Over the years, these labels have served the basis for devising also approximate distance labels, and compact routing schemes, which are arguably the \\emph{grand finale} of the distributed representation schemes. \n\n\nOur goal in this paper is to provide \\emph{fault-tolerant} analogs for the above mentioned schemes, while paying a small overhead in terms of space and other complexity aspects. Several notions of fault-tolerant labeling and routing schemes have been addressed in the literature; starting with the earlier introduction of FT routing schemes by Dolev \\cite{dolev1984new}, to the more recent formulations of forbidden-set labeling and routing schemes by Courcelle et al. \\cite{courcelle2007forbidden,CourcelleT07}. Despite much activity revolving these topics, FT labeling and routing schemes with \\emph{sub-linear} space are currently known only for a limited collection of graph families. We next elaborate more on the state-of-the-art affairs, and our main objectives.\n\n\n\\paragraph{Fault-Tolerant Connectivity and Distance Labeling.} \nFT connectivity labeling schemes, also known in the literature as \\emph{forbidden-set} labeling \\cite{CourcelleT07}, assign labels to the vertices and the edges of the graph such that given the labels of a vertex pair $s,t$, and the labels of the faulty edges $F$, one can determine if $s$ and $t$ are connected in $G \\setminus F$. \n\n\n\n %\n\nSince their introduction, efficient FT labeling schemes have been devised only for a restricted collection of graph families such as graphs with bounded tree-width and planar graphs \\cite{CourcelleT07,AbrahamCGP16}. In the lack of any FT connectivity labeling schemes for general graphs with sub-linear label length (for any $f\\geq 2$ faults\\footnote{While there is no \\emph{explicit} construction of FT labeling for general graphs, for $f=1$, the centralized distance sensitivity oracle of \\cite{khanna2010approximate} might be modified to provide approximate distance labels against a single fault.}), we ask:\n\n\\begin{question}\\label{q:label}\nIs it possible to design FT connectivity labeling scheme resilient to at most $f$ edge faults, for general graphs with label length of $\\operatorname{\\text{{\\rm poly}}}(\\log n)$ bits, or even $\\operatorname{\\text{{\\rm poly}}}(\\log n,f)$ bits? \n\\end{question}\n\nFT connectivity labels are also closely related to \\emph{connectivity sensitivity oracles} \\cite{patrascu2007planning}, which are low-space centralized data-structures that handle efficiently $\\langle s, t, F \\rangle$ connectivity queries using $S(n)$ space. Our main goal is in providing a \\emph{distributed} variant of such constructions, e.g., where each vertex or edge in the graph ``holds\" only $S(n)\/n$ bits of information, such that an $\\langle s,t, F \\rangle$ query can be addressed using only the information stored by $s,t$ and $F$. \n\nAn important step towards designing FT compact routing schemes involves the computation of \\emph{FT approximate distance labels}. In this setting, given the labels of $s,t$ and the faulty edges $F$, it is required to report an approximation for the $s$-$t$ shortest path distance in $G \\setminus F$.\nFT approximate distance labels can be viewed as the distributed analog of $f$-FT \\emph{distance sensitivity oracles} \\cite{khanna2010approximate,WeimannY10}. \nThese are global succinct data-structures that given an $\\langle s,t,F \\rangle$ query report fast an estimate for the approximate $s$-$t$ distance in $G \\setminus F$. Our goal is to provide FT approximate labeling schemes that match the state-of-the-art space vs. stretch tradeoff of the centralized data structures.\n\n\n\n\\paragraph{Fault-Tolerant Routing.} A desirable requirement in most communication networks is to provide efficient routing protocols in the presence of faults. Specifically, an $f$-FT routing protocol is a distributed algorithm that, for any set of at most $f$ faulty edges $F$, allows a vertex $s$ to route a message to a destination vertex $t$ along an approximate $s$-$t$ shortest path in $G \\setminus F$ (without knowing $F$ in advance). The routing scheme consists of two algorithms: (i) a preprocessing algorithm which computes (succinct) routing tables and labels for each vertex in the graph; and (ii) a routing algorithm that given the received message and the routing table of vertex $v$ determines the next-hop (specified as a port number) on the $v$-$t$ (approximate) shortest path in $G \\setminus F$. The efficiency of the scheme is determined by the tradeoff between the \\emph{stretch} (i.e., the ratio between the weighted length of the $s$-$t$ route in $G\\setminus F$ to the corresponding shortest path distance) and the \\emph{space} of the routing tables, labels and messages. \nWhile the stretch vs. space tradeoff of routing schemes is fully understood in the non-faulty setting, the corresponding bounds in the FT setting are still far from optimal.\nSo far, in all the prior schemes, the space of the individual routing tables could be linear in the worst case, even when allowing a large stretch bound. This is in strike contrast to the standard (non-faulty) compact routing schemes, e.g., by Thorup and Zwick \\cite{thorup2001compact}, which provide each vertex a table of $\\widetilde{O}(n^{1\/k})$ bits, while guaranteeing a route stretch of $2k-1$. The current large gap in the quality of FT routing schemes compared to their non-faulty counterparts leads to the following question.\n\n\\begin{question}\\label{q:route}\nIs it possible to design $f$-fault-tolerant compact routing scheme for general graphs with \\emph{sub-linear} table size and with a sub-logarithmic stretch?\n\\end{question}\n\n\n\n\n\\input{result.tex}\n\n\n\n\\section{Introduction}\nDistributed graph representation is concerned with augmenting each vertex (and possibly also edges) with useful and low-space information in order to efficiently address various graph queries in a distributed manner. As the\nvertices and edges of the network may occasionally fail or malfunction, it is desirable\nto make these representations robust against failures. In this paper, we provide new constructions of succinct \\emph{labeled-based distributed data structures} that can handle connectivity, distance queries and routing in the presence of edge failures. \n\nConnectivity labels are short names attached to each vertex in the $n$-vertex input graph $G$, such that given the labels of a pair of vertices $s$ and $t$ (and no any other information), it is possible to deduce if $s$ and $t$ are connected in $G$. The primary complexity measure of the  labeling scheme is the label length (maximum length of a label). In general, labels can be viewed as the \\emph{logical} names of the vertices \\cite{kannan1992implicat,peleg2005informative}, as they are considerably more informative than the physical names that usually correspond to arbitrary $O(\\log n)$-bit identifiers. For example, in routing applications the label of the vertex is treated as its ``address\". It is quite immediate to provide connectivity labeling schemes of logarithmic length. Over the years, these labels have served the basis for devising also approximate distance labels, and compact routing schemes, which are arguably the \\emph{grand finale} of the distributed representation schemes. \n\nOur goal in this paper is to provide \\emph{fault-tolerant} analogs for the above mentioned schemes, while paying a small overhead in terms of space and other complexity aspects. Fault-tolerant (FT) connectivity labeling scheme, also known in the literature as \\emph{forbidden-set} labeling \\cite{CourcelleT07}, assigns labels to the vertices and the edges of the graph such that given the labels of a vertex pair $s,t$, and the labels of the faulty edges $F$, one can determine if $s$ and $t$ are connected in $G \\setminus F$. Several notions of fault-tolerant labeling and routing schemes have been addressed in the literature; starting with the earlier introduction of FT routing schemes by Dolev \\cite{dolev1984new}, to the more recent formulations of forbidden-set labeling and routing schemes by Courcelle et al. \\cite{courcelle2007forbidden,CourcelleT07}. Despite much activity revolving these topics, FT labeling and routing schemes with \\emph{sub-linear} space are currently known only for a limited collection of graph families. We next elaborate more on the state-of-the-art affairs, and our main objectives. \\mtodo{the discussion of related work in the next few pages is quite long. We can consider to have a shorter intro, mainly defining the main problems addressed, and have later a detailed related work section.}\n\n\n\n %\n\n\\paragraph{Fault-Tolerant Connectivity Labeling.} FT labels for connectivity were introduced by \\cite{courcelle2007forbidden} \\mtext{under the term \\emph{forbidden-set labeling}. Forbidden set refers to a subset $F$ of at most $f$ edges, such that given the labels of $s,t$ and $F$ one should determine if $s$ and $t$ are connected in $G \\setminus F$. The forbidden edge set can be treated in this context as faulty edges\\footnote{For routing, the forbidden-set scheme is slightly weaker than FT scheme as explained later.}.} \\mtodo{not sure that this is the best place for discussing the forbidden-set name, maybe it can fit better in a related work section.} Since their introduction, efficient FT labeling schemes have been devised only for a restricted collection of graph families. \\textbf{MP: For example, Courcelle et al. \\cite{CourcelleT07} presented a labeling scheme with logarithmic label length for the families of $n$-vertex graphs with bounded clique-width, tree-width and planar graphs. For $n$-vertex graphs with doubling dimension at most $\\alpha$, Abraham et al. \\cite{AbrahamCGP16} designed FT labeling schemes with label length $O((1 + 1\/\\epsilon)^{2\\alpha}\\log n)$ that output $(1+\\epsilon)$ approximation of the shortest path distances under faults.} In the lack of any FT connectivity labeling schemes for general graphs with sub-linear label length (for any $f\\geq 2$ faults\\footnote{While there is no \\emph{explicit} construction of FT labeling for general graphs, for $f=1$, the centralized distance sensitivity oracle of \\cite{khanna2010approximate} might be modified to provide approximate distance labels against a single fault.}), we ask:\n\n\\begin{question}\\label{q:label}\nIs it possible to design FT connectivity labeling scheme resilient to at most $f$ edge faults, for general graphs with label length of $\\operatorname{\\text{{\\rm poly}}}(\\log n)$ bits, or even $\\operatorname{\\text{{\\rm poly}}}(\\log n,f)$ bits? \n\\end{question}\n\n\\mtodo{The next pargarph is a bit long, we can consider shortning it, and move the more detailed discussion to a related work section. Maybe just focus on the centralized data structures here and not on the certificates?} \\mertodo{Yes, I agree, modifying accordingly.} \nFT connectivity labels are also closely related to \\emph{sensitivity connectivity oracles}, which are low-space centralized data-structure that handle efficiently $\\langle s, t, F \\rangle$ connectivity queries. \\textbf{MP: The first construction of these oracles was given by Patrascu and Thorup \\cite{patrascu2007planning} providing an $S(n)=\\widetilde{O}(fn)$ space oracle that answers $\\langle s,t, F \\rangle$ connectivity queries in $\\widetilde{O}(f)$ time. The state-of-the-art bounds of these oracles are given by Duan and Pettie \\cite{DuanConnectivitySODA17}.} \nOur main goal is in providing a \\emph{distributed} variant of such constructions, e.g., where each vertex or edge in the graph ``holds\" only $S(n)\/n$ bits of information, such that an $\\langle s,t, F \\rangle$ query can be addressed using only the information stored by $s,t$ and $F$. \n\n\\paragraph{Fault-Tolerant Approximate Distance Labeling.} An important step towards designing FT compact routing schemes involves the computation of \\emph{FT approximate distance labels}. In this setting, given the labels of $s,t$ and the faulty edges $F$, it is required to report an approximation for the $s$-$t$ shortest path distance in $G \\setminus F$. \\mtodo{Also the part that starts here may fit better in a related work section (in this case, we will probably need to remove also the question from here - seems that one of the reviwers supported removing the questions in any case).} FT approximate distance labels can be viewed as the distributed analog of $f$-FT \\emph{distance sensitivity oracles} \\cite{khanna2010approximate,WeimannY10}. \nThese are global succinct data-structures that given an $\\langle s,t,F \\rangle$ query report fast an estimate for the approximate $s$-$t$ distance in $G \\setminus F$. \\textbf{MP: Chechik et al. \\cite{chechik2012f} presented the first randomized construction resilient to $f$ edge faults. \nSpecifically, for any $n$-vertex weighted graph, stretch parameter $k$, and a fault bound $f$, they provide a data-structure with $O(f k n^{1+1\/k}\\log(nW))$ space, query time of $\\widetilde{O}(|F|)$, and $O(f k)$ stretch, where $W$ is the weight of the heaviest edge in the graph. Their solution is based on an elegant transformation that converts the FT connectivity oracle of \\cite{patrascu2007planning} into an FT approximate distance oracle.} Our goal is to provide FT approximate labeling schemes that match the state-of-the-art space vs. stretch tradeoff provided by the oracles of \\cite{chechik2012f}, we ask:\n\n\\begin{question}\\label{q:dist-label}\nIs it possible to design FT approximate distance labels with space vs. stretch tradeoff that match the state-of-the-art bounds of the \\emph{centralized} sensitive oracles, e.g., of \\cite{chechik2012f}?\n\\end{question}\n\n\\textbf{MP: While the main focus of this paper is in approximate distances, sensitivity oracles that report (possibly near) exact distances under faults have been studied also thoroughly in e.g., \\cite{demetrescu2002oracles,bernstein2008improved,duan2009dual,WeimannY10,GrandoniW12,ChechikCFK17,van2019sensitive}. Since reporting exact distances requires linear label length already in the fault-free setting \\cite{gavoille2004distance}, we focus on the approximate relaxation, where there is still hope to obtain labels of polylogarithmic length.}\n\\mtodo{Same comment for this paragraph. Also, there is a recent paper about FT exact distance labels in planar graphs \\cite{DBLP:journals\/corr\/abs-2102-07154} that we should probably mention, they show that any directed weighted planar\ngraph admits fault-tolerant distance labels of size $O(n^{2\/3})$. There are also some references cited in their paper that maybe we should discuss as well, for example see the paragraph ``Forbidden-set distance labeling schemes'' in page 2 here: https:\/\/arxiv.org\/pdf\/2102.07154.pdf} \\mertodo{I actually preferred not open up the discussion on special graph families, e.g., planar graphs, graphs with bounded dimension, etc. We do mention it for the direct setting of routing or labeling, but I do not think it should be mentioned for oracles for the following reason. For labels, there was no prior work for general graphs, but for oracles we do have such works so no need to add the extra overhead of special graph families.} \n\n\\paragraph{Fault-Tolerant Routing.} A desirable requirement in most communication networks is to provide efficient routing protocols in the presence of faults. Specifically, an $f$-FT routing protocol is a distributed algorithm that, for any set of at most $f$ faulty edges $F$, allows a vertex $s$ to route a message to a destination vertex $t$ along an approximate $s$-$t$ shortest path in $G \\setminus F$ (without knowing $F$ in advance). The routing scheme consists of two algorithms: (i) a preprocessing algorithm which computes (succinct) routing tables and labels for each vertex in the graph; and (ii) a routing algorithm that given the received message and the routing table of vertex $v$ determines the next-hop (specified as a port number) on the $v$-$t$ (approximate) shortest path in $G \\setminus F$. The efficiency of the scheme is determined by the tradeoff between: \n\\begin{enumerate}[noitemsep]\n\\item the \\emph{stretch} of the route, i.e., the ratio between the length of the route to the $s$-$t$ distance in $G \\setminus F$. \n\\item the \\emph{space} of the routing tables, routing labels and messages. \n\\end{enumerate}\nWhile the stretch vs. space tradeoff of routing schemes is fully understood in the non-faulty setting, the corresponding bounds in the FT setting are still far from optimal. \\mtodo{Also the part that starts here can fit better in a related work section. Also, a reviwer suggested to discuss also \\cite{rajan2012space}. It seems that this work appeard after Chechik et al., and focus on the case of a single edge failure. They show (Theorem 1) a routing scheme with routing tables of size $\\widetilde{O}(k \\deg(v)+ n^{1\/k})$ size per vertex, $O(k^2)$ stretch and $O(k+\\log{n})$ size header that handle a failure of one edge.} \\textbf{MP: The first formalization of FT routing schemes was given by the influential works of Dolev \\cite{dolev1984new} and Peleg \\cite{peleg1987fault}. These earlier works presented the first non-trivial solutions for general graphs supporting at most $\\lambda$ faulty edges, where $\\lambda$ is the edge-connectivity of the graph. Their routing labels had linear size, providing $s$-$t$ routes of possibly linear length (even in cases where the surviving $s$-$t$ path is of $O(1)$ length). In competitive FT routing schemes, it is required to provide $s$-$t$ routes of length that competes with the shortest $s$-$t$ path in $G \\setminus F$, even in cases where $G \\setminus F$ is not connected. Competitive FT routing schemes \\cite{peleg2009good} for general graphs were given by Chechik et al. \\cite{ChechikLPR10,chechik2012f} for the special case of $f\\leq 2$ faults. \nSpecifically, for a given stretch parameter $k$, they gave a routing scheme with a total space bound of $\\widetilde{O}(n^{1+1\/k})$ bits, polylogarithmic-size labels and messages, and a routing \\emph{stretch} of $O(k)$. \nThis scheme was extended later on for any $f$ by Chechik \\cite{chechik2011fault}, at the cost of increasing the routing stretch to $O(f^2(f+\\log^2 n)k)$.}\nSo far, in all these prior schemes, the space of the individual routing tables could be linear in the worst case, even when allowing a large stretch bound. This is in strike contrast to the standard (non-faulty) compact routing schemes, e.g., by Thorup and Zwick \\cite{thorup2001compact}, which provide each vertex a table of $\\widetilde{O}(n^{1\/k})$ bits, while guaranteeing a route stretch of $2k-1$. The current large gap in the quality of FT routing schemes compared to their non-faulty counterparts leads to the following question.\n\n\\begin{question}\\label{q:route}\nIs it possible to design $f$-fault-tolerant compact routing scheme for general graphs with \\emph{sub-linear} table size and with a sub-logarithmic stretch?\n\\end{question}\n\n\\textbf{MP: A more relaxed setting of FT routing scheme which has been studied in the literature is given by the \\emph{forbidden set routing schemes}, introduced by Courcelle and Twigg \\cite{CourcelleT07}. In that setting, it is assumed that the routing protocol knows in advance the set of faulty edges $F$. In contrast, in the FT routing setting, the failing edges are a-priori unknown to the routing algorithm,  and can only be detected upon arriving one of their endpoints. Forbidden set routing schemes have been devised to the same class of restricted graph families as obtained for the forbidden set labeling setting \\cite{CourcelleT07,AbrahamCGP16,abraham2012fully}.}\n\n\n\n\\input{result.tex}\n\n\n\n\\subsection{Connectivity Labels Based on Graph Sketches}\\label{sec:ftconn-sketch}\nIn this section, we show the following:\n\\begin{theorem}\nFor every undirected $n$-vertex graph $G=(V,E)$, a positive integer $f$, there is a randomized $f$-FT connectivity labels $\\mathsf{ConnLabel}\\xspace_{G}: V \\cup E \\to \\{0,1\\}^{\\ell}$ of length $\\ell=O(\\log^3 n)$ bits. The decoding time of the scheme is $\\widetilde{O}(f)$, and the computation time for assigning the labels is $\\widetilde{O}(m+n)$.\n\\end{theorem}\nIn Section \\ref{sec:label-alg}, we present the labeling algorithm which assigns labels based on the notion of graph sketches. In Section \\ref{sec:dec-alg} we present the decoding algorithm that given the label information determines if $s$ and $t$ are connected in $G \\setminus F$. When the graph $G$ is clear from the context, we may omit it and simply write $\\mathsf{ConnLabel}\\xspace$. \n\n\n\n\\subsubsection{The Labeling Algorithm}\\label{sec:label-alg}\nGiven a connected graph $G$, let $T$ be an arbitrary rooted spanning tree in $G$ that is used throughout this section. In our future applications of this labeling scheme (e.g., routing), both the graph $G$ and the tree $T \\subseteq G$ will be given as input to the labeling algorithm. In the latter case, we denote the output labels by $\\mathsf{ConnLabel}\\xspace_{G,T}$. Throughout, all vertices have unique ids $\\operatorname{ID}(v)$ between $\\{1,\\ldots,n \\}$.  \n\n\\paragraph{Extended Edge Identifiers.} In our algorithm it is important to distinguish between an identifier of a single edge to the bitwise XOR of several edges. For this purpose, we define for each edge $e$ an extended edge identifier $\\operatorname{EID}_T(e)$ that allows distinguishing between these cases, and serves as the identifier of the edge.\nThe extended edge identifier $\\operatorname{EID}_T(e)$ consists of a (randomized) unique distinguishing identifier $\\operatorname{UID}(e)$, as well as additional tree related information that facilitates the decoding procedure. The computation of $\\operatorname{UID}(e)$ is based on the notion of $\\epsilon$-\\emph{bias} sets \\cite{naor1993small}. The construction is randomized and guarantees that, w.h.p., the XOR of the $\\operatorname{UID}$ part of each given subset of edges $S \\subseteq E$, for $|S|\\geq 2$, is not a legal $\\operatorname{UID}$ identifier of any edge.\nLet $\\mathsf{XOR}\\xspace(S)$ be the bitwise XOR of the extended identifiers of edges in $S$, i.e., $\\mathsf{XOR}\\xspace(S)=\\oplus_{e \\in S} \\operatorname{EID}_T(e)$. In addition, let $\\mathsf{XOR}\\xspace_U(S)=\\oplus_{e \\in S} \\operatorname{UID}(e)$. Missing proofs are deferred to Appendix \\ref{sec:miss-proof}.\n\n\\begin{lemma}[Modification of Lemma 2.4 in \\cite{GhaffariP16}]\n\\label{cl:epsbias}\nThere is an algorithm that creates a collection $\\mathcal{I}=\\{\\operatorname{UID}(e_1), \\ldots, \\operatorname{UID}(e_{M})\\}$ of $M=\\binom{n}{2}$ random identifiers for all possible edges $(u,v)$, each of $O(\\log n)$-bits using a seed $\\mathcal{S}_{ID}$ of $O(\\log^2 n)$ bits. These identifiers are such that for each subset $E' \\subseteq E$, where $|E'|\\neq 1$, we have $\\Pr[\\mathsf{XOR}\\xspace_U(E') \\in \\mathcal{I}] \\leq 1\/n^{10}$. In addition, given the identifiers $\\operatorname{ID}(u), \\operatorname{ID}(v)$ of the edge $e=(u,v)$ endpoints, and the seed $\\mathcal{S}_{ID}$, one can determine $\\operatorname{UID}(e)$ in $\\widetilde{O}(1)$ time.\n\\end{lemma}\n\\def\\APPENDUNIQUEID{\n\\begin{proof}[Proof of Lemma \\ref{cl:epsbias}]\nThe lemma is proved in \\cite{GhaffariP16}, the only part that is not discussed there is the time to determine $\\operatorname{UID}(e)$ that follows from \\cite{naor1993small}. \nBy Theorem 3.1 of \\cite{naor1993small}, given the seed $\\mathcal{S}_{ID}$ and the edge identifier $e_j=(\\operatorname{ID}(u), \\operatorname{ID}(v))$, determining the $i^{th}$ bit of $\\operatorname{UID}(e_{j})$ can be done in $O(\\log n)$ time. Thus, determining all $O(\\log n)$ bits, takes $O(\\log^2 n)$ time. \n\\end{proof}\n\nFor every vertex $v \\in G$, let $\\mathsf{ANC}\\xspace_T(v)$ be the ancestor label of $v$ computed for the given tree $T$ using Lemma \\ref{anc_labels}. The extended identifier $\\operatorname{EID}_T(e)$ is given by\n\\begin{equation}\\label{eq:extend-ID}\n\\operatorname{EID}_T(e)=[\\operatorname{UID}(e), \\operatorname{ID}(u), \\operatorname{ID}(v), \\mathsf{ANC}\\xspace_T(u), \\mathsf{ANC}\\xspace_T(v)]~.\n\\end{equation}\nThe identifiers of $\\operatorname{ID}(u), \\operatorname{ID}(v)$ are used in order to verify the validity of the unique identifier $\\operatorname{UID}(e)$.  \nWhen the tree $T$ is clear from the context, we might omit it and simply write $\\operatorname{EID}(e)$. As we will see, the labeling scheme will store the seed $\\mathcal{S}_{ID}$ as part of the labels of the tree edges. \n\n\n\\paragraph{Fault-Tolerant Labels via Graph Sketches.} \nGraph sketches are a tool to identify outgoing edges. We start by providing an intuition for them. Say that $S$ is a connected component, and that there are $2^j$ edges outgoing from $S$. If we sample all edges in the graph with probability $1\/2^j$, there is a constant probability that exactly one outgoing edge from $S$ is sampled, and our goal is to find it using local information stored at the vertices of $S$. This information is the \\emph{sketch}. \nThe sketch of each vertex stores the bitwise XOR of sampled edges adjacent to it. Now looking at the XOR of all the sketches of vertices of $S$ allows to detect an outgoing edge. This holds as any sampled edge that has both endpoints in $S$ gets cancelled out, and we are left with the XOR of sampled edges outgoing from $S$. If there is exactly one outgoing edge, we find it. To increase the success probability we can repeat the process $O(\\log{n})$ times. We define sets of vertices $E_{i,j}$, where for $i \\in \\{1, \\ldots, c \\log n\\}$, the set $E_{i,j}$ is obtained by sampling each edge with probability $2^{-j}$. Since we repeat the process $O(\\log{n})$ times for each $j$, then w.h.p we can use the sketches to identify outgoing edge from any component. To use this approach in our context, it is crucial to be able to simulate the sampling process using a small random seed. To do this, we follow \\cite{DuanConnectivityArxiv16,DuanConnectivitySODA17} and use pairwise independent hash functions to decide whether to include edges in sampled sets.\nWe choose $L=c\\log n$ \npairwise independent hash functions $h_1, \\ldots, h_{L}:\\{0,1\\}^{\\Theta(\\log n)} \\to \\{0, \\ldots, 2^{\\log m}-1\\}$,\nand for each $i \\in \\{1, \\ldots, L\\}$ and $j \\in [0,\\log m]$, define the edge set \n$$E_{i,j} =\\{ e \\in E ~\\mid~ h_i(e) \\in [0,2^{\\log m-j})\\}~.$$ \nEach of these hash functions can be defined using a random seed of logarithmic length \\cite{TCS-010}. Thus, a \nrandom seed $\\mathcal{S}_h$ of length $O(L \\log n)$ can be used to determine the collection of all these $L$ functions. As observed in \\cite{DuanConnectivityArxiv16,GibbKKT15}, pairwise independence is sufficient to guarantee that for any set $E' \\subset E$ and any $i$, there exists an index $j$, such that with constant probability $\\mathsf{XOR}\\xspace(E' \\cap E_{i,j})$ is the name (extended identifier) of one edge in $E'$, for a proof see Lemma 5.2 in  \\cite{GibbKKT15}.\n\\begin{lemma}\\label{lem:hitting-pairwise}\nFor any edge set $E'$ and any $i$, with constant probability there exists a $j$ satisfying that $|E' \\cap E_{i,j}|=1$.\n\\end{lemma}\n\n\nWe also need to be able to tell that a bit string of $\\mathsf{XOR}\\xspace(E' \\cap E_{i,j})$ is a legal edge ID or not. Here we exploit the extended ids. See Appendix \\ref{sec:miss-proof} for a proof.\n\\begin{lemma} \\label{lemma_unique}\nGiven the seed $\\mathcal{S}_{ID}$, one can determine in $\\widetilde{O}(1)$ time if $\\mathsf{XOR}\\xspace(E' \\cap E_{i,j})$ corresponds to a single edge ID in $G$ or not, w.h.p.\n\\end{lemma}\n\\def\\APPENDLEMMUNIQUE{\n\\begin{proof}[Proof of Lemma \\ref{lemma_unique}]\nLet $X=\\mathsf{XOR}\\xspace(E' \\cap E_{i,j})$. Letting $E''=E' \\cap E_{i,j}$, then $X$ can be written as the concatenation of $\\mathsf{XOR}\\xspace_1(E'')$ and $\\mathsf{XOR}\\xspace_2(E'')$, where $\\mathsf{XOR}\\xspace_1(E'')=\\mathsf{XOR}\\xspace_U(E'')$ is the bit-wise XOR of the unique identifiers $\\operatorname{UID}(e)$ for $e \\in E''$ and $\\mathsf{XOR}\\xspace_2(E'')$ is the bit-wise XOR of the remaining information in the extended identifiers of $E''$.  We now show how using the seed and $\\mathsf{XOR}\\xspace_2(E'')$, one can test the validity of $\\mathsf{XOR}\\xspace_1(E'')$.\nThe algorithm detects the case that $|E''| \\geq 2$ as follows. First, in the case that $E''$ is a single edge, $\\mathsf{XOR}\\xspace_2(E'')$ should contain legal ids $\\operatorname{ID}(u),\\operatorname{ID}(v)$. If this is not the case, it follows that $|E''| \\neq 1$. If $\\mathsf{XOR}\\xspace_2(E'')$ contains legal ids $\\operatorname{ID}(u),\\operatorname{ID}(v)$, we use them and the seed $\\mathcal{S}_{ID}$ to determine $\\operatorname{UID}(e)$ for $e = (u,v)$, and we check if $\\mathsf{XOR}\\xspace_1(E'')=\\operatorname{ID}_1(e)$. We have two options, either $E'' = \\{e\\}$ is the single edge $e$, in which case $\\mathsf{XOR}\\xspace_U(E'')=\\operatorname{UID}(e) \\in \\mathcal{I}$, and the verification succeeds. Otherwise $|E''| \\geq 2$, in which case, from Lemma \\ref{cl:epsbias}, $\\Pr[\\mathsf{XOR}\\xspace_U(E'') \\in \\mathcal{I}] \\leq 1\/n^{10}$, hence w.h.p $\\mathsf{XOR}\\xspace_U(E'') \\neq \\operatorname{UID}(e) \\in \\mathcal{I}$ and we identify that $|E''| \\geq 2$.\n\\end{proof}\n\n\nFor each vertex $v$ and indices $i,j$, let $E_{i,j}(v)$ be the edges incident to $v$ in $E_{i,j}$. \nThe $i^{th}$ \\emph{basic sketch unit} of each vertex $v$ is then given by:\n\\begin{equation}\n\\label{eq:vsketch}\n\\mathsf{Sketch}\\xspace_{G,i}(v)=[\\mathsf{XOR}\\xspace(E_{i,0}(v)),\\ldots,\\mathsf{XOR}\\xspace(E_{i,\\log m}(v))].\n\\end{equation}\nThe sketch of each vertex $v$ is defined by a concatenation of $L=\\Theta(\\log n)$ basic sketch units: \n$$\\mathsf{Sketch}\\xspace_G(v)=[\\mathsf{Sketch}\\xspace_{G,1}(v),\\mathsf{Sketch}\\xspace_{G,2}(v), \\ldots\\mathsf{Sketch}\\xspace_{G,L}(v)]~.$$ \nFor every subset of vertices $S$, let \n$\\mathsf{Sketch}\\xspace_G(S)=\\oplus_{v \\in S}\\mathsf{Sketch}\\xspace_G(v).$ When the graph $G$ is clear from the context, we may omit it and write $\\mathsf{Sketch}\\xspace_{i}(v)$ and $\\mathsf{Sketch}\\xspace(v)$. \n\nWe are now ready to define the fault-tolerant connectivity labels of vertices and edges. \nThe label of each vertex $u$ is given by:\n\\begin{equation}\\label{eq:conn-vertex}\n\\mathsf{ConnLabel}\\xspace_{G,T}(u)=\\langle \\mathsf{ANC}\\xspace_T(u), \\operatorname{ID}(u) \\rangle~,\n\\end{equation}\nwhere $\\mathsf{ANC}\\xspace_T(u)$ is the ancestry label of $u$ with respect to the tree $T$. \nFor every $u \\in V(T)$, let $T_u$ be the subtree rooted at $u$.  The label $\\mathsf{ConnLabel}\\xspace_{G,T}(e)$ of each \\emph{edge} $e=(u,v)$ is given by:\n\\begin{equation*}\n    \\mathsf{ConnLabel}\\xspace_{G,T}(e)=\n    \\begin{cases}\n      \\langle \\operatorname{EID}_T(e), \\mathsf{Sketch}\\xspace(V(T_u)), \\mathsf{Sketch}\\xspace(V(T_v)), \\mathsf{Sketch}\\xspace(V), \\mathcal{S}_{ID}, \\mathcal{S}_h\\rangle ,& \\mbox{~for~} e \\in T \\\\\n     \\langle \\operatorname{EID}_T(e) \\rangle,& \\mbox{~Otherwise}.\n    \\end{cases}\n\\end{equation*}\n\nWe complete this subsection by bounding the label size and computation time of the labeling algorithm. For proofs see Appendix \\ref{sec:miss-proof}. \n\\begin{claim}\\label{cl:label-length}\nThe label length is $O(\\log^3 n)$ bits.\n\\end{claim}\n\\def\\APPENDLABELCONSISE{\n\\begin{proof}[Proof of Claim \\ref{cl:label-length}]\nThe label size is dominated by the sketching information $\\mathsf{Sketch}\\xspace(V(T_u))$, which is made of a concatenation of the bitwise XOR of $O(\\log n)$ basic sketch units $\\mathsf{Sketch}\\xspace_i(u)$. By Eq. (\\ref{eq:vsketch}), each unit has $O(\\log^2 n)$ bits, and thus overall, the label has $O(\\log^3 n)$ bits.\n\\end{proof}\n\n\nWe show that assigning the labels takes  $\\widetilde{O}(m+n)$ time.\n\\begin{claim}\\label{cl:time-conn-labelsketch}\nThe time complexity of the labeling algorithm is $\\widetilde{O}(m+n).$\n\\end{claim}\n\\def\\APPENDCONNLABELSKETCH{\n\\begin{proof}[Proof of Claim \\ref{cl:time-conn-labelsketch}]\nTo compute the labels of vertices we assign ids to vertices in $O(n)$ time, and compute ancestry labels in $O(n)$ time using Lemma \\ref{anc_labels}. To compute the extended identifiers $\\operatorname{EID}_T(e)$, we also choose the random seed $\\mathcal{S}_{ID}$ and compute $\\operatorname{UID}(e)$ using Lemma \\ref{cl:epsbias}, this takes $\\widetilde{O}(1)$ time per edge, and $\\widetilde{O}(m)$ time for all edges. Lastly, we should compute the sketch values $\\mathsf{Sketch}\\xspace(V(T_u))$. For this, first, we choose the random seed $\\mathcal{S}_h$, and compute the values $\\mathsf{Sketch}\\xspace_G(v)$. For this, we should identify for each vertex the adjacent edges in $E_{i,j}$. For each edge we can identify the sets it belongs to in $\\widetilde{O}(1)$ time using Fact \\ref{fc:pairwise}. This allows us computing the sketch values of all vertices in $\\widetilde{O}(m+n)$ time. We can then compute the values $\\mathsf{Sketch}\\xspace(V(T_u))$ by scanning the tree in $\\widetilde{O}(n)$ time.   \n\\end{proof}\n\nFinally, the subsequent decoding algorithm will be based on the following useful property of the graph sketches, stored by our labels. \n\\begin{lemma}\\label{lem:sketch-property}\nFor any subset $S$, given one basic sketch unit $\\mathsf{Sketch}\\xspace_i(S)$ and the seed $\\mathcal{S}_{ID}$ one can compute, with constant probability, an outgoing edge $E(S, V \\setminus S)$ if such exists. The complexity is $\\widetilde{O}(1)$ time.\n\\end{lemma}\n\\def\\APPENDSKETCHPROP{\n\\begin{proof}[Proof of Lemma \\ref{lem:sketch-property}]\nThe proof follows from Lemma \\ref{lem:hitting-pairwise}. Note that by definition of the sketch values $\\mathsf{Sketch}\\xspace_i(S)=\\oplus_{v \\in S}\\mathsf{Sketch}\\xspace_i(v)=[\\mathsf{XOR}\\xspace(E_{i,0}(S)),\\ldots,\\mathsf{XOR}\\xspace(E_{i,\\log m}(S))],$ where $E_{i,j}(S)$ are the outgoing edges from $S$ in $E_{i,j}$ (edges that have both endpoints in $S$ are cancelled out by the XOR operation). Let $E'$ be all the outgoing edges from $S$. From Lemma \\ref{lem:hitting-pairwise}, with constant probability there exists a $j$ such that $|E' \\cap E_{i,j}|=1$. In this case, $\\mathsf{XOR}\\xspace(E_{i,j}(S))$ corresponds to an extended id of a single outgoing edge from $S$. We can check if this happens in $\\widetilde{O}(1)$ time using Lemma \\ref{lemma_unique}.   \n\\end{proof}\n\n\n\n\n\\subsubsection{The Decoding Algorithm} \\label{sec:dec-alg}\nWe next describe the decoding algorithm where given a triplet $s,t, F \\in V \\times V \\times E^f$ along with their labels, it determines whether $s$ and $t$ are connected in $G\\setminus F$, w.h.p. \nThe decoding algorithm has four key steps: The first step identifies the at most $f+1$ components $\\mathcal{C}_0=\\{C_1,\\ldots, C_\\ell\\}$ of $T \\setminus F$, as well as the components of $s$ and $t$ in $\\mathcal{C}_0$. The second step uses the label information to compute the sketch value $\\mathsf{Sketch}\\xspace(C_i)$ of each component $C_i \\in \\mathcal{C}_0$. The third step modifies this sketch information into $\\mathsf{Sketch}\\xspace_{G \\setminus F}(C_i)$, by subtracting the information related to the faulty edges. The forth and final step uses the sketch information in order to simulate $L=O(\\log n)$ steps of the Boruvka algorithm. At the end of these steps, the decoding algorithm identifies the connected components of both $s$ and $t$ in $G \\setminus F$. In the case where $s$ and $t$ are indeed connected in $G \\setminus F$, the algorithm also outputs a succinct representation of an $s$-$t$ path in $G \\setminus F$. This extra information would be used later on by our compact routing scheme. We next describe these steps in details. \n\n\\paragraph{Step 1: Identification of the connected components $\\mathcal{C}_0$ in $T \\setminus F$.} \nLet $F_T=F \\cap T$ be the faulty tree edges and let $F_{NT}=F \\setminus F_T$ be the faulty non-tree edges. Let $Q=\\{s,t\\} \\cup V(F_T)$.  Each component $C_i$ of $T \\setminus F$ will be identified by the maximum vertex ID in $C_i \\cap V(F_T)$. Note that in the case where $F_T=\\emptyset$, $T \\setminus F=T$ and thus $s$ and $t$ are connected iff $s,t \\in V(T)$.  From now on, we therefore assume that $F_T \\neq \\emptyset$. \n\nWe next show that although we do not have full information about the tree $T$ and the vertices of each connected component, the ancestry labels of $V(F_T)$ give us enough information to identify the connected components of $T \\setminus F$. Additionally, given an ancestry label of a vertex $u$, we can identify the connected component of $u$. To obtain this, it is helpful to look at the \\emph{component tree} that is obtained by contracting each connected component of $T \\setminus F$ to one vertex, as follows. Let $\\ell = |F_T|+1.$ The component tree $T_C = (\\mathcal{C}_0, E_C)$ is a tree of $\\ell$ vertices representing the connected components in $T \\setminus F$, and $|F_T|=\\ell-1$ edges corresponding to the edges of $F_T$. There is an edge $(C_i,C_j) \\in E_C$ iff there is an edge $(u,v) \\in F_T$ where $u \\in C_i, v \\in C_j$. See Figure \\ref{componentTreePic} for an illustration.  \n\n\n\\setlength{\\intextsep}{0pt}\n\\begin{figure}[h]\n\\centering\n\\setlength{\\abovecaptionskip}{-2pt}\n\\setlength{\\belowcaptionskip}{6pt}\n\\includegraphics[scale=0.55]{componentTree.pdf}\n \\caption{Illustration of the component tree where $F=\\{e_1,e_2,e_3,e_4\\}$. Each connected component of $T \\setminus F$ is contracted to one vertex on the right.}\n\\label{componentTreePic}\n\\end{figure}\n\nWe can construct the tree $T_C$ using the ancestry labels of the edges $F_T$.  For this, for each edge $e \\in F_T$ we just need to identify the set of edges from $F_T$ above $e$ in $T$. Moreover, for a given vertex $v$, its connected component is exactly determined by the set of edges in $F_T$ above it in $T$, which can again be identified using the ancestry labels of $v \\cup V(F_T)$. In particular, we can identify the connected components of $s$ and $t$. The component tree can be constructed in $O(f^2)$ time by checking for any pair of edges $e,e' \\in F_T$, if $e$ is above $e'$ in the tree. We next show a faster algorithm taking only $\\widetilde{O}(f)$ time by exploiting properties of the ancestry labels. Moreover, we show that the component of each vertex can be identified in $O(\\log{f})$ time.\n\n\\begin{claim} \\label{claim_component_tree}\nThe component tree can be constructed in $O(f \\log{f})$ time. Additionally, given $\\mathsf{ANC}\\xspace_T(v)$, we can identify the connected component of $v$ in $T \\setminus F$ in $O(\\log{f})$ time.\n\\end{claim}\n\n\\begin{proof}\nOur algorithm uses ancestry labels based on DFS from \\cite{kannan1992implicat}. In this scheme, the label of each vertex $v$ is composed of two numbers $(DFS_1(v),DFS_2(v))$ that represent the first and last times a DFS scan of the tree visits $v$. A vertex $u$ is an ancestor of a vertex $v$ iff the interval $(DFS_1(u),DFS_2(u))$ contains the interval $(DFS_1(v),DFS_2(v))$. To build the component tree, we sort the labels of $V(F_T)$, as described next. First, for each component $C \\in T \\setminus F$, we use the highest vertex in the component to represent the component. For the highest component, this is the root $r$. For any other component, we have that the highest vertex of the component, $v$, is in $V(F_T)$. This holds as the edge connecting $v$ to its parent $p(v)$ is necessarily in $F_T$ (otherwise, $v$ is not the highest vertex in its component), see Figure \\ref{componentTreePic} for illustration. Hence, for any edge $(v,p(v)) \\in F_T$, we have that the vertex $v$ represents one component (we can identify which of the vertices is the parent using the ancestry labels). Hence, we have $|F_T|+1$ vertices $v_i$ representing the components $C_i$ of the component tree, and we also know the ancestry labels $(DFS_1(v_i),DFS_2(v_i))$ of all vertices $v_i$, except $r$. For $r$ we can use the label $(1,M)$ where $M$ is a number greater than all values $DFS_2(v_i)$ of other vertices. We next use these labels to determine the structure of the component tree.\nFor this, we create for each vertex $v_i$ two tuples: $(DFS_1(v_i),v_i,1),(DFS_2(v_i),v_i,2)$, and we sort the $2(|F_T|+1)$ tuples according to their first coordinate. This takes $O(f \\log{f})$ time. We next scan the sorted list, and when we reach the tuple $(DFS_1(v_i),v_i,1)$, we identify the parent of $v_i$ in the component tree, as follows. The first tuple is $(1,r,1)$ and $r$ is set to be the root of the component tree. For a vertex $v_i \\neq r$, we identify its parent when we reach $(DFS_1(v_i),v_i,1)$. Let $(DFS_b(u),u,b)$ be the last tuple before $(DFS_1(v_i),v_i,1)$ in the sorted order. If $b=1$, then $u$ is the parent of $v_i$ in the component tree. If $b=2$, let $w$ be the parent of $u$ in the component tree, then $w$ is also the parent of $v$ in the component tree. Additionally, $w$ was already computed as $(DFS_1(u),u,1)$ appears before $(DFS_1(v_i),v_i,1)$. Hence, we can find the parent of $v$ in $O(1)$ time using the tuple before it. Scanning the list takes $O(f)$ time, and after it we know for each component its parent in the component tree, which gives the complete structure of the tree. We next prove the correctness of the algorithm.  \n\nWe first discuss the case that $b=1$. Here $(DFS_1(u),u,1)$ is the last tuple before $(DFS_1(v_i),v_i,1)$. This means that $u$ is necessarily an ancestor of $v$, because the entry $(DFS_1(v_i),v_i,1)$ is between the entries $(DFS_1(u),u,1)$ and $(DFS_2(u),u,2)$, and the DFS scan traverses exactly the subtree of $u$ in the time interval $(DFS_1(u),DFS_2(u))$, implying that $v_i$ is a child of $u$. Moreover, this is the closest ancestor to $v_i$ among the vertices $\\{v_1,v_2,...,v_{\\ell}\\} \\setminus \\{v_i\\}$, as the DFS scan traverses the ancestors of $v_i$ from the highest to the lowest. It follows that $u$ represents the closest component $C$ above $v_i$ in the component tree, as needed.  \n\nWe next discuss the case that $b=2$. Here $(DFS_2(u),u,2)$ is the last tuple before $(DFS_1(v_i),v_i,1)$. Note that now $u$ is not an ancestor of $v_i$, as the DFS scan finished scanning the subtree of $u$ before reaching $v_i$, but we claim that $u$ and $v_i$ have the same parent in the component tree. For this, we show they have exactly the same ancestors in the set $\\{v_1,v_2,...,v_{\\ell}\\} \\setminus \\{u,v_i\\}.$ For any ancestor $w\\neq u$ of $u$, we have that $DFS_1(w) < DFS_1(u) < DFS_2(u) < DFS_2(w)$. As $(DFS_1(v_i),v_i,1)$ is the first tuple after $(DFS_2(u),u,2)$, it must hold that $DFS_1(w) < DFS_1(v_i) <  DFS_2(w)$, implying that $v_i$ is a child of $w$ as needed. Similarly, any ancestor $w \\neq v_i$ of $v_i$ is also an ancestor of $u$, as we have $DFS_1(w) < DFS_2(u) < DFS_1(v_i) < DFS_2(v_i) < DFS_2(w)$.\nHence, the parent of $u$ in the component tree is also the parent of $v_i$ in the component tree, as needed.  \n \nLastly, we show that using similar ideas we can also identify the component of a vertex $v$ in $T \\setminus F$. We create for $v$ the tuple, $(DFS_1(v),v,1)$, and use binary search to find the last tuple smaller or equal to it in the sorted list we computed before, denote it by $(DFS_b(u),u,b)$. \nIf $b=1$ then $v$ is in the component of $u$, and else it is in the component of the parent of $u$ (that was computed before). The complexity of the binary search is $O(\\log{f})$, we next prove correctness. \nOne special case is that $v$ is a root of one of the components in the component tree. In this case, the entry $(DFS_b(u),u,b)$ we find is equal to $(DFS_1(v),v,1)$, and $u=v$ is indeed the component of $v$. Otherwise, $v$ is an internal vertex in its component, and the root of the component is the closest ancestor to $v$ in $\\{v_1,...,v_{\\ell}\\}$.\nIf $b=1$, then as shown before, $u$ is the closest ancestor to $v$ in the component tree, as needed. If $b=2$, then as shown before, $u$ is not an ancestor of $v$, but has exactly the same ancestors in the component tree. Hence, the root $w$ of the component above $u$ is the root of the component of $v$, as needed.    \n\\end{proof} \n\n\n\\paragraph{Step 2: Computing the sketch values of each component $\\mathcal{C}_0$ in $G$.} \nFor each component $C_j \\in \\mathcal{C}_0$ the algorithm computes $\\mathsf{Sketch}\\xspace_G(C_j)$ using the sketch information of the vertices in $V(F_T)$.  The basic observation here is the following. Given $S' \\subset S$ and $\\mathsf{Sketch}\\xspace(S), \\mathsf{Sketch}\\xspace(S')$, it holds that $\\mathsf{Sketch}\\xspace(S \\setminus S')=\\mathsf{Sketch}\\xspace(S) ~\\oplus~ \\mathsf{Sketch}\\xspace(S')$. To compute the sketch values, first, we define for each component a temporary value $\\mathsf{Sketch}\\xspace'_G(C_j)$ as follows. Let $v_j$ be the highest vertex (closest to the root in $T$) in the component $C_j$. For the component of the root $r$, this is $r$. For any other component $C_j$, let $(C_j,p(C_j))$ be the edge connecting $C_j$ to its parent in the component tree. This edge corresponds to an edge $(v_j,p(v_j)) \\in F_T$, where $v$ is the highest vertex in $C_j$. We define $\\mathsf{Sketch}\\xspace'_G(C_j) = \\mathsf{Sketch}\\xspace_G(V(T_{v_j}))$.\nSince $(v_j,p(v_j)) \\in F_T$, the sketch information $\\mathsf{Sketch}\\xspace'_G(C_j)$ can be obtained from the label of the tree edge $(v_j,p(v_j))$. We also know the temporary sketch value of the component of $r$, as $\\mathsf{Sketch}\\xspace_G(V_{r})=\\mathsf{Sketch}\\xspace_G(V)$ is part of the labels of all tree edges (and we assume that $F_T \\neq \\emptyset$). We next use the temporary sketch values to compute the sketch values of components using the following claim.\n\n\\begin{claim}\nLet $C_j$ be a component in $T \\setminus F$. If $C_j$ is a leaf in the component tree, we have $\\mathsf{Sketch}\\xspace_G(C_j) = \\mathsf{Sketch}\\xspace'_G(C_j).$ Otherwise, let $D=\\{D_1,...,D_t\\}$ be the children of $C_j$ in the component tree and let $\\mathsf{Sketch}\\xspace'(D)=\\oplus_{1 \\leq i \\leq t} \\mathsf{Sketch}\\xspace'_G(D_i)$, then $\\mathsf{Sketch}\\xspace_G(C_j) = \\mathsf{Sketch}\\xspace'_G(C_j) \\oplus \\mathsf{Sketch}\\xspace'(D).$ \n\\end{claim}\n\n\\begin{proof}\nIt holds that $\\mathsf{Sketch}\\xspace_G(C_j) = \\oplus_{v \\in C_j} \\mathsf{Sketch}\\xspace_G(v)$. By definition, $\\mathsf{Sketch}\\xspace'_G(C_j)=\\mathsf{Sketch}\\xspace_G(V(T_{v_j})) = \\oplus_{v \\in V(T_{v_j})} \\mathsf{Sketch}\\xspace_G(v)$ is the XOR of sketches of all vertices in the subtree of $v_j$. As $v_j$ is the highest vertex in $C_j$, if $C_j$ is a leaf component in the component tree, then the vertices in $C_j$ are exactly the vertices in $T_{v_j}$, and the claim follows. Otherwise, the vertices in $C_j$ are all vertices in $T_{v_j}$ that are not contained in any component below $C_j$. Hence, to compute the value $\\mathsf{Sketch}\\xspace_G(C_j)$, we should subtract from $\\mathsf{Sketch}\\xspace_G(V(T_{v_j}))$ the sketch values of vertices in components below $C_j$. Let $D_1,\\ldots,D_t$ be the children of $C_j$ in the component tree, and let $u_1,\\ldots,u_t$ be the highest vertices in the components $D_1,\\ldots,D_t$, respectively. Any vertex that is in some component below $C_j$ is in exactly one of the subtrees $T_{u_1},\\ldots,T_{u_t}$. Hence the sketch value of vertices in components below $C_j$ equals $\\oplus_{1 \\leq i \\leq t} \\mathsf{Sketch}\\xspace_G(V(T_{u_i}))= \\oplus_{1 \\leq i \\leq t} \\mathsf{Sketch}\\xspace'_G(D_i)=\\mathsf{Sketch}\\xspace'(D)$. To conclude, we get $\\mathsf{Sketch}\\xspace_G(C_j)=\\mathsf{Sketch}\\xspace_G(V(T_{v_j})) \\oplus \\mathsf{Sketch}\\xspace'(D)=\\mathsf{Sketch}\\xspace'_G(C_j) \\oplus \\mathsf{Sketch}\\xspace'_G(D)$, as needed.\n\\end{proof}\n\nTo conclude, from the values $\\mathsf{Sketch}\\xspace'_G(C_j)$, we can easily compute the values $\\mathsf{Sketch}\\xspace_G(C_j)$. The complexity is $\\widetilde{O}(f)$, as for each component, the sketch $\\mathsf{Sketch}\\xspace'(C_j)$ participates in two computations, and we have at most $O(f)$ components and the sketches have poly-logarithmic size.\n\n\n\\paragraph{Step 3: Computing the sketch values of each component $\\mathcal{C}_0$ in $G \\setminus F$.} \nFor each faulty edge $e \\in F$ (both tree and non-tree edges), our goal is to subtract the sketch information of $e$ from the corresponding components of the endpoint of $e$. The step does not require the label information of the edges, and it would be sufficient to know only the seed $\\mathcal{S}_h$ that determines the sampling of edges into the sketches, and the extended identifier of the failing edges. Since $F_T \\neq \\emptyset$, the algorithm holds the seed $\\mathcal{S}_h$ (from the label of an edge $e \\in F_T$), and it has the extended identifiers of all edges in $F$ as part of their labels. \n\nUsing the extended identifier of the faulty edge $e=(u,v)$, one can determine in $O(\\log{f})$ time the components in $\\mathcal{C}_0$ to which its endpoints belong, from Claim \\ref{claim_component_tree}. Using the identifier $\\operatorname{EID}(e)$ and the seed $\\mathcal{S}_h$, one can determine all the indices of the sketch to which the edge $e$ was sampled in $\\widetilde{O}(1)$ time using Fact \\ref{fc:pairwise}.\nLetting $C_u, C_v$ be the components of $u$ and $v$ in $T \\setminus F$, respectively. If $C_u \\neq C_v$, then the values $\\mathsf{Sketch}\\xspace_G(C_u),\\mathsf{Sketch}\\xspace_G(C_v)$ are updated by XORing them with the matrix that contains the extended identifier $\\operatorname{EID}(e)$ in the relevant positions. The complexity is poly-logarithmic, as the matrix has poly-logarithmic size. In the case that $C_u=C_v$, as $e$ is an internal edge in the component, it is not part of $\\mathsf{Sketch}\\xspace_G(C_u)$, and there is no need to update the value. Overall, doing the computation for all edges in $F$ takes $\\widetilde{O}(f)$ time.\nFrom that point on, all sketches of the components $\\mathcal{C}_0$ can be treated as sketches that have been computed in $G \\setminus F$.  \n\n\n\\paragraph{Step 4: Simulating the Boruvka algorithm.} Finally, our goal is to determine the identifiers of the maximal connected components of $s$ and $t$ of $G \\setminus F$. The input to this step is the identifiers of the components $\\mathcal{C}_0=\\{C_{1}, \\ldots, C_k\\}$ in $T \\setminus F$, along with their sketch information in $G \\setminus F$. While the algorithm does not have information on the vertices of each component, it knows the component identifier of each vertex in $Q$. \n\nThe algorithm consists of $L=O(\\log n)$ phases of the Boruvka algorithm. Each phase $i \\in \\{1,\\ldots, L\\}$ will be given as input a partitioning $\\mathcal{C}_i=\\{C_{i,1}, \\ldots, C_{i,k_i}\\}$ of (not necessarily maximal) connected components in $G \\setminus F$.\nThese components are identified by an $O(\\log n)$ bit identifier, where for each vertex in $Q$, the algorithm receives its unique component identifier in  $\\mathcal{C}_i$. In addition, the algorithm receives the sketch information of the components $\\mathcal{C}_i$ in $G \\setminus F$. The output of the phase is a partitioning $\\mathcal{C}_{i+1}$, along with their sketch information in $G \\setminus F$ and the identifiers of the components for each vertex in $U$. A component $C_{i,j} \\in \\mathcal{C}_i$ is \\emph{growable} if it has at least one non-faulty outgoing edge to a vertex in $V \\setminus C_{i,j}$. That is, the component is growable if it is strictly contained in some maximal connected component in $G \\setminus F$. Letting $N_i$ denote the number of growable components in $\\mathcal{C}_i$, the output partitioning $\\mathcal{C}_{i+1}$ of the $i^{th}$ step guarantees that $N_{i+1}\\leq N_i \/2$ w.h.p. To obtain outgoings edges from the growable components in $\\mathcal{C}_i$, the algorithm uses the $i^{th}$ basic-unit sketch $\\mathsf{Sketch}\\xspace_i(C_{i,j})$ of each $C_{i,j} \\in \\mathcal{C}_i$. By Lemma \\ref{lem:sketch-property}, from every growable component in $\\mathcal{C}_i$, we get one outgoing edge $e'=(x,y)$ with constant probability. Using the extended edge identifier of $e'$ the algorithm can also detect the component $C_{i,j'}$ to which the second endpoint, say $y$, of $e'$ belongs using Claim \\ref{claim_component_tree}.\nThat label allows us to compute the component of $y$ in the initial partitioning $T \\setminus F$, i.e., the component $C_{0,q}$ of $y$ in $\\mathcal{C}_0$. Thus $y$ belongs to the unique component $C_{i,j'} \\in \\mathcal{C}_i$ that contains \n$C_{0,q}$. \n\n\n\nAs noted in prior works \\cite{ahn2012analyzing,kapron2013dynamic,DuanConnectivityArxiv16}, it is important to use fresh randomness (i.e., independent sketch information) in each of the Boruvka phases. The reason is that the cut query, namely, asking for a cut edge between $S$ and $V \\setminus S$, should not be correlated with the randomness of the sketches. Note that indeed the components of $\\mathcal{C}_i$ are correlated with the randomness of the first $(i-1)$ basic sketch units of the vertices. Thus, in phase $i$ the algorithm uses the $i^{th}$ basic sketch units of the vertices (which are independent of the other sketch units) to determine the outgoing edges of the components in $\\mathcal{C}_i$.\n\n\nThe algorithm then computes the updated sketches of the merged components. This is done by XORing over the sketches of the components in $\\mathcal{C}_i$ that got merged into a single component in \n$\\mathcal{C}_{i+1}$. In expectation, the number of growable components is reduced by factor $2$ in each phase. Thus after $O(\\log n)$ phases, the expected number of growable components is at most $1\/n^5$, and using Markov inequality, we conclude that w.h.p there are no growable components. The final partitioning $\\mathcal{C}_L$ corresponds w.h.p to the maximal connected components in $G \\setminus F$. The pair $s$ and $t$ are connected in $G \\setminus F$ only if the components $C_s,C_t$ of $s,t$ respectively in $T \\setminus F$ are connected in the final component decomposition.\nWe next show that the complexity of the algorithm is $\\widetilde{O}(f)$. This is also the decoding time of the whole algorithm, as all steps take $\\widetilde{O}(f)$ time, as discussed above. \n\n\\begin{claim}\\label{cl:complexity-step-four}\nThe complexity of step 4 is $\\widetilde{O}(f)$.\n\\end{claim}\n\n\\begin{proof\nThe algorithm has $O(\\log{n})$ phases, where in each phase the following is computed. First, given the sketch values of the current components we identify outgoing edges from the components. This takes $\\widetilde{O}(1)$ time per component from Lemma \\ref{lem:sketch-property}, and $\\widetilde{O}(f)$ time for all components, as we have at most $f+1$ components. Next, for each outgoing edge we identify the components it connects using its ancestry labels, this takes $\\widetilde{O}(1)$ time per edge using Claim \\ref{claim_component_tree}. Then, we merge components accordingly and compute the sketch values of the new components by XORing the sketch values of merged components. Overall this takes $\\widetilde{O}(f)$ time, as we have at most $O(f)$ merges. In more detail, we can use a union-find data structure to implement the merges, where every time we merge components we compute the sketch value of the new component. We also maintain for each original component in $T \\setminus F$ its current component in phase $i$, this allows us to learn the current components connected by an outgoing edge $e$. This information can be maintained as follows. Let $C$ be a component in $T \\setminus F$, and assume we know the component $C_{i,j}$ it belongs to at the beginning of phase $i$. After the merges of phase $i$, $C_{i,j}$ joins some component $C_{i+1,j'}$ of phase $i+1$. We can use the find operation to identify the id of the new component. Overall, we have $O(f)$ merges and $O(f)$ find operations to identify for each component $C \\in T \\setminus F$, the corresponding component $C_{i+1,j'}$ it belongs to, hence the complexity is bounded by $\\widetilde{O}(f)$. \n\\end{proof}\n\nFinally, we show that the decoding algorithm can be slightly modified to output a compressed encoding of an $s$-$t$ path in $G \\setminus F$, using $O(f\\log n)$ bits. This encoding is represented by an $s$-$t$ path $\\widehat{P}$ that has two type of edges, appearing in an alternate manner on  $\\widehat{P}$: $G$-edges and edges $e'=(u,v)$ such that the $u$-$v$ tree path is intact in $T \\setminus F$. See Figure \\ref{fig:succ-paths}. \n\\begin{lemma}\\label{lem:useful-recovery-edges}\nConsider a triplet $s,t,F$ such that $s$ and $t$ are connected in $G \\setminus F$. \nThe decoding algorithm can also output a set of at most $f$ recovery edges $R$ such $(T \\setminus F) \\cup R$ is a spanning tree. In addition, it outputs a labeled $s$-$t$ path $\\widehat{P}$ of length $O(f)$ that provides a succinct description of the $s$-$t$ path. The edges of $\\widehat{P}$ are labeled by $0$ and $1$, where $0$-labeled edges correspond to $G$-edges and $1$-labeled edges $e=(x,y)$ correspond to $x$-$y$ paths in $T \\setminus F$. \n\\end{lemma}\n\\begin{proof}\nLet $C_s, C_t$ be the components of $s$ and $t$ in the initial partitioning $\\mathcal{C}_0$. In Step $4$ of the decoding algorithm, the Boruvka algorithm is simulated up to the point that $C_s$ and $C_t$ are connected. Therefore, the algorithm has computed a path $P$ that connects the components $C_s$ and $C_t$. Each vertex on that path corresponds to a component in $\\mathcal{C}_0$, and each edge corresponds to an outgoing edge (discovered using the sketch information). Since $\\mathcal{C}_0$ has at most $f+1$ components, $|P|\\leq f+1$. \nEach such edge $e' \\in P$ corresponds to an edge in $G$. Let $e_1=(x_1,y_1),\\ldots, e_k=(x_k,y_k)$ be the $G$-edges corresponding to the edges of $P$ ordered from $C_s$ to $C_t$. Letting $y_0=s$ and $x_{k+1}=t$, we get that \n$y_i$ and $x_{i+1}$ belong to the same component in $\\mathcal{C}_0$, for every $i \\in \\{0,\\ldots, k\\}$. \nThe labeled path is given by $\\widehat{P}=[s, x_1,y_1, x_2, y_2, \\ldots y_k,t]$ where the edges $(y_i,x_{i+1})$ are labeled $1$ and the edges $(x_{i}, y_i)$ are labeled $0$. Each $0$-labeled edge is a real edge in $G$, and each $1$-labeled edge $(x_{i}, y_i)$ corresponds to a tree path $\\pi(x_i, y_i)$ in $T \\setminus F$. \n\\end{proof}\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.40]{suc-path.pdf}\n\\caption{\\sf Shown is a tree $T$ with faulty edges $e_1,\\ldots, e_4$. The $s$-$t$ path in $G \\setminus F$ is represented by the path $\\widehat{P}=[s,v_1]\\circ (v_1,v_2) \\circ [v_2,v_3] \\circ (v_3,r) \\circ [r,v_4] \\circ (v_4, v_5) \\circ [v_5,t]$. The recovery edges $(v_1,v_2), (v_3,r)$ and $(v_4, v_5)$ are shown in dashed lines.  \\label{fig:succ-paths}\n}\n\\end{center}\n\\end{figure}\n\\subsection{Connectivity Labels Based on Graph Sketches}\\label{sec:ftconn-sketch}\n\n\\mtodo{I'm not sure if we want to have the statement of the theorem with respect to a tree. Maybe only have a variant with a tree where it's needed (for the routing)}\n\nIn this section, we show the following:\n\\begin{theorem}\nFor every undirected $n$-vertex graph $G=(V,E)$ and a spanning tree $T \\subseteq G$, a positive integer $f$, there is a randomized $f$-FS connectivity labels $\\mathsf{ConnLabel}\\xspace_{G,T}: V \\cup E \\to \\{0,1\\}^{Q}$ of length $Q=O(\\log^3 n)$ bits. \n\\textbf{MP: add later the computation time of the scheme and the decoding time.}\n\\end{theorem}\nIn Section \\ref{sec:label-alg}, we present the labeling algorithm which assigns labels based on the notion of graph sketches. In Section \\ref{sec:dec-alg} we present the decoding algorithm that given the label information determines if $s$ and $t$ are connected in $G \\setminus F$. When the graph $G$ and the spanning tree $T$ are clear from the context, we may omit it and simply write $\\mathsf{ConnLabel}\\xspace$. \n\n\n\n\\subsubsection{The Labeling Algorithm}\\label{sec:label-alg}\nGiven a graph $G$, let $T$ be an arbitrary rooted tree in $G$ that is used throughout this section.\nFor any node $u \\in V(T)$, let $V_u$ be the subset of nodes in the subtree of $T$ rooted at $u$. The algorithm starts by computing ancestry labels $\\mathsf{ANC}\\xspace(u)$ for all nodes $u$ given the tree $T$ using Lemma \\ref{anc_labels}. Additionally, we assign to all vertices unique ids $\\operatorname{ID}(v)$ between $\\{1,...,n \\}.$\n\n\\paragraph{Extended Edge Identifiers.} Each edge $e$ in $G$ is assigned to a unique identifier $ID_T(e)$ that is made of two parts: a \\emph{distinguishing} part $ID_{1,T}(e)$ (that does not depend on $T$) and a \\emph{logical part} $ID_{2,T}(e)$ that depends on $T$; each of $O(\\log n)$ bits. The distinguishing part $ID_{1,T}(e)$ is defined in a way that guarantees that the XOR of several identifiers does not correspond to a legal identifier of any edge $e \\in G$ w.h.p. The logical part $ID_{2,T}(e)$ contains auxiliary information that aids the decoding algorithm. \nThe computation of the distinguishing parts $ID_{1,T}(e)$ is based on the notion of $\\epsilon$-\\emph{bias} sets \\cite{naor1993small}. The construction is randomized and guarantees that, w.h.p., the XOR of the $ID_{1,T}$ part of each given subset of edges $S \\subseteq E$, for $|S|\\geq 2$, is not a legal $ID_{1,T}$ identifier of any edge.\nLet $\\mathsf{XOR}\\xspace(S)$ be the bitwise XOR of the extended identifiers of edges in $S$, i.e., $\\mathsf{XOR}\\xspace(S)=\\oplus_{e \\in S} \\operatorname{ID}_T(e)$. In addition, let $\\mathsf{XOR}\\xspace_1(S)=\\oplus_{e \\in S} \\operatorname{ID}_{1,T}(e)$.\n\n\\begin{lemma}[Modification of Lemma 2.4 in \\cite{GhaffariP16}]\n\\label{cl:epsbias}\nThere is an algorithm that creates a collection $\\mathcal{I}=\\{\\operatorname{ID}_1(e_1), \\ldots, \\operatorname{ID}_1(e_{M})\\}$ of $M=\\binom{n}{2}$ random identifiers for all possible edges $(u,v)$, each of $O(\\log n)$-bits using a seed $\\mathcal{S}_{ID}$ of $O(\\log^2 n)$ bits. These identifiers are such that for each subset $E' \\subseteq E$, where $|E'|\\neq 1$, we have $\\Pr[\\mathsf{XOR}\\xspace_1(E') \\in \\mathcal{I}] \\leq 1\/n^{10}$. In addition, given the identifiers $\\operatorname{ID}(u), \\operatorname{ID}(v)$ of the edge $e=(u,v)$ endpoints, and the seed $\\mathcal{S}_{ID}$, one can determine $\\operatorname{ID}_1(e)$ in $\\widetilde{O}(1)$ time.\n\\end{lemma}\n\\begin{proof}\nThe lemma is proved in \\cite{GhaffariP16}, the only part that is not discussed there is the time to determine $\\operatorname{ID}_1(e)$ that follows from \\cite{naor1993small}. \nBy Theorem 3.1 of \\cite{naor1993small}, given the seed $\\mathcal{S}_{ID}$ and the edge identifier $e_j=(\\operatorname{ID}(u), \\operatorname{ID}(v))$, determining the $i^{th}$ bit of $\\operatorname{ID}_1(e_{j})$ can be done in $O(\\log n)$ time. Thus, determining all $O(\\log n)$ bits, takes $O(\\log^2 n)$ time. \n\\end{proof}\nThe second part of the identifier $\\operatorname{ID}_{2,T}(e)$ is given by $\\operatorname{ID}_{2,T}(e)=[\\operatorname{ID}(u), \\operatorname{ID}(v), \\mathsf{ANC}\\xspace_T(u), \\mathsf{ANC}\\xspace_T(v)]$. \nThe identifiers of $\\operatorname{ID}_T(u), \\operatorname{ID}_T(v)$ are used in order to verify the validity of the first part of the label $\\operatorname{ID}_1(e)$. When the tree $T$ is clear from the context, we might omit it and simply write $\\operatorname{ID}(e)$. \n\n\\paragraph{Graph Sketches.}\nGraph sketches are a powerful tool to identify outgoing edges \\cite{kapron2013dynamic,ahn2012analyzing}. \\mtodo{I can add some references later, maybe in the related work section} We start by illustrating the intuition behind them. For a vertex $v$, let $\\mathsf{BaseSketch}\\xspace(v)$ be the bitwise xor of all IDs of edges adjacent to $v$. If we take a subset of vertices $S$, and define  $\\mathsf{BaseSketch}\\xspace(S) = \\oplus_{v \\in S} \\mathsf{BaseSketch}\\xspace(v)$, we can see that all edges that have both endpoints in $S$ are cancelled, and we are left with the xor of outgoing edges from $S$. If there is only one such edge, we get its id. In the case there are more outgoing edges, we can use sampling to identify one outgoing edge. We next formalize this idea and show how to use it in our labels. \n\n\\paragraph{Forbidden-Set Labels via Graph Sketches.} \nThe graph sketches are based on random sub-sampling of the graph edges with logarithmic number of scales, i.e., with probability of $2^{-i}$ for every $i \\in \\{1,\\ldots, m\\}$. For our purposes and similarly to \\cite{DuanConnectivityArxiv16,DuanConnectivitySODA17}, we use pairwise independent hash functions to decide whether to include edges in sampled sets. Choose $L=c\\log n$ \npairwise independent hash functions $h_1, \\ldots, h_{L}:\\{0,1\\}^{2\\log n} \\to \\{0, \\ldots, 2^{\\log m}-1\\}$, and for each $i \\in \\{1, \\ldots, L\\}$ and $j \\in [0,\\log m]$, define the edge set \n$$E_{i,j} =\\{ e \\in E ~\\mid~ h_i(e) \\in [0,2^{\\log m-j})\\}~.$$ \nEach of these hash functions can be defined using a random seed of logarithmic length \\cite{TCS-010}. Thus, a \nrandom seed $\\mathcal{S}_h$ of length $O(L \\log n)$ can be used to determine the collection of all these $L$ functions. As observed in \\cite{DuanConnectivityArxiv16,GibbKKT15}, pairwise independence is sufficient to guarantee that for any set $E' \\subset E$ and any $i$, there exists a $j$ such that with constant probability $\\mathsf{XOR}\\xspace(E' \\cap E_{i,j})$ is the name (extended identifier) of one edge in $E'$, for a proof see Lemma 5.2 in  \\cite{GibbKKT15}.\n\\begin{lemma}\\label{lem:hitting-pairwise}\nFor any edge set $E'$ and any $i$, with constant probability there exists a $j$ satisfying that $|E' \\cap E_{i,j}|=1$.\n\\end{lemma}\n\n\nWe also need to be able to tell that a bit string of $\\mathsf{XOR}\\xspace(E' \\cap E_{i,j})$ is a legal edge ID or not.\n\\textbf{MP: all these lemmas and claims can (should) be moved to the subsequent analysis section.}\n\n\\begin{lemma}\nGiven the seed $\\mathcal{S}_{ID}$, one can determine in $\\widetilde{O}(1)$ time if $\\mathsf{XOR}\\xspace(E' \\cap E_{i,j})$ corresponds to a single edge ID in $G$ or not, w.h.p.\n\\end{lemma}\n\\begin{proof}\nLet $X=\\mathsf{XOR}\\xspace(E' \\cap E_{i,j})$. Letting $E''=E' \\cap E_{i,j}$, then $X$ can be written as the concatenation of $\\mathsf{XOR}\\xspace_1(E'')$ and $\\mathsf{XOR}\\xspace_2(E'')$. \nUsing the seed and $\\mathsf{XOR}\\xspace_2(E'')$, one can test the validity of $\\mathsf{XOR}\\xspace_1(E'')$.\nThe algorithm detects the case that $|E''| \\geq 2$ as follows. First, in the case that $E''$ is a single edge, $\\mathsf{XOR}\\xspace_2(E'')$ should contain legal ids $\\operatorname{ID}(u),\\operatorname{ID}(v)$. If this is not the case, it follows that $|E''| \\neq 1$. If $\\mathsf{XOR}\\xspace_2(E'')$ contains legal ids $\\operatorname{ID}(u),\\operatorname{ID}(v)$, we use them and the seed $\\mathcal{S}_{ID}$ to determine $\\operatorname{ID}_1(e)$ for $e = (u,v)$, and we check if $\\mathsf{XOR}\\xspace_1(E'')=\\operatorname{ID}_1(e)$. We have two options, either $E'' = \\{e\\}$ is the single edge $e$, in which case $\\mathsf{XOR}\\xspace_1(E'')=\\operatorname{ID}_1(e) \\in \\mathcal{I}$, and the verification succeeds. Otherwise $|E''| \\geq 2$, in which case, from Lemma \\ref{cl:epsbias}, $\\Pr[\\mathsf{XOR}\\xspace_1(E'') \\in \\mathcal{I}] \\leq 1\/n^{10}$, hence w.h.p $\\mathsf{XOR}\\xspace_1(E'') \\neq \\operatorname{ID}_1(e) \\in \\mathcal{I}$ and we identify that $|E''| \\geq 2$.\n\\end{proof}\nFor each vertex $v$ and indices $i,j$, let $E_{i,j}(v)$ be the edges incident to $v$ in $E_{i,j}$. \nThe $i^{th}$ \\emph{basic sketch unit} of each node $v$ is then given by:\n\\begin{equation}\n\\label{eq:vsketch}\n\\mathsf{Sketch}\\xspace_i(v)=[\\mathsf{XOR}\\xspace(E_{i,0}(v)),\\ldots,\\mathsf{XOR}\\xspace(E_{i,\\log m}(v))].\n\\end{equation}\nThe sketch of each node $v$ is defined by a concatenation of $L=\\Theta(\\log n)$ basic sketch units: \n$$\\mathsf{Sketch}\\xspace(v)=[\\mathsf{Sketch}\\xspace_{1}(v),\\mathsf{Sketch}\\xspace_{2}(v), \\ldots\\mathsf{Sketch}\\xspace_{L}(v)]~.$$ \nFor every subset of vertices $S$, let \n$\\mathsf{Sketch}\\xspace(S)=\\oplus_{v \\in S}\\mathsf{Sketch}\\xspace(v).$ \n\nWe are now ready to define the forbidden set \\mtodo{fault-tolerant?} connectivity labels of vertices and edges. \nThe label of each vertex $u$ is given by:\n$$\\mathsf{ConnLabel}\\xspace_{G,T}(u)=\\langle \\mathsf{Sketch}\\xspace(V(T(u))), \\mathsf{Sketch}\\xspace(V), \\mathsf{ANC}\\xspace_T(u), \\mathcal{S}_{ID}, \\mathcal{S}_h\\rangle,$$ \nwhere $\\mathsf{ANC}\\xspace_T(u)$ is the ancestry label of $u$ with respect to the tree $T$.\nThe label $\\mathsf{ConnLabel}\\xspace_{G,T}(e)$ of each \\emph{edge} $e=(u,v)$ is as given by:\n\\begin{equation*}\n    \\mathsf{ConnLabel}\\xspace(e)=\n    \\begin{cases}\n      \\langle \\mathsf{ConnLabel}\\xspace_{G,T}(u), \\mathsf{ConnLabel}\\xspace_{G,T}(v)\\rangle ,& \\mbox{~for~} e \\in T \\\\\n     \\langle ID_T(e) \\rangle,& \\mbox{~Otherwise}.\n    \\end{cases}\n\\end{equation*}\n\n\n\\begin{claim}\nThe label length is $O(\\log^3 n)$ bits.\n\\end{claim}\n\\begin{proof}\nThe label size is dominated by the sketching information $\\mathsf{Sketch}\\xspace(V_u)$, which is made of a concatenation of the bitwise XOR of $O(\\log n)$ basic sketch units $\\mathsf{Sketch}\\xspace_i(u)$. By Eq. (\\ref{eq:vsketch}), each unit has $O(\\log^2 n)$ bits, and thus overall, the label has $O(\\log^3 n)$ bits.\n\\end{proof}\n\n\\mtodo{I think maybe now we want something slightly different, that given a basic sketch unit, we can find with constant probability an outgoing edge? Also, should add a proof.}\n\n\\begin{lemma}\\label{lem:sketch-property}\nFor any subset $S$, given $\\mathsf{Sketch}\\xspace(S)$ one can compute, w.h.p., an outgoing edge $E(S, V \\setminus S)$ if such exists. \n\\end{lemma}\n\n\n\\subsubsection{The Decoding Algorithm} \\label{sec:dec-alg}\nWe next describe the decoding algorithm where given every triplet $s,t, F \\in V \\times V \\times E^f$ along with their labels, it determines whether $s$ and $t$ are connected in $G\\setminus F$, w.h.p. For our decoding algorithm it would actually be sufficient to get as input:\n\\begin{enumerate}[noitemsep]\n\\item the connectivity labels of $s,t$, the labels of the faulty tree-edge $F \\cap E(T)$, and in addition,\n\\item the extended identifiers of the faulty non-tree edges $F \\setminus T$ (which are part of the labels)\\footnote{This property will be important later on for obtaining the compact routing schemes.}. \n\\end{enumerate}\nThe decoding algorithm has four key steps: The first step identifies the at most $f+1$ components $\\mathcal{C}_0=\\{C_1,\\ldots, C_\\ell\\}$ of $T \\setminus F$, as well as the components of $s$ and $t$ in $\\mathcal{C}_0$. The second step uses the label information to compute the sketch value $\\mathsf{Sketch}\\xspace(C_i)$ of each component $C_i \\in \\mathcal{C}_0$. The third step modifies this sketch information into $\\mathsf{Sketch}\\xspace_{G \\setminus F}(C_i)$, by subtracting the information related to the faulty edges. The forth and final step uses the sketch information in order to simulate $L=O(\\log n)$ steps of the Boruvka algorithm. At the end of these steps, the decoding algorithm identifies the connected components of both $s$ and $t$ in $G \\setminus F$. In the case where $s$ and $t$ are indeed connected in $G \\setminus F$, the algorithm also outputs a succinct representation of an $s$-$t$ path in $G \\setminus F$. This extra information would be used later on by our compact routing scheme. We next describe these steps in details. \n\n\\paragraph{Step 1: Identification of the connected components $\\mathcal{C}_0$ in $T \\setminus F$.} \nLet $F_T=F \\cap T$ be the faulty tree edges and let $F_{NT}=F \\setminus F_T$ be the faulty non-tree edges. Let $U=\\{s,t\\} \\cup V(F_T)$.  Each component $C_i$ of $T \\setminus F$ will be identified by the maximum vertex ID in $C_i \\cap U$. Note that the non-tree faulty edges $F_{NT}$ have no impact on the components of $T \\setminus F$ (thus their labels are indeed not needed for that step).\nWe next show that although we do not have full information about the tree $T$ and the vertices of each connected component, the ancestry labels of $V(F_T)$ give us enough information to identify the connected components of $T \\setminus F$. Additionally, given an ancestry label of a vertex $u$, we can identify the connected component of $u$. To obtain this, it is helpful to look at the \\emph{component tree} that is obtained by contracting each connected component of $T \\setminus F$ to one vertex, as follows. Let $\\ell = |F_T|+1.$ The component tree $T_C = (\\mathcal{C}_0, E_C)$ is a tree of $\\ell$ vertices representing the connected components in $T \\setminus F$, and $|F_T|=\\ell-1$ edges corresponding to the edges of $F_T$. There is an edge $\\{C_i,C_j\\} \\in E_C$ iff there is an edge $\\{u,v\\} \\in F_T$ where $u \\in C_i, v \\in C_j$. See Figure \\ref{componentTreePic} for an illustration.  We can construct the tree $T_C$ using the ancestry labels of the edges $F_T$. For this we just need to identify for any edge in $F_T$ the set of edges from $F_T$ above it in $T$. Moreover, for a given vertex $v$, its connected component is exactly determined by the set of edges in $F_T$ above it in $T$, which can again be identified using the ancestry labels of $v \\cup V(F_T)$. In particular, we can identify the connected components of $s$ and $t$. \\mtodo{add the time complexity, we can probably get a faster algorithm if needed.}\n\n\\setlength{\\intextsep}{0pt}\n\\begin{figure}[h]\n\\centering\n\\setlength{\\abovecaptionskip}{-2pt}\n\\setlength{\\belowcaptionskip}{6pt}\n\\includegraphics[scale=0.55]{componentTree.pdf}\n \\caption{Illustration of the component tree where $F=\\{e_1,e_2,e_3,e_4\\}$. Each connected component of $T \\setminus F$ is contracted to one vertex on the right.}\n\\label{componentTreePic}\n\\end{figure}\n \n\n\\paragraph{Step 2: Computing the sketch values of each component $\\mathcal{C}_0$ in $G$.} \nFor each component $C_j \\in \\mathcal{C}_0$ the algorithm computes $\\mathsf{Sketch}\\xspace_G(C_j)$ using the label information of the nodes in $U$.  The basic observation here is the following. Given $S' \\subset S$ and $\\mathsf{Sketch}\\xspace(S), \\mathsf{Sketch}\\xspace(S')$, it holds that $\\mathsf{Sketch}\\xspace(S \\setminus S')=\\mathsf{Sketch}\\xspace(S) ~\\mathsf{XOR}\\xspace~ \\mathsf{Sketch}\\xspace(S')$. To compute the sketch values, first, we define for each component a temporary value $\\mathsf{Sketch}\\xspace'_G(C_j)$ as follows. Let $v_j$ be the highest vertex in the component $C_j$. For the component of the root $r$, this is $r$. For any other component $C_j$, let $\\{C_j,p(C_j)\\}$ be the edge connecting $C_j$ to its parent in the component tree. This edge corresponds to an edge $\\{v,p(v)\\} \\in F_T$, where $v$ is the highest vertex in $C_j$. We define $\\mathsf{Sketch}\\xspace'_G(C_j) = \\mathsf{Sketch}\\xspace_G(V_{v_j})$. Note that this value is part of the label of the vertex $v_j$. For any $v_j \\neq r$, we have $v_j \\in V(F_T)$, and we also learn the identity of $v_j$ when constructing the component tree in Step 1, hence we know $\\mathsf{Sketch}\\xspace'_G(C_j)$. We also know the temporary sketch value of the component of $r$, as $\\mathsf{Sketch}\\xspace_G(V_{r})=\\mathsf{Sketch}\\xspace_G(V)$ is part of the labels of all vertices. We next use the temporary sketch values to compute the sketch values of components using the following claim.\n\n\\begin{claim}\nLet $C_j$ be a component in $T \\setminus F$. If $C_j$ is a leaf in the component tree, we have $\\mathsf{Sketch}\\xspace_G(C_j) = \\mathsf{Sketch}\\xspace'_G(C_j).$ Otherwise, let $D=\\{D_1,...,D_t\\}$ be the children of $C_j$ in the component tree and let $\\mathsf{Sketch}\\xspace'(D)=\\oplus_{1 \\leq i \\leq t} \\mathsf{Sketch}\\xspace'_G(D_i)$, then $\\mathsf{Sketch}\\xspace_G(C_j) = \\mathsf{Sketch}\\xspace'_G(C_j) \\oplus \\mathsf{Sketch}\\xspace'(D).$ \n\\end{claim}\n\n\\begin{proof}\nIt holds that $\\mathsf{Sketch}\\xspace_G(C_j) = \\oplus_{v \\in C_j} \\mathsf{Sketch}\\xspace_G(v)$. By definition, $\\mathsf{Sketch}\\xspace'_G(C_j)=\\mathsf{Sketch}\\xspace_G(V_{v_j}) = \\oplus_{v \\in V_{v_j}} \\mathsf{Sketch}\\xspace_G(v)$ is the xor of sketches of all vertices in the subtree of $v_j$. As $v_j$ is the highest vertex in $C_j$, if $C_j$ is a leaf component in the component tree, then the vertices in $C_j$ are exactly the vertices in $V_{v_j}$, and the claim follows. Otherwise, the vertices in $C_j$ are all vertices in $V_{v_j}$ that are not contained in any component below $C_j$. Hence, to compute the value $\\mathsf{Sketch}\\xspace_G(C_j)$, we should subtract from $\\mathsf{Sketch}\\xspace_G(V_{v_j})$ the sketch values of vertices in components below $C_j$. Let $D_1,...,D_t$ be the children of $C_j$ in the component tree, and let $u_1,...,u_t$ be the highest vertices in the components $D_1,...,D_t$. Any vertex that is in some component below $C_j$ is in exactly one of the subtrees $V_{u_1},...,V_{u_t}$. Hence the sketch value of vertices in components below $C_j$ equals $\\oplus_{1 \\leq i \\leq t} \\mathsf{Sketch}\\xspace_G(V_{u_i})= \\oplus_{1 \\leq i \\leq t} \\mathsf{Sketch}\\xspace'_G(D_i)=\\mathsf{Sketch}\\xspace'(D)$. To conclude, we get $\\mathsf{Sketch}\\xspace_G(C_j)=\\mathsf{Sketch}\\xspace_G(V_{v_j}) \\oplus \\mathsf{Sketch}\\xspace'(D)=\\mathsf{Sketch}\\xspace'_G(C_j) \\oplus \\mathsf{Sketch}\\xspace'(D)$, as needed.\n\\end{proof}\n\nTo conclude, from the values $\\mathsf{Sketch}\\xspace'_G(C_j)$, we can easily compute the values $\\mathsf{Sketch}\\xspace_G(C_j)$. The complexity is $\\tilde{O}(f)$, as for each component, the sketch $\\mathsf{Sketch}\\xspace'(C_j)$ participates in two computations, and we have at most $O(f)$ components and the sketches have poly-logarithmic size.\n\n\n\\paragraph{Step 3: Computing the sketch values of each component $\\mathcal{C}_0$ in $G \\setminus F$.} \nFor each faulty edge $e \\in F$ (both tree and non-tree edges), our goal is to subtract the sketch information of $e$ from the corresponding components of the endpoint of $e$. The step does not require the label information of the edges, and it would be sufficient to know only the seed $\\mathcal{S}_h$ that determines the sampling of edges into the sketches, and the extended identifier of the failing edges. \n\nUsing the extended identifier of the faulty edge $e=(u,v)$, one can determine the components in $\\mathcal{C}_0$ to which its endpoints belong from the ancestry labels of $u$ and $v$, as explained in Step 1. Using the identifier $\\operatorname{ID}(e)$ and the seed $\\mathcal{S}_h$, one can determine all the indices of the sketch to which the edge $e$ was sampled. \\mtodo{what is the complexity of this? MP: It should be $\\widetilde{O}(1)$ using Fact \\ref{fc:pairwise}.}\nLetting $C_u, C_v$ be the components of $u$ and $v$ in $T \\setminus F$, respectively. The values $\\mathsf{Sketch}\\xspace_G(C_u),\\mathsf{Sketch}\\xspace_G(C_v)$ are updated by XORing them with the matrix that contains the identifier $\\operatorname{ID}(e)$ in the relevant positions. \\mtext{The complexity is poly-logarithmic, as the matrix has poly-logarithmic size.}\nFrom that point on, all sketches of the components $\\mathcal{C}_0$ can be treated as sketches that have been computed in $G \\setminus F$.  \n\n\n\\paragraph{Step 4: Simulating the Boruvka algorithm.} Finally, our goal is to determine the identifiers of the maximal connected components of $s$ and $t$ of $G \\setminus F$. The input to this step is the identifiers of the components $\\mathcal{C}_0=\\{C_{1}, \\ldots, C_k\\}$ in $T \\setminus F$, along with their sketch information in $G \\setminus F$. While the algorithm does not have information on the nodes of each component, it knows the component identifier of each node in $U$. \n\nThe algorithm consists of $L=O(\\log n)$ phases of the Boruvka algorithm. Each phase $i \\in \\{1,\\ldots, L\\}$ will be given as input a partitioning $\\mathcal{C}_i=\\{C_{i,1}, \\ldots, C_{i,k_i}\\}$ of (not necessarily maximal) connected components in $G \\setminus F$.\nThese components are identified by an $O(\\log n)$ bit identifier, where for each vertex in $U$, the algorithm receives its unique component identifier in  $\\mathcal{C}_i$. In addition, the algorithm receives the sketch information of the components $\\mathcal{C}_i$ in $G \\setminus F$. The output of the phase is a partitioning $\\mathcal{C}_{i+1}$, along with their sketch information in $G \\setminus F$ and the identifiers of the components for each node in $U$. A component $C_{i,j} \\in \\mathcal{C}_i$ is \\emph{growable} if it has at least one non-faulty outgoing edge to a node in $V \\setminus C_{i,j}$. That is, the component is growable if it is strictly contained in some maximal connected component in $G \\setminus F$. Letting $N_i$ denote the number of growable components in $\\mathcal{C}_i$, the output partitioning $\\mathcal{C}_{i+1}$ of the $i^{th}$ step guarantees that $N_{i+1}\\leq N_i \/2$ w.h.p. To obtain outgoings edges from the growable components in $\\mathcal{C}_i$, the algorithm uses the $i^{th}$ basic-unit sketch $\\mathsf{Sketch}\\xspace_i(C_{i,j})$ of each $C_{i,j} \\in \\mathcal{C}_i$. By Lemma \\ref{lem:sketch-property}, from every growable component in $\\mathcal{C}_i$, we get one outgoing edge $e'=(x,y)$ with constant probability. Using the extended edge identifier of $e'$ the algorithm can also detect the component $C_{i,j'}$ to which the second endpoint, say $y$, of $e'$ belongs. Specifically, this is done using the ancestry label of the detected edge $e'$. That label allows us to compute the component of $y$ in the initial partitioning $T \\setminus F$, i.e., the component $C_{0,q}$ of $y$ in $\\mathcal{C}_0$. Thus $y$ belongs to the unique component $C_{i,j'} \\in \\mathcal{C}_i$ that contains \n$C_{0,q}$. \n\n\n\nAs noted in prior works \\cite{ahn2012analyzing,kapron2013dynamic,DuanConnectivityArxiv16}, it is important to use fresh randomness (i.e., independent sketch information) in each of the Boruvka phases. The reason is that the cut query, namely, asking for a cut edge between $S$ and $V \\setminus S$, should not be correlated with the randomness of the sketches. Note that indeed the components of $\\mathcal{C}_i$ are correlated with the randomness of the first $(i-1)$ basic sketch units of the vertices. Thus, in phase $i$ the algorithm uses the $i^{th}$ basic sketch units of the vertices (which are independent of the other sketch units) to determine the outgoing edges of the components in $\\mathcal{C}_i$.\n\n\nThe algorithm then computes the updated sketches of the merged components. This is done by xoring over the sketches of the components in $\\mathcal{C}_i$ that got merged into a single component in \n$\\mathcal{C}_{i+1}$. In expectation, the number of growable components is reduced by factor $2$ in each phase. Thus after $O(\\log n)$ phases, the expected number of growable components is at most $1\/n^5$, and using Markov inequality, we conclude w.h.p there are no growable components. The final partitioning $\\mathcal{C}_L$ corresponds w.h.p to the maximal connected components in $G \\setminus F$. The pair $s$ and $t$ are connected in $G \\setminus F$ only if the components $C_s,C_t$ of $s,t$ respectively in $T \\setminus F$ are connected in the final component decomposition.\n\\mtodo{add complexity.}\n\nFinally, we show that the decoding algorithm can be slightly modified to output a compressed encoding of an $s$-$t$ path in $G \\setminus F$, using $O(f\\log n)$ bits. This encoding is represented by an $s$-$t$ path $\\widehat{P}$ that has two type of edges, appearing in an alternate manner on  $\\widehat{P}$: $G$-edges and edges $e'=(u,v)$ such that the $u$-$v$ tree path is intact in $T \\setminus F$. See Figure \\ref{fig:succ-paths}. \n\\begin{lemma}\\label{lem:useful-recovery-edges}\nConsider a triplet $s,t,F$ such that $s$ and $t$ are connected in $G \\setminus F$. \nThe decoding algorithm can also output a set of at most $f$ recovery edges $Q$ such $(T \\setminus F) \\cup Q$ is a spanning tree. In addition, it outputs a labeled $s$-$t$ path $\\widehat{P}$ of length $O(f)$ that provides a succinct description of the $s$-$t$ path. The edges of $\\widehat{P}$ are labeled by $0$ and $1$, where $0$-labeled edges correspond to $G$-edges and $1$-labeled edges $e=(x,y)$ correspond to $x$-$y$ paths in $T \\setminus F$. \n\\end{lemma}\n\\begin{proof}\nLet $C_s, C_t$ be the components of $s$ and $t$ in the initial partitioning $\\mathcal{C}_0$. In Step $4$ of the decoding algorithm, the Boruvka algorithm is simulated up to the point that $C_s$ and $C_t$ are connected. Therefore, the algorithm has computed a path $P$ that connects the components $C_s$ and $C_t$. Each node on that path corresponds to a component in $\\mathcal{C}_0$, and each edge corresponds to an outgoing edge (discovered using the sketch information). Since $\\mathcal{C}_0$ has at most $f+1$ components, $|P|\\leq f+1$. \nEach such edge $e' \\in P$ corresponds to an edge in $G$. Let $e_1=(x_1,y_1),\\ldots, e_k=(x_k,y_k)$ be the $G$-edges corresponding to the edges of $P$ ordered from $C_s$ to $C_t$. Letting $y_0=s$ and $x_{k+1}=t$, we get that \n$y_i$ and $x_{i+1}$ belong to the same component in $\\mathcal{C}_0$, for every $i \\in \\{0,\\ldots, k\\}$. \nThe labeled path is given by $\\widehat{P}=[s, x_1,y_1, x_2, y_2, \\ldots y_k,t]$ where the edges $(y_i,x_{i+1})$ are labeled $1$ and the edges $(x_{i}, y_i)$ are labeled $0$. Each $0$-labeled edge is a real edge in $G$, and each $1$-labeled edge $(x_{i}, y_i)$ corresponds to a tree path $\\pi(x_i, y_i)$ in $T \\setminus F$. \n\\end{proof}\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.40]{suc-path.pdf}\n\\caption{\\sf Shown is a tree $T$ with faulty edges $e_1,\\ldots, e_4$. The $s$-$t$ path in $G \\setminus F$ is represented by the path $\\widehat{P}=[s,v_1]\\circ (v_1,v_2) \\circ [v_2,v_3] \\circ (v_3,r) \\circ [r,v_4] \\circ (v_4, v_5) \\circ [v_5,t]$. The recovery edges $(v_1,v_2), (v_3,r)$ and $(v_4, v_5)$ are shown in dashed lines.  \\label{fig:succ-paths}\n}\n\\end{center}\n\\end{figure}\n\\section{Fault-Tolerant (FT) Connectivity Labels}\n\n\nWe next discuss two labeling schemes for connectivity that are based on two different approaches. The first one uses the \\emph{cycle space sampling} technique to try to find cuts that disconnect $s$ and $t$. The second one uses \\emph{graph sketches} to try to find a path that connects $s$ and $t$. Since the second approach allows to find a path between $s$ and $t$ if exists, it is also useful later for routing. In terms of label size, the first approach gives labels of size $O(f + \\log{n})$, which is near-optimal if the number of failures is $f=O(\\log{n})$. On the other hand, the second scheme gives labels of size $O(\\log^3{n})$, which is better when the number of failures is large.\nWe next discuss the labeling schemes. During this section, we assume that the input graph $G$ is connected. If not, we can add to the label of each vertex and edge the id of their connected component in $G$, and apply the labeling scheme to each one of the connected components separately. \n\n\\subsection{Connectivity Labels Based on Cycle Space Sampling}\n\n\\subsubsection{The Labeling Algorithm}\n\nOur labels are composed of two ingredients, that we review next.\n\n\\paragraph{Cycle Space Labels.}\nThe cycle space sampling technique, introduced in \\cite{pritchard2011fast}, allows to give the edges of a graph short labels that allow to detect cuts in the graph. For a set of vertices $S$, $\\delta(S)$ is the set of edges with exactly one endpoint in $S$. A subset of edges $F$ is called an \\emph{induced edge cut} if $F = \\delta(S)$ for some $S$.\nThe following is shown in \\cite{pritchard2011fast} (see Corollary 2.9). \n\n \\cycle*\n\n\\remove{\n\\begin{restatable}{lemma}{cycle} \\label{cycle_space_lemma}\nThere is an algorithm that assigns the edges of a graph $G=(V,E)$ $b$-bit labels $\\phi(e)$ such that given a subset of edges $F \\subseteq E$, we have:\n$$Pr[\\Moplus_{e \\in F} \\phi(e) = 0] = \\left\\{\n                \\begin{array}{ll}\n                  1,\\ if\\ F\\ is\\ an\\ induced\\ edge\\ cut\\\\\n                  2^{-b},\\ otherwise\n                \\end{array}\n              \\right. $$ \nWhere $0$ is the all-zero vector. The time complexity for assigning the labels is $O((m+n)b)$.\n\\end{restatable}\n}\n\nFor an overview of the technique, see Appendix \\ref{sec:cycle_space_overview}. \nIn our algorithm, given a subset of edges $F$ of size at most $f$, we want to be able to check for any subset $F' \\subseteq F$ if $F'$ is an induced edge cut. To support all these $2^f$ queries w.h.p we choose $b=f+ c \\log{n}$ for a constant $c$. This guarantees that the probability of error is at most $\\frac{2^f}{2^{f+c\\log{n}}}=\\frac{1}{n^c}$. This will guarantee that given a query $\\langle s,t,F \\rangle$, our algorithm answers correctly w.h.p. We remark that if we increase the size of labels to $O(f \\log{n})$ we can get an algorithm that is correct for \\emph{all} queries w.h.p. \nThe reason is that we can then check for any subset of edges $F$ of size at most $f$ if $F$ is an induced edge cut. As the number of subsets of size at most $f$ is bounded by $O(n^f)$, we get that the labels are correct for all such subsets w.h.p.\n\n\n\\paragraph{Ancestry Labels.} Our second ingredient are ancestry labels for trees.\nTo use them, we first fix a spanning tree $T$ of the graph rooted at $r$. The goal is to assign vertices short labels, such that given the labels of $u$ and $v$, we can infer if $u$ is an ancestor of $v$ in $T$. A simple labeling scheme based on a DFS scan solves the problem with labels of size $2 \\lceil \\log{n} \\rceil$ per vertex \\cite{kannan1992implicat}, the time for assigning the labels is $O(n)$ for the DFS scan of the tree. Labeling schemes with improved label size appear in \\cite{abiteboul2006compact,alstrup2002improved,fraigniaud2010compact,fraigniaud2010optimal}.\n\n\\begin{lemma} \\label{anc_labels}\nFor every tree $T$, there is an algorithm that assigns the vertices $u$ of the tree labels $\\mathsf{ANC}\\xspace_T(u)$ of $O(\\log{n})$ bits, such that given the labels of $u$ and $v$ we can infer if $u$ is an ancestor of $v$ in $T$ in $O(1)$ time. The time for assigning the labels is $O(n)$. \n\\end{lemma}\n\n\\paragraph{The Final Labels.}\nOur final labels contain the following ingredients:\n\\begin{enumerate}\n\\item The label of the edge $e=(u,v)$ is composed of $(\\phi(e),\\mathsf{ANC}\\xspace_T(u),\\mathsf{ANC}\\xspace_T(v),j)$, where $j$ is a bit indicating if $e$ is a tree edge in $T$. In total, the label size is $O(f + \\log{n})$.\n\\item The label of a vertex $v$ is its ancestry label $\\mathsf{ANC}\\xspace_T(v)$ of size $O(\\log{n})$ bits.\n\\end{enumerate}\n\nAs discussed, the time for assigning the labels is $O((m+n)b)=\\widetilde{O}((m+n)f)$, as $b=f+c\\log{n}$.\nWe next explain how we use these labels to check FT connectivity.\n\n\n\\subsubsection{The Decoding Algorithm}\n\nWe next discuss several observations that allow us to check if $s$ and $t$ are disconnected by $F$.\n\n\\begin{claim} \\label{obs_induced}\nThe vertices $s$ and $t$ are disconnected by $F$ if an only if there is an induced edge cut $F' \\subseteq F$ that disconnects $s$ and $t$.\n\\end{claim}\n\n\\begin{proof}\nFirst, if $F' \\subseteq F$ disconnects $s$ and $t$, then clearly $F$ disconnects $s$ and $t$.\nOn the other hand, if $s$ and $t$ are disconnected by $F$, let $F' \\subseteq F$ be a minimal set of edges whose removal disconnects $s$ and $t$. We show that $F'$ is an induced edge cut. Let $V_s$ be the vertices in the connected component of $s$ in $G \\setminus F'$. We show that all edges in $F'$ are between $V_s$ and $V \\setminus V_s$, implying that $F'$ is an induced edge cut. Assume to the contrary that there is an edge $e \\in F'$ with both endpoints in one of the sides, say $V_s$, then $F' \\setminus \\{e\\}$ is still a cut that disconnects $s$ and $t$ (as $V_s$ is still disconnected from the rest of the graph if we add $e$), contradicting the minimality of $F'$. A symmetric argument shows that $e$ cannot have both its endpoints in $V \\setminus V_s$.\n\\end{proof}\n\nWe next show that given an induced edge cut $F'$, there is a simple way to determine the two sides of the cut induced by $F'$ (see Figure \\ref{cutSidesPic} for illustration). For a vertex $v$ and an induced edge cut $F'$, we denote by $n_v(F')$ the number of edges from $F'$ in the path from the root $r$ to $v$ in the spanning tree $T$. We show the following.\n\n\\remove{\n\\begin{claim}\nLet $F'$ be an induced edge cut, and let $T$ be a spanning tree with root $r$. Let $V_0$ be all the vertices $v$ where in the path from $r$ to $v$ there is an even number of edges from $F'$, and let $V_1 = V \\setminus V_0$. Then $(V_0,V_1)$ is the induced edge cut defined by $F'$.\n\\end{claim}\n}\n\n\\begin{claim} \\label{obs_cut_sides}\nLet $F'$ be an induced edge cut. Let $$V_0=\\{v \\in V|\\ n_v(F')\\ is \\ even \\},$$ $$V_1=\\{v \\in V|\\ n_v(F')\\ is \\ odd \\}.$$ Then $(V_0,V_1)$ is the induced edge cut defined by $F'$.\n\\end{claim}\n\n\\begin{proof}\nSince $F'$ is an induced edge cut, the endpoints of every edge in $F'$ are on different sides of the cut. Hence, if we scan the tree $T$ from the root to the leaves, every time we reach an edge from $F'$ we change the side of the cut. It follows that one side of the cut contains all vertices $v$ such that $n_v(F')$ is even, and the other side has all vertices $v$ such that $n_v(F')$ is odd. Hence $V_0,V_1$ are the two sides of the cut.\n\\end{proof}\n\n\\setlength{\\intextsep}{0pt}\n\\begin{figure}[h]\n\\centering\n\\setlength{\\abovecaptionskip}{-2pt}\n\\setlength{\\belowcaptionskip}{6pt}\n\\includegraphics[scale=0.55]{cutSides2.pdf}\n \\caption{Here $F'=\\{e_1,e_2,e_3,e_4\\}$ is an induced edge cut. On the right, you can see the partition into sides in the tree. Every time we reach an edge from $F'$, we change the side of the cut.}\n\\label{cutSidesPic}\n\\end{figure}\n\nFrom Claims \\ref{obs_induced} and \\ref{obs_cut_sides}, we get the following.\n\n\\begin{corollary} \\label{cor_ft_connectivity}\nThe vertices $s$ and $t$ are disconnected by $F$ if an only if there is an induced edge cut $F' \\subseteq F$, such that one of the values $n_s(F'),n_t(F')$ is even and the other is odd. \n\\end{corollary}\n\nThis gives a simple approach to detect if $s$ and $t$ are disconnected by $F$. We go over all subsets $F' \\subseteq F$, for each one of them we first check if $F'$ is an induced edge cut using the cycle space labels. Second, if $F'$ is an induced edge cut, we compute the values $n_s(F'),n_t(F')$, if the number is even for one of them and odd for the second, we deduce that $F'$ disconnects $s$ and $t$. Note that we can use the ancestry labels to compute the values $n_s(F'),n_t(F')$. For example, for computing $n_s(F')$ we should check how many edges in $F'$ are in the tree path between $r$ to $s$. For this, for each tree edge $e=(u,v)$ in $F'$, we check if it is above $s$ in the tree, which happens if and only if both $u$ and $v$ are ancestors of $s$.\nThis simple approach requires time exponential in $|F|$ for going over all subsets of $F$, we next show a faster way to check the same condition.\n\n\\subsubsection{Faster Decoding Algorithm}\n\nWe next show that checking the condition from Corollary \\ref{cor_ft_connectivity} boils down to solving a system of linear equations.\nFirst, note that from Lemma \\ref{cycle_space_lemma}, w.h.p, a set of edges $F' \\subseteq F$ is an induced edge cut iff $\\Moplus_{e \\in F'} \\phi(e) = 0$. Hence, if we want to check if there is a non-empty subset $F' \\subseteq F$ that is an induced edge cut it is equivalent to checking if there exists a binary vector $x=(x_1,...,x_f) \\neq 0$ such that $\\Moplus_{1 \\leq i \\leq f} x_i \\phi(e_i) = 0$, where $\\{e_1,...e_f\\}$ are the edges of $F$. Or equivalently checking if the vectors $\\{\\phi(e)\\}_{e \\in F}$ are linearly dependant. To check the condition from Corollary \\ref{cor_ft_connectivity}, we generalize this idea. \n\nLet $b = O(f + \\log{n})$ be the size of the cycle space labels.\nGiven a triplet $(s,t,F)$, we assign for each edge $e \\in F$, a binary vector $\\phi'(e)$ of length $b+2$, as follows.\n\\begin{enumerate}\n\\item If $e$ is a tree edge which is in the tree path $r-s$ but not in the path $r-t$, then $\\phi'(e)=10\\phi(e)$.\n\\item If $e$ is a tree edge which is in the tree path $r-t$ but not in the path $r-s$, then $\\phi'(e)=01\\phi(e)$.\n\\item In all other cases, $\\phi'(e)=00\\phi(e).$ \n\\end{enumerate}\n\nWe denote by $w_1,w_2$ binary vectors of length $b+2$ such that $w_1=100..0,w_2=010...0$ (all right entries are equal to 0).\nWe show that the condition from Corollary \\ref{cor_ft_connectivity} holds iff there is a binary vector $x=(x_1,...,x_f)$ and $j \\in \\{1,2\\}$ such that $$\\Moplus_{1 \\leq i \\leq f} x_i \\phi'(e_i) = w_j.$$ This holds iff there is a solution to at least one of the linear systems $Ax=w_1,Ax=w_2$, where $A$ is a $(b+2) \\times f$ matrix that has the vectors $\\{\\phi'(e)\\}_{e \\in F}$ as its column vectors, and $x,w_1,w_2$ are column vectors. All operations are modulo 2.\n\n\\begin{lemma}\nWith high probability, the vertices $s$ and $t$ are disconnected by $F$ if an only if there is a binary vector $x=(x_1,...,x_f)$ and $j \\in \\{1,2\\}$ such that $\\Moplus_{1 \\leq i \\leq f} x_i \\phi'(e_i)  = w_j.$\n\\end{lemma}\n\n\\begin{proof}\nWe assume for the proof that the cycle space labels are correct, i.e., a set of edges $F' \\subseteq F$ is an induced edge cut iff $\\Moplus_{e \\in F'} \\phi(e) = 0$. This happens w.h.p from Lemma \\ref{cycle_space_lemma} and the choice of $b=O(f + \\log{n})$.\n\nFirst we show that if $s$ and $t$ are disconnected by $F$, the condition of the lemma holds.\nFrom Corollary \\ref{cor_ft_connectivity}, $s$ and $t$ are disconnected by $F$ iff there is an induced edge cut $F' \\subseteq F$, such that one of the values $n_s(F'),n_t(F')$ is even and the other is odd. Denote by $n'_s(F')$ the number of edges from $F'$ in the $r-s$ tree path that are not in the $r-t$ path, and denote by $n'_t(F')$ the number of edges from $F'$ in the $r-t$ tree path that are not in the $r-s$ path. Note that if one of the values $n_s(F'),n_t(F')$ is even and the other is odd, then also one of $n'_s(F'),n'_t(F')$ is even and the other is odd, as if we denote by $y$ the number of edges from $F'$ that are in both $r-s$ and $r-t$, we get that $n'_s(F') = n_s(F') - y, n'_t(F') = n_t(F') - y$. Assume first that $n'_s(F')$ is even and $n'_t(F')$ is odd. Let $x$ be the characteristic vector of $F'$. We show that $\\Moplus_{1 \\leq i \\leq f} x_i \\phi'(e_i)  = w_2$. First, as $F'$ is an induced edge cut, we have that $\\Moplus_{e \\in F'} \\phi(e) = 0$. Hence, the $b$ last bits of $\\Moplus_{1 \\leq i \\leq f} x_i \\phi'(e_i)$ are equal to 0 as needed. $F'$ has even number of edges that are in the path $r-s$ and not $r-t$, as the labels $\\phi'(e)$ of all these edges start in $10$, the XOR of the first 2 bits of these edges sums to $00$.  $F'$ has odd number of edges that are in the path $r-t$ but not $r-s$. The labels of all these edges start in $01$, as there is an odd number of them, the XOR of the first 2 bits of these edges sums to $01$. All other edges have labels that start in $00$, hence the XOR of their first 2 bits sums to $00$. Overall we get that $\\Moplus_{1 \\leq i \\leq f} x_i \\phi'(e_i)=\\Moplus_{e \\in F'} \\phi'(e)=010...0=w_2$. The case that $n'_s(F')$ is odd and $n'_t(F')$ is even is symmetric and results in the equation $\\Moplus_{1 \\leq i \\leq f} x_i \\phi'(e_i)=100...0=w_1.$\n\nOn the other hand, if we have that $\\Moplus_{1 \\leq i \\leq f} x_i \\phi'(e_i)=w_j$ for a binary vector $x=(x_1,...,x_f)$ and $j \\in \\{1,2\\}$, we can build from it $F'$ that satisfies the condition in Corollary \\ref{cor_ft_connectivity}, as follows. We define $F'$ to be all edges $e_i \\in F$ such that $x_i =1$. Since $\\Moplus_{1 \\leq i \\leq f} x_i \\phi'(e_i)=\\Moplus_{e \\in F'} \\phi'(e)=w_j$, we have that $\\Moplus_{e \\in F'} \\phi(e) = 0$, hence $F'$ is an induced edge cut. Additionally if $w_j=w_2$, it implies that the XOR of the first 2 bits of labels $\\{\\phi'(e)\\}_{e \\in F'}$ are equal to $01$. By the definition of the labels, this can only happen if $n'_s(F')$ is even and $n'_t(F')$ is odd. Similarly, if $w_j=w_1$, then $n'_s(F')$ is odd and $n'_t(F')$ is even. In both cases we get that one of the values $n_s(F'),n_t(F')$ is even and the other is odd, hence $s$ and $t$ are disconnected by $F$ from Corollary \\ref{cor_ft_connectivity}. \n\\end{proof}\n\nTo conclude, the question if $s$ and $t$ are disconnected by $F$ boils down to checking if there is a solution to at least one of the linear systems $Ax=w_1,Ax=w_2$, where $A$ is a $(b+2) \\times f$ matrix, and $b=O(f + \\log{n})$. Note that we can construct the labels $\\phi'(e)$ and hence the matrix $A$ given the labels of $s,t,F$. For this, we need the labels $\\phi(e)$ of edges in $F$, and also to distinguish for each edge in $F$ if it is in the $r-s,r-t$ paths in the tree. The latter can be deduced from the ancestry labels of $s,t,F$ and from the bits indicating which edges in $F$ are tree edges. A tree edge $e=(u,v) \\in F$ is in the $r-s$ path iff both $u$ and $v$ are ancestors of $s$, this can be checked in $O(1)$ time using the ancestry labels of $u,v,s$. Hence we can build the matrix $A$ in $O(fb)$ time. To check if the linear systems have a solution we can use Gaussian elimination, that takes $O(MN^2)$ time for $M \\times N$ matrix, in our case this is $O((f+\\log{n})f^2)$. Alternatively, we can use $O(N^{\\omega})$ algorithms for $N \\times N$ matrices, where $\\omega$ is the exponent of matrix multiplication.\nFor this, we add zero columns to our matrix $A$ to make it a $(b+2) \\times (b+2)$ matrix $A'$ and increase the length of $x$ to $b+2$, the new system $A'x=w_i$ has a solution iff the original system $Ax=w_i$ has a solution. The complexity here is $O((b+2)^{\\omega})=O((f+\\log{n})^{\\omega})$.\nThis gives the following. \n \n\\begin{theorem}\nThere is a randomized $f$-FT connectivity labeling scheme that assigns the edges and vertices of the graph labels of size $O(\\log{n})$ bits per vertex and $O(f + \\log{n})$ bits per edge. The decoding time of the scheme is $\\min\\{O((f+\\log{n})f^2),O((f+\\log{n})^{\\omega})\\}$. The time complexity for assigning the labels is $\\widetilde{O}((m+n)f).$\n\\end{theorem}  \n\n\\remove{\n\\begin{theorem}\nWe can assign the edges and vertices of the graph labels of size $O(\\log{n})$ bits per vertex and $O(f + \\log{n})$ bits per edge, such that given the labels of $(s,t,F)$ we can check if $s$ and $t$ are disconnected by $F$ in $O(f^3 \\log{n})$ \\mtodo{check the complexity} time, w.h.p.\n\\end{theorem} \n} \n\n\\section{Missing Proofs}\\label{sec:miss-proof}\n\n\n\\APPENDUNIQUEID\n\n\\APPENDLEMMUNIQUE\n\n\\APPENDLABELCONSISE\n\n\\APPENDSKETCHPROP\n\n\\APPENDCONNLABELSKETCH\n\\section{Preliminaries}\nGiven a graph $G=(V,E)$, and vertex $u \\in V$, let $\\deg(u,G)$ be the degree of $u$ in $G$. \nGiven a tree $T$ and $u, v \\in T$, denote the $u$-$v$ path in $T$ by $\\pi(u,v,T)$. When the tree $T$ is clear from the context, we may omit it and write $\\pi(u,v)$. For a (possibly weighted) subgraph $G' \\subseteq G$ and a vertex pair $s,t \\in V$, let $\\mbox{\\rm dist}_{G'}(s,t)$ denote the length of the $s$-$t$ shortest path in $G'$. \n\n\\paragraph{Fault-Tolerant Labeling Schemes.}\nFor a given graph $G$, let $\\Pi: V\\times V \\times \\mathcal{G} \\to \\mathbb{R}_{\\geq 0}$\nbe a function defined on pairs of vertices and a subgraph $G' \\subset G$, where $\\mathcal{G}$ is the family of all subgraphs of $G$. For an integer parameter $f\\geq 1$, an $f$-\\emph{fault-tolerant labeling scheme} for a function $\\Pi$ and a graph family $\\mathcal{F}$ is a pair of functions $(L_{\\Pi},D_{\\Pi})$. The function $L_{\\Pi}$ is called the \\emph{labeling function}, and $D_{\\Pi}$ is called the \\emph{decoding function}. For every graph $G$ in the family $\\mathcal{F}$, the labeling function $L_{\\Pi}$ associates with each vertex $u \\in V(G)$ and every edge $e \\in E(G)$, a label $L_{\\Pi}(u,G)$ (resp., $L_{\\Pi}(e,G)$). It is then required that given the labels of any triplets $s,t, F \\in V \\times V \\times E^f$, the decoding function $D_{\\Pi}$ computes $\\Pi(s,t, G \\setminus F)$.  The primary complexity measure of a labeling scheme is the \\emph{label length}, measured by the length (in bits) of the largest label it assigns to some vertices (or edges) in $G$ over all graphs $G \\in \\mathcal{F}$. An $f$-FT connectivity labeling scheme is required to output YES iff $s$ and $t$ are connected in $G \\setminus F$.  In $f$-FT \\emph{approximate distance labeling scheme} it is required to output an estimate for the $s$-$t$ distance in the graph $G \\setminus F$. Formally, an $f$-FT labeling scheme is $q$\\emph{-approximate} if the value $\\delta(s,t,F)$ returned by the decoder algorithm satisfies that $\\mbox{\\rm dist}_{G \\setminus F}(s,t)\\leq \\delta(s,t,F) \\leq q \\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$.  Throughout the paper we provide randomized labeling schemes which provide a high probability guarantee of correctness for any fixed triplet $\\langle s,t, F \\rangle$. \n\n\n\\paragraph{Fault-Tolerant Routing Schemes.} In the setting of FT routing scheme, one is given a pair of source $s$ and destination $t$ as well as $F$ edge faults, which are initially unknown to $s$. The routing scheme consists of \\emph{preprocessing} and \\emph{routing} algorithms. The preprocessing algorithm defines labels $L(u)$ to each of the vertices $u$, and a header $H(M)$ to the designated message $M$. In addition, it defines for every vertex $u$ a routing table $R(u)$. The labels and headers are usually required to be short, i.e., of poly-logarithmic bits. \nThe routing procedure determines at each vertex $u$ the port-number on which $u$ should send the messages it receives. The computation of the next-hop is done by considering the header of the message $H(M)$, the label of the source and destination $L(s)$ and $L(t)$ and the routing table $R(u)$. The routing procedure at vertex $u$ might also edit the header of the message $H(M)$. The failing edges are not known in advance and can only be revealed by reaching (throughout the message routing) one of their endpoints. The \\emph{space} of the scheme is determined based on maximal length of message headers, labels and the individual routing tables. The stretch of the scheme is measured by the ratio between the length of the path traversed until the message arrived its destination and the length of the shortest $s$-$t$ path in $G \\setminus F$. In the more relaxed setting of \\emph{forbidden-set routing schemes} the failing edges are given as input to the routing algorithm.\n\n\n\\subsection{Additional Related Work}\n\n\\paragraph{Fault-Tolerant Labeling Schemes.} FT labels for connectivity were introduced by \\cite{courcelle2007forbidden} under the term \\emph{forbidden-set labeling}. Forbidden set refers to a subset $F$ of at most $f$ edges, such that given the labels of $s,t$ and $F$ one should determine if $s$ and $t$ are connected in $G \\setminus F$. The forbidden edge set can be treated in this context as faulty edges\\footnote{For routing, the forbidden-set scheme is slightly weaker than FT scheme as explained later.}.\nPrevious works study FT connectivity labels only in restricted graph families. For example, Courcelle et al. \\cite{CourcelleT07} presented a labeling scheme with logarithmic label length for the families of $n$-vertex graphs with bounded clique-width, tree-width and planar graphs. For $n$-vertex graphs with doubling dimension at most $\\alpha$, Abraham et al. \\cite{AbrahamCGP16} designed FT labeling schemes with label length $O((1 + 1\/\\epsilon)^{2\\alpha}\\log n)$ that output $(1+\\epsilon)$ approximation of the shortest path distances under faults. Recently, \\cite{DBLP:journals\/corr\/abs-2102-07154} studied FT exact distance labels in planar graphs, and show that any directed weighted planar graph admits fault-tolerant distance labels of size $O(n^{2\/3})$.\n\n\\paragraph{Connectivity and Distance Sensitivity Oracles.} \nConnectivity and distance sensitivity oracles are centralized data structures that support connectivity or distance queries in the presence of failures. \nThe first construction of connectivity sensitivity oracles was given by Patrascu and Thorup \\cite{patrascu2007planning} providing an $S(n)=\\widetilde{O}(fn)$ space oracle that answers $\\langle s,t, F \\rangle$ connectivity queries in $\\widetilde{O}(f)$ time. The state-of-the-art bounds of these oracles are given by Duan and Pettie \\cite{DuanConnectivitySODA17}.\nChechik et al. \\cite{chechik2012f} presented the first randomized construction of distance sensitivity oracle resilient to $f$ edge faults. \nSpecifically, for any $n$-vertex weighted graph, stretch parameter $k$, and a fault bound $f$, they provide a data-structure with $O(f k n^{1+1\/k}\\log(nW))$ space, query time of $\\widetilde{O}(|F|)$, and $O(f k)$ stretch, where $W$ is the weight of the heaviest edge in the graph. Their solution is based on an elegant transformation that converts the FT connectivity oracle of \\cite{patrascu2007planning} into an FT approximate distance oracle.\n\nWhile the main focus of this paper is in approximate distances, sensitivity oracles that report (possibly near) exact distances under faults have been studied also thoroughly in e.g., \\cite{demetrescu2002oracles,bernstein2008improved,duan2009dual,WeimannY10,GrandoniW12,ChechikCFK17,van2019sensitive}. Since reporting exact distances requires linear label length already in the fault-free setting \\cite{gavoille2004distance}, we focus on the approximate relaxation, where there is still hope to obtain labels of polylogarithmic length.\n\n\\paragraph{Fault-Tolerant Routing Schemes.}\nThe first formalization of FT routing schemes was given by the influential works of Dolev \\cite{dolev1984new} and Peleg \\cite{peleg1987fault}. These earlier works presented the first non-trivial solutions for general graphs supporting at most $\\lambda$ faulty edges, where $\\lambda$ is the edge-connectivity of the graph. Their routing labels had linear size, providing $s$-$t$ routes of possibly linear length (even in cases where the surviving $s$-$t$ path is of $O(1)$ length). In competitive FT routing schemes, it is required to provide $s$-$t$ routes of length that competes with the shortest $s$-$t$ path in $G \\setminus F$, even in cases where $G \\setminus F$ is not connected. Competitive FT routing schemes \\cite{peleg2009good} for general graphs were given by Chechik et al. \\cite{ChechikLPR10,chechik2012f} for the special case of $f\\leq 2$ faults. \nSpecifically, for a given stretch parameter $k$, they gave a routing scheme with a total space bound of $\\widetilde{O}(n^{1+1\/k})$ bits, polylogarithmic-size labels and messages, and a routing \\emph{stretch} of $O(k)$. \nThis scheme was extended later on for any $f$ by Chechik \\cite{chechik2011fault}, at the cost of increasing the routing stretch to $O(f^2(f+\\log^2 n)k)$. For a single edge failure, \\cite{rajan2012space} showed a routing scheme with routing tables of size $\\widetilde{O}(k \\deg(v)+ n^{1\/k})$ size per vertex, $O(k^2)$ stretch and $O(k+\\log{n})$ size header.\n\n\\paragraph{Forbidden Set Routing.}\nA more relaxed setting of FT routing scheme which has been studied in the literature is given by the \\emph{forbidden set routing schemes}, introduced by Courcelle and Twigg \\cite{CourcelleT07}. In that setting, it is assumed that the routing protocol knows in advance the set of faulty edges $F$. In contrast, in the FT routing setting, the failing edges are a-priori unknown to the routing algorithm,  and can only be detected upon arriving one of their endpoints. Forbidden set routing schemes have been devised to the same class of restricted graph families as obtained for the forbidden set labeling setting \\cite{CourcelleT07,AbrahamCGP16,abraham2012fully}.\n\\subsection{Our Results}\nWe provide space-efficient labeling and routing schemes for any $n$-vertex graph. Our schemes are \\emph{randomized} and provide a high probability guarantee\\footnote{As standard, we use the term high-probability to indicate success guarantee of $1-1\/n^c$ for any given constant $c>1$.} for any given triplet $\\langle s, t, F\\rangle$. In other words, the schemes can faithfully support polynomially many queries\\footnote{The same type of guarantee is provided in the centralized sensitivity oracles, e.g., of \\cite{DuanConnectivitySODA17}. Providing a high probability guarantee over all possible triplets is possible upon increasing the space bound by a factor of $f$ (largest number of faults supported).}. \n\nOur first key result presents two independent schemes for FT connectivity labels. These are the first FT connectivity labels for \\emph{general graphs}.\nThese two constructions yield the following theorem, addressing Question \\ref{q:label}: \n\n\\begin{theorem}\\label{thm:conn-labels}[FT Connectivity Labeling Schemes, Informal]\nFor any $n$-vertex graph and a bound $f$ on the number of edge faults, there is a \\emph{randomized} $f$-FT connectivity labeling scheme with label length of $O(\\min\\{f+\\log n, \\log^3 n\\})$ bits. The labels are computed in $\\widetilde{O}(m)$ time, and the decoding algorithm takes $\\operatorname{\\text{{\\rm poly}}}(f,\\log n)$ time. \n\\end{theorem}\n\nBy the tightness of the label length of fault-free connectivity labels,\nour scheme is optimal for $f=O(\\log n)$. Moreover, the label length is nearly-optimal for any $f$. Our actual scheme provides more information then merely a single bit (connected or not connected). Specifically, we augment the connectivity labels with additional information so that the decoding algorithm, given the labels of $s,t$ and $F$, can also output a succinct description of an $s$-$t$ path in $G\\setminus F$ (if such a path exists). This succinct path representation finds applications in the context of our FT routing schemes. \n\nWe next consider the task of reporting also approximate $s$-$t$ distances in $G\\setminus F$ using the labels of $s,t$ and $F$. We employ the reduction of Chechik et al. \\cite{chechik2012f} to convert the FT connectivity labels into FT approximate distance labels, providing nearly the same space vs. stretch tradeoff as in the centralized data-structures of \\cite{chechik2012f}. Specifically, we show:\n\n\\begin{theorem}\\label{thm:dist-labels}[FT Approximate Distance Labeling Schemes]\nFor any $n$-vertex (possibly weighted) graph, a bound $f$ on the number of edge faults, and a stretch parameter $k$, there is a randomized $f$-FT approximate distance labeling scheme with label length of $O(k \\cdot n^{1\/k}\\cdot \\log (nW) \\cdot \\log^3 n)$. Given the labels of $s,t$ and $F$ the scheme returns a distance estimate \n$$\\mbox{\\rm dist}_{G\\setminus F}(s,t)\\leq \\delta(s,t,F) \\leq (8k-2)(|F|+1)\\mbox{\\rm dist}_{G\\setminus F}(s,t)~.$$\n\\end{theorem}\n\nFor the purpose of routing, we exploit the extra information provided by our connectivity labels, in order to output, in addition to the distance estimate $\\delta(s,t,F)$, also a succinct description of the approximate $s$-$t$ \nshortest path in $G \\setminus F$. Our second key result provides FT compact routing schemes, with an almost optimal tradeoff between the space and stretch, for constant number of faults $f$. We answer Question \\ref{q:route} by showing:\n\n\\begin{theorem}\\label{thm:routing}[FT Compact Routing]\nFor every integers $k,f$, there exists an $f$-sensitive compact routing scheme that given a message $M$ at the source vertex $s$ and the routing label of the destination $t$, in the presence of at most $f$ faulty edges $F$ (unknown to $s$) routes $M$ from $s$ to $t$ in a distributed manner over a path of length at most $32k (|F|+1)^2\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$. The routing labels have $\\widetilde{O}(f)$ bits, the table size of each vertex is $\\widetilde{O}(f^3 \\cdot n^{1\/k} \\log(nW))$, the header size (also known as message size) is bounded by $\\widetilde{O}(f^3)$ bits. \n\\end{theorem}\nThis improves over the state-of-the-art construction of Chechik \\cite{chechik2011fault} that \nobtained routing schemes with stretch of $O(f^2(f+\\log^2 n)k)$ and tables of size $O(\\deg(v)n^{1\/k}\\log{(nW)})$ for every vertex $v$. We note that the construction of Chechik \\cite{chechik2011fault} has a bounded global space of $\\widetilde{O}(n^{1+1\/k} \\log{(nW)})$, but the individual tables might have even super-linear space (e.g., when $k=O(1)$ and $\\deg(v)=O(n)$).  For the special case of $f=2$, Chechik et al. \\cite{ChechikLPR10,chechik2012f} provide a stretch bound of $O(k)$, and total space of $\\widetilde{O}(n^{1+1\/k} \\log{(nW)})$, where the space of each table is bounded by $O(\\deg(v)n^{1\/k})$, thus super-linear in the worst case. Our scheme provides an improved bound on the individual tables, nearly matching the fault-free constructions for $f=O(1)$. We also show an improved scheme if one only aims to optimize for the global space, rather than optimizing for the largest table size for a vertex. For comparison of our results to prior work see Table \\ref{table_routing}.\n\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{ |p{4.2cm}|p{4cm}|p{6cm}|p{0.5cm}|}\n \\hline\t\n \\multicolumn{4}{|c|}{Constructions of Fault-Tolerant Routing Schemes}\\\\\n \\hline\n  Reference & Stretch & Table Size & $|F|$ \\\\\n \\hline\n  Rajan \\cite{rajan2012space} & $O(k^2)$ & $\\widetilde{O}(k \\deg(v)+ n^{1\/k})$ per vertex & 1\\\\\n  Chechik et al. \\cite{chechik2012f} & $O(k)$ & $\\widetilde{O}(n^{1+1\/k} \\log(nW))$ total size & 2\\\\\n  Chechik \\cite{chechik2011fault}  & $O(|F|^2(|F|+\\log^2 n)k)$ & $\\widetilde{O}(n^{1+1\/k} \\log(nW))$ total size & $f$\\\\\n  Chechik \\cite{chechik2011fault}  & $O(|F|^2(|F|+\\log^2 n)k)$ & $\\widetilde{O}(\\deg(v)n^{1\/k} \\log(nW))$ per vertex & $f$\\\\\n  \\textbf{Here} & $O(|F|^2 k)$ & $\\widetilde{O}(f \\cdot n^{1+1\/k} \\log(nW))$ total size & $f$\\\\\n  \\textbf{Here} & $O(|F|^2 k)$ & $\\widetilde{O}(f^3 \\cdot n^{1\/k} \\log(nW))$ per vertex & $f$\\\\\n \\hline\n\n\n \\end{tabular}\n \\caption{Comparison between FT routing schemes with a set of failures $F$}\n\\label{table_routing}\n\\end{table}\n\nFinally, we provide a lower bound result on the minimal stretch regardless for the \\emph{space} of the routing scheme, e.g., even if all vertices store all the graph edges. \n\n\\begin{theorem}[Stretch Lower-Bound for FT Routing]\\label{thm:lb-routing}\nAny FT routing randomized scheme resilient to $f$ faults induces an expected stretch of $\\Omega(f)$ regardless of the size of the routing tables and labels. In particular, this holds even if each routing table contains a complete information on the graph. \n\\end{theorem}\n\n\n\\paragraph{Open Problems.} Our work leaves several interesting open ends. One natural direction is to provide labeling and routing schemes resilient to \\emph{vertex} faults. The major challenge in handling vertex failure is that even a single faulty vertex might disconnect the graph into $\\Omega(n)$ disconnected components. Another interesting direction is to derandomize our constructions. Currently there are no deterministic constructions of FT labeling schemes for general graphs. Finally, it will be also important to provide FT distance approximate labeling schemes whose stretch bound is independent in the number of faults $f$. This problem is also open in the corresponding setting of approximate distance sensitive oracles. \n\n\\section{Compact Routing Schemes}\nIn this section, we explain how to use our FT distance labels to provide compact and low stretch routing schemes. This is the first scheme to provide an almost tight tradeoff between the space and the multiplicative stretch, for a constant number of faults $f=O(1)$.  Throughout this section, tree routing operations are performed by using the tree routing scheme of Thorup and Zwick \\cite{thorup2001compact}.\n\\begin{fact}\\label{fc:route-trees}[Routing on Trees]\\cite{thorup2001compact}\nFor every $n$-vertex tree $T$, there exists a routing scheme that assigns each vertex $v \\in V(T)$ a label $L_T(v)$ of $(1+o(1))\\log n$ bits. Given the label of a source vertex\nand the label of a destination, it is possible to compute, in constant time, the port number of the edge from the source that heads in the direction of the destination.\n\\end{fact}\n\n\n\nWe slightly modify the connectivity label of the edges and vertices by augmenting them with routing information. \nFirst, we augment the extended identifier of an edge (see Eq. (\\ref{eq:extend-ID})) with port information and tree routing information, by having:\n\\begin{equation}\\label{eq:edge-extended-routing}\n\\operatorname{EID}_T(e)=[\\operatorname{UID}(e), \\operatorname{ID}(u), \\operatorname{ID}(v), \\mathsf{ANC}\\xspace_T(u), \\mathsf{ANC}\\xspace_T(v), \\mbox{\\tt port}(u,v), \\mbox{\\tt port}(v,u), L_T(u), L_T(v)]~,\n\\end{equation}\nwhere $\\mbox{\\tt port}(u,v)$ is the port number of the edge $(u,v)$ for $u$, and the labels $L_T(u), L_T(v)$ are the tree routing labels taken from Fact \\ref{fc:route-trees}. \nWe then slightly modify the connectivity label of Eq. (\\ref{eq:conn-vertex}) to include also the tree label $L_T(u)$from Fact \\ref{fc:route-trees}, by defining \n\\begin{equation}\\label{eq:conn-vertex-label-routing}\n\\mathsf{ConnLabel}\\xspace_{G,T}(u)=\\langle \\mathsf{ANC}\\xspace_T(u), \\operatorname{ID}(u), L_T(u)\\rangle~.\n\\end{equation}\n\nThroughout this section, when applying the connectivity labels from Section \\ref{sec:ftconn-sketch} on a graph $G$ with a spanning tree $T$, we use these modified extended identifiers and labels. This will also be the basis for the application of the distance labels of Section \\ref{sec:ft-distance}. \nSimilarly to the distance labels of Section \\ref{sec:ft-distance}, we will apply the connectivity labels with respect to the different trees of the tree cover as discussed in Section \\ref{sec:ft-distance}. \nLet $T_{i,j} \\in \\mathsf{TC}\\xspace_i$, recall that $G_{i,j}=G[V(T_{i,j})]$ and that $\\mathcal{T}=\\bigcup_{i=1}^K \\mathsf{TC}\\xspace_i$ for $K=O(\\log (nW))$. \n\n\\begin{lemma}\\label{lem:succint_path_routing}\nConsider a triplet $s,t,F$ such that $s,t,F \\in G_{i,j}$. \\\\\nGiven the connectivity labels $\\{\\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(w)\\}_{w \\in F \\cup \\{s,t\\}}$, we can determine w.h.p if $s$ and $t$ are connected in $G_{i,j} \\setminus F$. If they are connected, we can output a labeled $s$-$t$ path $\\widehat{P}$ of length $O(f)$ that provides a succinct description of the $s$-$t$ path in $G_{i,j} \\setminus F$. The edges of $\\widehat{P}$ are labeled by $0$ and $1$, where $0$-labeled edges correspond to $G_{i,j}$-edges and $1$-labeled edges $e=(x,y)$ correspond to $x$-$y$ paths in $T_{i,j} \\setminus F$. For each $G_{i,j}$-edge, the succinct path description has the port information of the edge, and for each $x-y$ path, the description has the tree routing labels $L_{T_{i,j}}(x),L_{T_{i,j}}(y)$.\nThe length of the $s$-$t$ path encoded by $\\widehat{P}$ is bounded by $(4k-1)(|F|+1)\\cdot 2^i$. \n\\end{lemma}\n\n\\begin{proof}\nThe proof follows the proof of Lemma \\ref{lem:useful-recovery-edges}.\nUsing $\\{\\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(w)\\}_{w \\in F \\cup \\{s,t\\}}$, the decoding algorithm of Section \\ref{sec:ftconn-sketch} determines if $s$ and $t$ are connected in $G_{i,j} \\setminus F$. If they are connected, then from Lemma \\ref{lem:useful-recovery-edges}, we get a succinct description of the $s$-$t$ path in $G_{i,j} \\setminus F$. We next show that the algorithm indeed has the relevant port and tree routing information. For this note that all the vertices in the path $\\widehat{P}$ obtained by Lemma \\ref{lem:useful-recovery-edges} are either $s$ and $t$ or endpoints of the $|F|$ recovery edges found in the algorithm. The labels of $s$ and $t$ contain the tree routing information $L_{T_{i,j}}(s)$ and $L_{T_{i,j}}(t)$, and when the algorithm finds a recovery edge, it learns about its extended id $\\operatorname{EID}_{T_{i,j}}(e)$ that has the port information and tree routing information of its endpoints. Any $G_{i,j}$-edge in $\\widehat{P}$ is a recovery edge, hence the algorithm has its port information, and for any $x$-$y$ path in $T_{i,j} \\setminus F$, the algorithm has the tree routing labels $L_{T_{i,j}}(x),L_{T_{i,j}}(y)$, as needed.\nThe stretch analysis follows the stretch analysis in Section \\ref{sec:ft-distance}. It is based on the fact that $\\widehat{P}$ has as most $|F|+1$ subpaths in $T_{i,j} \\setminus F$, each of length at most $(4k-2)2^i$, and at most $|F|$ recovery edges of weight at most $2^i$.\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{Forbidden Set Routing (Faulty Edges are Known)}\\label{sec:routing-known}\nWe start by describing the routing scheme in the forbidden set setting, where the faulty edges $F$ are known to the source vertex $s$. We show the following.\n\n\\begin{theorem}\\label{thm:routing-known}[Forbidden-Set Routing]\nFor every integers $k,f$, there exists an $f$-sensitive compact routing scheme that given a message $M$ at the source vertex $s$, a label of the destination $t$, and labels of at most $f$ forbidden edges $F$ (known to $s$), routes $M$ from $s$ to $t$ in a distributed manner over a path of length at most $(8k-2)(|F|+1)\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$. The table size of each vertex is bounded by $\\widetilde{O}(n^{1\/k} \\log{(nW)})$. The header size of the messages is bounded by $\\widetilde{O}(f)$ bits. The labels of vertices and edges have size $\\widetilde{O}(n^{1\/k} \\log(nW))$.\n\\end{theorem}\n\n\\begin{proof}\nThe algorithm is based on the distance labels from Section \\ref{sec:ft-distance} using the slightly modified connectivity labels (augmented with port and tree roting information). Recall that the distance labels are based on applying fault-tolerant connectivity labels on different graphs $G_{i,j}$, we use the slightly modified connectivity labels and the corresponding distance labels. \nThe routing table of each vertex $u$ consists of its distance label $\\mathsf{DistLabel}\\xspace(u)$. The label of an edge $e$ is $\\mathsf{DistLabel}\\xspace(e)$. Each distance label has $\\widetilde{O}(n^{1\/k} \\log(n W))$ bits. \n\nIn the routing algorithm, the vertex $s$ is given the label $\\mathsf{DistLabel}\\xspace(t)$, and the labels $\\{\\mathsf{DistLabel}\\xspace(e)\\}_{e \\in F}$, and it needs to route a message to $t$ in the graph $G \\setminus F$. \nRecall that the algorithm from Section \\ref{sec:ft-distance} works in $K$ phases, where in phase $i$ it checks if $s$ and $t$ are connected in the graph $G_{i,i^*(s)} \\setminus F$ that contains the $2^i$-ball around $s$. Let $i$ be the first iteration where $s$ and $t$ are connected in $G_{i,i^*(s)} \\setminus F$ according to the algorithm, and denote $G_i = G_{i,i^*(s)},T_i = T_{i,i^*(s)}$, and let $F_i = F \\cap G_i$. The algorithm can also give a succinct description of an $s$-$t$ path in $G_i \\setminus F_i$ following Lemma \\ref{lem:succint_path_routing}. For this, note that we indeed have all the required information. The distance labels of edges in $F$ in particular contain the labels $\\{\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)\\}_{e \\in F_i}$, and we can also tell which edges of $F$ are in $G_i$ from the labels. Also, the labels of $s,t$ contain the information $\\operatorname{ID}_{T_i}(s),\\operatorname{ID}_{T_i}(t)$ if they are both in $T_i$ (otherwise, they are not connected in level $i$).\n\nThe path $\\widehat{P}$ as described in Lemma \\ref{lem:succint_path_routing} is composed of $O(|F|)$ parts, where segment $(x,y)$ in the path corresponds either to an edge in $G_i$ or to a tree path in $T_i \\setminus F$, it also has the relevant port and tree routing information. Our goal is to route a message according to this path. For this we add to the header of the message the description of $\\widehat{P}$, the indexes $(i,i^*(s))$ of the tree we explore and an index $1 \\leq q \\leq 2|F|+1$ that represents the segment of $\\widehat{P}$ we currently explore, initially $q=1$. Overall, the header size is $\\widetilde{O}(f)$. To route a message according to the path, we work as follows. The header specifies the current segment in $\\widehat{P}$. If the current segment corresponds to an edge $(x,y) \\in G$, then $x$ uses the port information to route the message to $y$ and increases the index $q$. Otherwise, the current segment represents a tree path $(x,y) \\in T_i$ and a vertex $u$ in this path uses its routing label in $T_i$ and the routing label of $y$ in $T_i$ (that is part of the header) to route the message towards $y$. When the message reaches $y$, it increases the index $q$. This completes the description of the routing process. The length of the path described is at most $(8k-2)(|F|+1)\\cdot \\mbox{\\rm dist}_{G\\setminus F}(s,t)$, as shown in Section  \\ref{sec:ft-distance}. \n\\end{proof}\n\n\n\n\\subsection{Fault-Tolerant Routing (Faulty Edges are Unknown)}\\label{sec:route-unknown}\nWe now consider the more involved setting where the set of failed edges $F$ are unknown to $s$. In this case, an edge $(u,v) \\in F$ is detected only when the message arrives, during the routing procedure, to one of the endpoints of $e$. Note that the routing scheme should, by definition, be prepared to any set of faulty edges $F$. However, the space bound of our scheme is required to be bounded by $\\widetilde{O}(f n^{1+1\/k})$, which is possibly much smaller than the number of graph edges $m$. This in particular implies that we cannot store the FT distance labels of all the graph edges. Nevertheless, we show that it is sufficient to explicitly store the labeling information for the tree edges in $\\mathcal{T}=\\bigcup_{i=1}^K \\mathsf{TC}\\xspace_i$. The required information for the failed non-tree edges would be revealed throughout the process, by applying the decoding algorithm of Lemma \\ref{lem:succint_path_routing}.\nOur routing scheme eventually routes the message along the $s$-$t$ path encoded by the FT distance labels of $s,t$ and $F$. However, since the labels of $F$ are unknown in advance, the routing scheme will detect these edges in a trail and error fashion which induces an extra factor of $f$ in the final multiplicative stretch. This extra $f$ factor is also shown to be essential, in the end of the section.\nWe proceed by describing the routing tables.  \n\n\n\n\\paragraph{The routing labels and tables.} For ease of presentation, we first describe a solution with a multiplicative stretch of $O(kf^2)$, and \\emph{global} space of $\\widetilde{O}(f K \\cdot n^{1+1\/k})$, but the individual tables of some of the vertices might be large. We later on improve the space of each table to $\\widetilde{O}(f^3 K \\cdot n^{1\/k})$ bits.\n\nRecall that $\\mathcal{T}=\\bigcup_i^{K} \\mathsf{TC}\\xspace_i$, for $K=O(\\log (nW))$ is a collection of tree covers in all $K=\\lceil \\log (nW) \\rceil$ distance scales, see Eq. (\\ref{eq:TC-i}). For every vertex $v$, let $\\deg_{\\mathcal{T}}(v)=\\sum_{T_{i,j} \\in \\mathcal{T}}\\deg(u,T_{i,j})$ be the sum of degrees of $u$ in the collection of trees  $\\mathcal{T}$. Recall that $G_{i,j}=G[V(T_{i,j})]$.\nFor the routing we apply the FT connectivity labels on the graphs $G_{i,j}$, similarly to Section \\ref{sec:ft-distance}. \n\\\\ \\\\\n\\noindent \\textbf{Routing labels.} The routing process uses at most $f'=f+1$ independent applications of randomized FT connectivity labels from Section \n\\ref{sec:ftconn-sketch}, applied on each one of the graphs $G_{i,j}$. \nIn more details, when we apply the labeling scheme on the graph $G_{i,j}$ with spanning tree $T_{i,j}$, we use $f'$ independent random seeds $\\mathcal{S}_h$ to determine the randomness of the sketches. \nHowever, the seed $\\mathcal{S}_{ID}$ used to determine the extended ids of edges in $G_{i,j}$ is fixed in the $f'$ applications, hence the extended identifiers of the edges (see Eq. (\\ref{eq:extend-ID})) are fixed in all the $f'$ applications, and we only use fresh randomness to compute the sketch information using $f'$ independent seeds $\\mathcal{S}^1_h,\\ldots, \\mathcal{S}^{f'}_h$. \nThis process is done independently on each one of the graphs $G_{i,j}$. \n\nDenote the output connectivity labels obtained by the ${\\ell}^{th}$ application of the scheme (using $\\mathcal{S}^\\ell_h$) on the graph $G_{i,j}$ by $\\mathsf{ConnLabel}\\xspace^{\\ell}_{G_{i,j},T_{i,j}}(w)$ for every $w \\in E(G_{i,j})\\cup V(G_{i,j})$. For every edge $e \\in G_{i,j}$, define its $T_{i,j}$ routing label by\n\\begin{equation}\\label{eq:route-edge-label}\n    L_{route,i,j}(e)=\n    \\begin{cases}\n      (\\mathsf{ConnLabel}\\xspace^1_{G_{i,j},T_{i,j}}(e),\\ldots,\\mathsf{ConnLabel}\\xspace^{f'}_{G_{i,j},T_{i,j}}(e)),& \\mbox{~for~} e \\in T_{i,j} \\\\\n     \\operatorname{EID}_{T_{i,j}}(e),& e \\in G_{i,j}\\setminus E(T_{i,j})~.\n    \\end{cases}\n\\end{equation}\nEvery $L_{route,i,j}(e)$ label has $O(f \\log^3 n)$ bits. \nIn our routing algorithms, the $T_{i,j}$ routing labels of the discovered faulty edges will be added to the header for the message in order to guide the routing process. \nWe now turn to define the routing labels of vertices. Recall that for a vertex $v$ and index $1 \\leq i \\leq K$, we denote by $i^*(v)$ an index such that the $2^i$-ball around $v$ is contained in $G_{i,i^*(v)}$.\nThe routing label $L_{route}(v)$ of $v$ \nFor every \\emph{vertex} $v$, the routing label of $v$ is given by\n\\begin{equation}\\label{eq:Label-route-vertex}\nL_{route}(v) = \\{(i^*(v), \\mathsf{ConnLabel}\\xspace^1_{G_{i,i^*(v)},T_{i,i^*(v)}}(v) | i \\in [1,K]\\}~.\n\\end{equation}\n\n\nNote that by definition, the connectivity labels of the \\emph{vertices} are the same in all $f'$ applications of the labeling algorithm, and therefore it is sufficient to include only one of these copies in the label. The size of the label is $O(K\\log{n})=O(\\log{n} \\log{nW})$.\n\\\\ \\\\\n\\noindent \\textbf{Routing tables.} The routing table $R_{route}(v)$ of a vertex $v$ has the following information for every tree \n$T_{i,j}$ such that $v \\in T_{i,j}$:\n\\begin{equation}\\label{eq:route-table-ij}\nR_{route,i,j}(v)=\\{L_{route,i,j}(e), e \\in E(v,T_{i,j})\\} \\cup \\{\\mathsf{ConnLabel}\\xspace^1_{G_{i,j},T_{i,j}}(v)\\}~,\n\\end{equation}\nwhere $E(v,T_{i,j})$ is the set of edges incident to $v$ in the tree $T_{i,j}$. \nThe final routing table is given by $R_{route}(v)=\\{R_{route,i,j}(v), (i,j) ~\\mid~ T_{i,j} \\in \\mathcal{T}, v\\in T_{i,j}\\}$.\n\nSince the connectivity labels are of size $\\widetilde{O}(f)$, and as each $v$ appears in $\\deg_{\\mathcal{T}}(v)$ trees, the size of the table is $\\widetilde{O}(f \\deg_{\\mathcal{T}}(v)).$ Since the total number of tree edges in $\\mathcal{T}$ is bounded by $\\widetilde{O}(K \\cdot n^{1+1\/k})$, this provides a global space bound of $\\widetilde{O}(fK \\cdot n^{1+1\/k})$ bits.\n\n\n\\paragraph{The routing algorithm.} In the routing algorithm, the source vertex $s$ initially gets the routing label $L_{route}(t)$ (Eq. (\\ref{eq:Label-route-vertex})) of the destination $t$ and its own routing table, $R_{route}(s)$, and its goal is to find the smallest radius graph $G_{i,j}$ such that $s$ and $t$ are connected in $G_{i,j} \\setminus F$, and use it for routing. \nAs the set $F$ is \\emph{not} known in advance, the algorithm works in $K= O(\\log{nW})$ phases, where in phase $i$ it tries to route a message in the graph $G_{i,i^*(t)}$ (which contains the entire $2^i$-radius ball of $t$). If $s$ and $t$ are connected in $G_{i,i^*(t)} \\setminus F$ the algorithm succeeds, and otherwise we proceed to the next phase, corresponding to the distance scale of $2^{i+1}$. \nWe next describe the algorithm for a single phase $i$, we denote $G_i = G_{i,i^*(t)}, T_i = T_{i,i^*(t)}$. Note that $s$ can deduce the index $i^*(t)$ from the routing label of $t$, and it can check if $s \\in T_i$ using its routing table. If $s \\not \\in T_i$, we proceed to the next phase.\n\nIf $s \\in T_i$, the routing procedure for phase $i$ has at most $|F|+1$ iterations. We maintain the following invariant in the beginning of each iteration $\\ell \\in \\{1,\\ldots, |F|+1\\}$: (i) the iteration starts at vertex $s$, (ii) the algorithm has already detected a subset of $\\ell-1$ faulty edges $F_\\ell \\subseteq F$, and (iii) the header contains the labels $\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)$ of all the edges $e \\in F_\\ell$. Each iteration $\\ell \\leq |F|+1$ terminates either at the destination vertex $t$, or at the source vertex $s$. In addition, w.h.p., if $s$ and $t$ are connected in $G_i \\setminus F$, iteration $|F|+1$ terminates at $t$. The invariant holds vacuously for iteration $1$.\n\nWe now describe the $\\ell^{th}$ iteration (of the $i^{th}$ phase) of the routing procedure given the invariant. The source vertex $s$ considers the $\\ell^{th}$ copy of the FT connectivity labels, $\\mathsf{ConnLabel}\\xspace^{\\ell}_{G_i,T_i}(e)$ for every $e\\in F_\\ell$. \nUsing the routing labels of the edges, that are part of the header, the routing label $L_{route}(t)$ (of Eq. (\\ref{eq:Label-route-vertex})) and the routing table $R_{route}(s)$, $s$ can apply the decoding algorithm of \nLemma \\ref{lem:succint_path_routing} to determine if $s$ and $t$ are connected in $G_i \\setminus F_\\ell$. \nIf the answer is no, the algorithm proceeds to the next phase $i+1$.\nOtherwise, by applying the decoding algorithm of Lemma \\ref{lem:succint_path_routing}, it computes the succinct path $\\widehat{P}_\\ell$. The path $\\widehat{P}_\\ell$ encodes an $s$-$t$ path in $G_i \\setminus F_\\ell$, that includes the relevant port and tree routing information of its vertices. The header of the message $H_\\ell$ then contains \n$$H_\\ell=\\langle \\widehat{P}_\\ell, i, i^*(t), \\{L_{route, i, i^*(t)}(e)\\}_{e \\in F_{\\ell}}, q \\rangle~,$$ where $q = O(f)$ is an index indicating the current segment of $\\widehat{P}_\\ell$ we explore. Note that the header $H_\\ell$ contains the $f$ copies of connectivity labels of the $F_{\\ell}$ edges, and not only the $\\ell^{th}$ copy.\nThe size of the header is $\\widetilde{O}(f^2)$, as the description of the path has size $\\widetilde{O}(f)$, and additionally we have at most $f$ faulty edges with labels of size $\\widetilde{O}(f)$.\nLet $P_\\ell$ be the $G$-path encoded by the path $\\mathcal{P}_\\ell$. The algorithm then routes the message along $P_\\ell$ in the same manner as in Sec. \\ref{sec:routing-known}. In the case where $P_{\\ell}\\cap F=\\emptyset$, the iteration successfully terminates at the destination vertex $t$. From now on, we consider the case that $P_{\\ell}$ contains at least one faulty edge. \n\nLet $e=(u,v)$ be the first edge (closest to $s$) on the path $P_\\ell$ that belongs to $F$. Since $P_\\ell \\cap F_\\ell=\\emptyset$, it holds that $e \\in F \\setminus F_\\ell$. Without loss of generality, assume that $u$ is closer to $s$ on $P_\\ell$. Thus the faulty edge $e$ is detected upon arriving to the vertex $u$. \nIn the case where $e$ is a \\emph{non-tree edge}, then it must be a $G$-edge on $\\widehat{P}_\\ell$. Since this path has the extended ids $\\operatorname{EID}_{T_i}(e)$ of its $G$-edges, and since the connectivity label of a non-tree edge $e$ is its extended identifier $\\operatorname{EID}_{T_i}(e)$ in all the $f'$ applications of the scheme on $G_i$\\footnote{This is because we use the same random seed $\\mathcal{S}_{ID}$ in all these applications.}, $u$ can add $L_{route,i,i^*(t)}(e)=\\operatorname{EID}_{T_i}(e)$ to the header of the message. Assume now that $e$ is a tree edge in $T_i$. The vertex $u$ then adds the routing label $L_{route,i,i^*(t)}(e)$ to the header of the message, as $e$ is a tree edge adjacent to $u$ it has this information in its routing table. Finally, it marks the header with the sign $R$, indicating that the message should now be routed in the reverse direction, until arriving $s$ again. This completes the description of iteration $\\ell$. It is easy to see that the invariant is maintained. If $s$ and $t$ are connected in $G_i \\setminus F$, after at most $f$ iterations all faulty edges are detected. In the last iteration, the path computed based on the labeling information is free from faulty edges, and the routing is completed (in the same manner as in Sec. \\ref{sec:routing-known}) at the destination $t$. We next bound the multiplicative stretch of the routing.\n\n\n\n\n\\begin{claim}\\label{cl:route-length}\nFix a set of faulty edges $F$, and let $s,t$ be vertices that are connected in $G \\setminus F$. Then, the message is routed from $s$ to $t$ within $32k (|F|+1)^2 \\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$ steps, w.h.p.\n\\end{claim}\n\\begin{proof}\nFirst note that since each iteration and each graph $G_{i}$ uses an independent set of FT connectivity labels, then in each phase and each iteration the decoding algorithm succeeds w.h.p. and outputs an $s$-$t$ path $\\widehat{P}_\\ell$ if exists. \n\nAssume that $\\mbox{\\rm dist}_{G \\setminus F}(s,t) \\in (2^{i-1},2^i]$. Then, $s$ and $t$ are connected in $G_i \\setminus F$, as $T_i = T_{i,i^*(t)}$ contains the $2^i$-ball around $t$. We show that the algorithm terminates at $t$ in phase $i$ or before it, and that in any phase $j \\leq i$, the routing algorithm traverses a path of length at most $2(4k-1)(|F|+1)^2 \\cdot 2^j$.\n\nLet $j \\leq i$. In the $\\ell$'th iteration of phase $j$, the algorithm first checks if $s$ and $t$ are connected in $G_j \\setminus F_{\\ell}$, where $F_{\\ell}$ is the set of currently detected faults. If the answer is no, the algorithm proceeds to the next phase. Otherwise, it tries to route a message from $s$ to $t$ on the path encoded by $\\widehat{P}_{\\ell}$. The length of the path is bounded by $(4k-1)(|F|+1)\\cdot 2^j$ from Lemma \\ref{lem:succint_path_routing}. The algorithm either succeeds, or finds a faulty edge on the way in which case it returns to $s$ by traversing the same path on the reverse direction. Overall, the algorithm traverses a path of length at most $2(4k-1)(|F|+1)\\cdot 2^j$, in this iteration. In all $|F|+1$ iterations of phase $j$, the length of the path explored is at most $2(4k-1)(|F|+1)^2 \\cdot 2^j$. Summing over all iterations $j \\leq i$, the stretch is bounded by $$\\sum_{j=1}^{i} 2(4k-1)(|F|+1)^2 \\cdot 2^j = 2(4k-1)(|F|+1)^2 \\sum_{j=1}^i 2^j \\leq 2^{i+2} (4k-1)(|F|+1)^2 \\leq 32k (|F|+1)^2 \\mbox{\\rm dist}_{G \\setminus F}(s,t).$$ The last inequality uses the fact that $2^{i-1} \\leq \\mbox{\\rm dist}_{G \\setminus F}(s,t).$ \n\nIn the $i$'th phase, since $s$ and $t$ are connected in $G_i \\setminus F$, then for any $F_{\\ell} \\subseteq F$, $s$ and $t$ are connected in $G_i \\setminus F_{\\ell}$, hence the algorithm always finds a path $\\widehat{P}_{\\ell}$. Hence, it either succeeds in routing the message to $t$ in one of the iterations (or one of the previous phases), or learns about all the failures $F$. In the latter case, in iteration $|F|+1$ it learns about a failure-free path $\\widehat{P}_{|F|+1}$, and the routing terminates at $t$. This completes the proof.\n\\end{proof}\nTo conclude, we have the following.\n\n\\begin{theorem}\nFor every integers $k,f$, there exists an $f$-FT compact routing scheme that given a message $M$ at the source vertex $s$ and a label $L_{route}(t)$ of the destination $t$, in the presence of at most $f$ faulty edges $F$ (unknown to $s$) routes $M$ from $s$ to $t$ in a distributed manner over a path of length at most $32k (|F|+1)^2\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$. \nThe global table size is $\\widetilde{O}(f \\cdot n^{1+1\/k} \\log{(nW)})$.\nThe header size of the messages is bounded by $\\widetilde{O}(f^2)$ bits, and the label size of vertices is $O(\\log{(nW)} \\log{n})$. \n\\end{theorem}\n\n\n\n\n\n\\paragraph{Improving the size of the routing tables.} \nSo far, we have described a routing scheme that consumes a total space of $\\widetilde{O}(f\\cdot n^{1+1\/k}\\log (nW))$ bits, and multiplicative stretch\n$32(|F|+1)^2 k$. We now explain the required modifications needed to providing routing tables with $\\widetilde{O}(f^3\\cdot n^{1\/k})$ bits per vertex. The most space consuming information for a vertex $u$ is the connectivity labeling\ninformation of the edges incident to $u$ in each of the trees $T_{i,j} \\in \\mathcal{T}$. As the degree of $u$ in some of the trees might be $\\Theta(n)$, it leads to tables of possible super-linear size. To reduce the space of the individual tables, we apply a load balancing idea which distributes the labeling information incident to \\emph{high}-degree vertices among their neighbors. \n\nInstead of storing the labeling information of $e=(u,v)$ at the routing tables of $u$ and $v$, we define \nfor every tree $T \\in \\mathcal{T}$ and an edge $e=(u,v) \\in T$, a subset $\\Gamma_T(e)$ of vertices that store the connectivity labeling information of $e$ in $T$. We will make sure that the information on some vertex in $\\Gamma_T(e)$ can be easily extracted in the routing procedure upon arriving one of its endpoints. In addition, we will make sure that each vertex stores the information only for a small number of edges in each of its trees. Consider an edge $e=(u,v)$ in a tree $T$, and assume, without loss of generality, that $u$ is the parent of $v$ in the tree $T$. In the case where $\\deg(u,T)\\leq f+1$, we simply let $\\Gamma_T(e)=\\{u,v\\}$. That is, the label of $e$ is stored by both endpoints of $e$ (as before). The interesting case is where $\\deg(u,T)\\geq f+2$, in which case, $u$ might not be able to store the label of $e$, and will be assisted by its other children as follows.  Let $Child(u,T)=[v_1,\\ldots, v_\\ell]$ be the lexicographically ordered list of the children of $u$ in $T$.  The algorithm partitions $Child(u,T)$ into consecutive blocks of size $f+1$ (the last block might have $2f+1$ vertices). Letting $[v_{q,1}, \\ldots, v_{q,f+1}] \\subseteq Child(u,T)$ be the block containing $v$, define\n$$\\Gamma_T(e)=\\{v_{q,1}, \\ldots, v_{q,f+1}\\}~.$$\nNote that in particular, $v \\in \\Gamma_T(e)$. Thus, the label of $e$ is stored by $v$ and $\\ell \\in [f,2f-1]$ additional children of $u$ in $T$. \n\nWe then modify the tree labels from Fact \\ref{fc:route-trees} to contain the port information of $\\Gamma_T(e)$. \nIn order to do that, we will be using the more relaxed variant of Fact \\ref{fc:route-trees}, we have:\n\\begin{claim}\\label{cl:route-trees-port}\nFor every $n$-vertex tree $T$, there exists a (deterministic) routing scheme that assigns each vertex $v \\in V(T)$ a label $L_T(v)$ of $O(f\\log^2 n)$ bits and table $R_T(v)$ of $O(f\\log n)$ bits. Given the label $L_T(t)$ of the target $t$  \nand the routing table $R_T(u)$, the vertex $u$ can compute in $\\widetilde{O}(f)$ time: (i) the port number of the edge $e=(u,v)$ on its tree path to $t$, and (ii) the port numbers of the neighbors of $u$ in the set $\\Gamma_T(e=(u,v))$. \n\\end{claim}\n\\begin{proof}\nThe proof follows by slightly modifying the simpler scheme of Fact \\ref{fc:route-trees} by \\cite{thorup2001compact}. Specifically, we will be using the routing scheme based on heavy-light tree decomposition. This scheme assigns each vertex $v$ labels of $O(\\log^2 n)$ bits that contain the port information of the at most $O(\\log n)$ light edges on the root to $v$ path in $T$.  The vertices are enumerated in DFS ordering, and the label of each vertex  contains its DFS range, and the specification of all light edges on its path in $T$ from the root, along with a port information of these edges.  The routing table of $v$ stores its DFS range, the port number of the (unique) heavy child of $v$ and also the port to its parent. In our modification, we augment the label of each vertex $u$ with the port information of $\\Gamma_T(e')$ for every light edge $e'$ appearing on the root to $u$ path in $T$. Since there are $O(\\log n)$ such light edges, the total label information is encoded in $O(f\\log^2 n)$ bits. The routing table $R_T(u)$ is augmented with the port information for the set $\\Gamma_T(e'')$, where $e''$ is the (unique) heavy child of $u$.  The routing scheme is then exactly as described at \\cite{thorup2001compact}, only that in addition to the port of the next-hop $e=(u,v)$, we also obtain the port information of $\\Gamma_T(e)$. This increases the labels and tables in the scheme of \\cite{thorup2001compact} by a factor of $O(f)$, the claim follows.  \n\\end{proof}\n\nSince the modified claim of tree routing defines now both tree routing labels and tables, we employ the following modifications. The extended identifier $\\operatorname{EID}_T(e)$ of an edge $e=(u,v)$ from Eq. (\\ref{eq:edge-extended-routing})\ncontains the modified tree labels and thus has $O(f\\log^2 n)$ bits.\n\nThe \\emph{routing labels} of Eq. (\\ref{eq:route-edge-label}) are defined in the same manner only using the modified extended edge identifiers. The routing label of each edge has $\\widetilde{O}(f^2)$ bits, and routing label of every vertex has $\\widetilde{O}(f)$ bits. \nWe are now ready to describe the more succinct \\emph{routing tables} of each vertex $v$. We modify the definition of Eq. (\\ref{eq:route-table-ij}) by letting:\n\\begin{equation*}\\label{eq:route-table-ij-mod}\nR_{route,i,j}(v)=\\{L_{route,i,j}(e), e \\in \\Gamma_{T_{i,j}}(e)\\} \\cup \\mathsf{ConnLabel}\\xspace^1_{G_{i,j},T_{i,j}}(v) \\cup R_{T_{i,j}}(v)~,\n\\end{equation*}\nthus the routing table $R_{route,i,j}(v)$ is augmented the tree routing tables $R_{T_{i,j}}(v)$ of Claim \\ref{cl:route-trees-port}. In addition, $R_{route}(v)=\\{R_{route,i,j}(v), (i,j) ~\\mid~ T_{i,j} \\in \\mathcal{T}, v\\in T_{i,j}\\}$ as before.\nWe therefore have:\n\\begin{claim}\\label{cl:route-balance-table}\nThe size of each routing table $R_{route}(v)$ is bounded by $\\widetilde{O}(f^3 K n^{1\/k})$ bits.\n\\end{claim}\n\\begin{proof}\nFor every tree $T_{i,j}$ containing $v$, $v$ stores the routing labels for the tree $T_{i,j}$ of all edges in the set $E'(v,T_{i,j})=\\{e \\in T_{i,j} ~\\mid~ v \\in  \\Gamma_{T_{i,j}}(e)\\}$. Since each connectivity label of an edge contains the modified tree labels from Fact \\ref{cl:route-trees-port}, it has $\\widetilde{O}(f)$ bits, and as the routing label for $T_{i,j}$ contains $O(f)$ copies of this label, overall each routing label of an edge has $\\widetilde{O}(f^2)$ bits. Observe that $|E'(v,T_{i,j})|=O(f)$ as each vertex stores the label of its parent in the tree, $O(f)$ child edges, and $O(f)$ child edges of its parent in the tree. Since each $v$ participates in $\\widetilde{O}(K n^{1\/k})$ trees, overall its routing table has $\\widetilde{O}(f^3 K n^{1\/k})$ bits, as required. \n\\end{proof}\n\nIt remains to explain the required modifications for the routing procedure over a tree $T_i=T_{i,i^*(t)}$. Upon arriving to a vertex $u$ incident to a faulty \\emph{tree} edge $e=(u,v)$ the procedure is as follows. If $e$ is a non-tree edge or if $u$ stores the connectivity label $\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)$\\footnote{This covers the cases where $v$ is either a parent of $u$ or else, it is one of the at most $f+1$ children of $u$ in $T_i$.}, then $u$ adds the routing label of the edge to the header, as before. In the remaining case it must hold that $e$ is the edge incident to $u$ on its tree path to some vertex $y$. By using the tree routing scheme of Claim \\ref{cl:route-trees-port} we have that given the tree routing labels  $L_{T_{i}}(u)$ and $L_{T_{i}}(y)$, the vertex $u$ can also obtain the port numbers of its $\\ell \\in [f,2f-1]$ children in $\\Gamma_{T_{i}}(e)$ that store the label $\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)$. Since there are at most $f$ edge faults in the network, and $\\Gamma_{T_{i,j}}(e)$ contains information on at least $f+1$ ports of $u$'s neighbors that contain the label of $e$, the vertex $u$ can access a non-faulty neighbor, say $w$, that has the label information of $e$. That vertex can then add the labeling information of $e$ to the header of the message, and the routing algorithm proceeds as before. Since we use the modified tree labels of Claim \\ref{cl:route-trees-port}, each connectivity label has $\\widetilde{O}(f)$ bits, and each routing label of an edge for a tree $T_{i,j}$ has $\\widetilde{O}(f^2)$ bits. Since the header stores the routing labels of $O(f)$ edges, it consists of $\\widetilde{O}(f^3)$ bits. \n\n\nThe stretch is still bounded by $32k (|F|+1)^2\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$, as we next explain. Recall that in the proof of Claim \\ref{cl:route-length}, we bounded the length of the path we explore in one iteration of the algorithm of phase $j$ by $2(4k-1)(|F|+1)2^j.$ In the new scheme, when we discover a faulty edge, the vertex $u$ may send messages to $|F|+1$ neighbors until it finds the label of the edge. This adds at most $2(|F|+1)2^j$ to the stretch, as the weight of edges in the tree of phase $j$ is at most $2^j$, and we may send messages in both directions. This gives that the length of the path we explore in one iteration is now at most $2(4k-1)(|F|+1)2^j+2(|F|+1)2^j=8k(|F|+1)2^j.$ The rest of the analysis proceeds as in the proof of Claim \\ref{cl:route-length}, and gives that the stretch is bounded by $32k (|F|+1)^2\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$ (we get the same bound as in the original proof we bounded $2(4k-1)$ with $8k$ during the analysis).\nWe therefore have:\n\\begin{theorem}\\label{thm:routing-unknown}[Fault-Tolerant Routing]\nFor every integers $k,f$, there exists an $f$-sensitive compact routing scheme that given a message $M$ at the source vertex $s$ and a label $L_{route}(t)$ of the destination $t$, in the presence of at most $f$ faulty edges $F$ (unknown to $s$) routes $M$ from $s$ to $t$ in a distributed manner over a path of length at most $32k (|F|+1)^2\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$. The routing labels have $\\widetilde{O}(f)$ bits, the table size of each vertex is $\\widetilde{O}(f^3 \\cdot n^{1\/k} \\log(nW))$. The header size of the messages is bounded by $\\widetilde{O}(f^3)$ bits. \n\\end{theorem}\n\n\n\\paragraph{Lower Bound.} Finally, we show that the price of not knowing the set of faulty edges $F$ in advance might indeed incur a multiplicative stretch of $\\Omega(f)$. \n\n\\begin{proof}[Proof of Theorem \\ref{thm:lb-routing}]\nConsider a graph that consists of $f+1$ vertex disjoint $s$-$t$ paths, each of length $L=\\Theta(n\/f)$. The last edge of each of the paths, except for one, is faulty. Assume that the non-faulty path is chosen uniformly at random. Since the routing scheme is oblivious to the faulty edges, it can discover a faulty edge only upon sending the message to one of the edge endpoints. The expected length of the routing is given by:\n$$\\frac{L}{f+1} +2L \\cdot \\left(1-\\frac{1}{f+1} \\right)\\cdot \\frac{1}{f} + \\ldots+ \\left(f+1\\right)L\\cdot \\prod_{i=0}^{f-1} \\left(1-\\frac{1}{f+1-i}\\right)=\\Omega(f L)~.$$ \nSince the $s$-$t$ shortest path under these faults is $L$, the proof follows. See Fig. \\ref{fig:LB-stretch} for an illustration.\n\\end{proof}\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.40]{lb.pdf}\n\\caption{\\sf Illustration for a stretch lower bound for any FT routing schemes. The $s$-$t$ pair are connected by $f+1$ vertex disjoint paths of length $L$. Since the faulty-edge is the last edge of the path, the routing requires $\\Omega(L)$ steps to discover a single faulty edge. As the non-faulty path is chosen uniformly at random, in expectation, the routing requires $\\Omega(fL)$ steps.  \\label{fig:LB-stretch}\n}\n\\end{center}\n\\end{figure}\n\n\n\\section{Compact Routing Schemes}\nIn this section, we explain how to use our FT-distance labels to provide compact and low stretch routing schemes. This is the first scheme to provide an almost tight tradeoff between the space and the multiplicative stretch, for a constant number of faults $f=O(1)$.  Throughout this section, tree routing operations are performed by using the tree routing scheme of Thorup and Zwick \\cite{thorup2001compact}.\n\\begin{fact}\\label{fc:route-trees}[Routing on Trees]\\cite{thorup2001compact}\nFor every $n$-vertex tree $T$, there exists a routing scheme that assigns each vertex $v \\in V(T)$ a label $L_T(v)$ of $(1+o(1))\\log n$ bits. Given the label of a source vertex\nand the label of a destination, it is possible to compute, in constant time, the port number of the edge from the source that heads in the direction of the destination.\n\\end{fact}\n\n\\textbf{MP: check if needed.} \\mtodo{I've added a discussion of the port information later.}\nIn our routing scheme, each vertex $u$ is required to compute the port number of the next-hop towards the target destination. Towards that goal, we modify the FT-distance labels of Sec. \\ref{sec:ft-distance}, by adding the port information to the extended identifier of the edges. That is, the extended identifier of each edge $e=(u,v)$ is augmented with a third field containing the port number of $v$ w.r.t $u$ and vice-versa. To avoid cumbersome notation, we still refer to this port-augmented FT-distance label of $u$ by $\\mathsf{DistLabel}\\xspace(u)$. \n\n\n\\mtodo{Added the part from here. This adds port and tree routing information to the ids, and adapts Lemma \\ref{lem:useful-recovery-edges} (finding succinct path description) to the context of routing.}\n\nWe slightly modify the connectivity label of the edges and vertices by augmenting them with routing information. \nSpecifically, we define the vertex identifier $\\operatorname{ID}_T(u)=[\\operatorname{ID}(u), L_T(u)]$ where $L_T(u)$ is taken from Fact \\ref{fc:route-trees}. We then slightly modify the \nthe connectivity label of Eq. (\\ref{{eq:conn-node}}) by defining \n$$\\mathsf{ConnLabel}\\xspace_{G,T}(u)=\\langle \\mathsf{ANC}\\xspace_T(u), \\operatorname{ID}_T(u) \\rangle~.$$\nIn addition, we also augment the extended identifier of an edge Eq. (\\ref{eq:extend-ID}) with port information and tree routing information, by having:\n$$\\operatorname{EID}(e)=[\\operatorname{UID}(e), \\operatorname{ID}(u), \\operatorname{ID}(v), \\mathsf{ANC}\\xspace_T(u), \\mathsf{ANC}\\xspace_T(v), \\mbox{\\tt port}(u,v), \\mbox{\\tt port}(v,u), L_T(u), L_T(v)]~,$$\nwhere $\\mbox{\\tt port}(u,v)$ is the port number of the edge $(u,v)$ for $u$. \nThroughout this section, when applying the connectivity labels from Section \\ref{sec:ftconn-sketch} on a graph $G$ with spanning tree $T$, we use these labels and extended identifiers to support routing.\nWe later apply the connectivity labels with respect to the different trees of the tree cover as discussed in Section \\ref{sec:ft-distance}. \nLet $T_{i,j} \\in \\mathsf{TC}\\xspace_i$, recall that $G_{i,j}=G[V(T_{i,j})]$ and that $\\mathcal{T}=\\bigcup_{i=1}^K \\mathsf{TC}\\xspace_i$ for $K=O(\\log (nW))$. \n\n\\begin{lemma}\\label{lem:succint_path_routing}\nConsider a triplet $s,t,F$ such that $s,t,F \\in G_{i,j}$. \nGiven $ID_{T_{i,j}}(s),ID_{T_{i,j}}(t)$ and the connectivity labels $\\{\\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(e)\\}_{e \\in F}$, we can determine w.h.p if $s$ and $t$ are connected in $G_{i,j} \\setminus F$. If they are connected, we can output a labeled $s$-$t$ path $\\widehat{P}$ of length $O(f)$ that provides a succinct description of the $s$-$t$ path in $G_{i,j} \\setminus F$. The edges of $\\widehat{P}$ are labeled by $0$ and $1$, where $0$-labeled edges correspond to $G_{i,j}$-edges and $1$-labeled edges $e=(x,y)$ correspond to $x$-$y$ paths in $T_{i,j} \\setminus F$. For each $G_{i,j}$-edge, the succinct path description has the port information of the edge, and for each $x-y$ path, the description has the tree routing labels $L_{T_{i,j}}(x),L_{T_{i,j}}(y)$.\nThe length of the $s$-$t$ path encoded by $\\widehat{P}$ is bounded by $(4k-1)(f+1)\\cdot 2^i$. \n\\end{lemma}\n\n\\mtodo{For the proof I use the fact that the decoding algorithm only uses the ancestry labels of $s,t$, we should probably add some remark on the topic also on the labels section.}\n\n\\begin{proof}\nThe proof follows the proof of Lemma \\ref{lem:useful-recovery-edges}. To see this, first note that the decoding algorithm from Section \\ref{sec:ftconn-sketch} mostly uses the connectivity labels of the faulty edges $F$, and it needs only the labels $\\mathsf{ANC}\\xspace(s),\\mathsf{ANC}\\xspace(t)$ of $s$ and $t$, to detect the connected components of $s$ and $t$ in $T_{i,j} \\setminus F$. \\mtodo{do we want to edit this there? at least have some remark} \nHence, using $ID_{T_{i,j}}(s),ID_{T_{i,j}}(t)$ and the connectivity labels $\\{\\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(e)\\}_{e \\in F}$, we can first check if $s$ and $t$ are connected in $G_{i,j} \\setminus F$ using the decoding algorithm of Section \\ref{sec:ftconn-sketch}. If they are connected, then from Lemma \\ref{lem:useful-recovery-edges}, we get a succinct description of the $s$-$t$ path in $G_{i,j} \\setminus F$. We next show that the algorithm indeed has the relevant port and tree routing information. For this note that all the vertices in the path $\\widehat{P}$ obtained by Lemma \\ref{lem:useful-recovery-edges} are either $s$ and $t$ or endpoints of the $f$ recovery edges found in the algorithm. The ids $ID_{T_{i,j}}(s),ID_{T_{i,j}}(t)$ of $s$ and $t$ have their tree routing information, and when the algorithm finds a recovery edge, it learns about its extended id $ID_{T_{i,j}}(e)$ that has the port information and tree routing information of its endpoints. Any $G_{i,j}$-edge in $\\widehat{P}$ is a recovery edge, hence the algorithm has its port information, and for any $x$-$y$ path in $T_{i,j} \\setminus F$, the algorithm has the tree routing labels $L_{T_{i,j}}(x),L_{T_{i,j}}(y)$, as needed.\nThe stretch analysis follows the stretch analysis in Section \\ref{sec:ft-distance}. It is based on the fact that $\\widehat{P}$ has as most $f+1$ subpaths in $T_{i,j} \\setminus F$, each of length at most $(4k-2)2^i$, and at most $f$ recovery edges of weight at most $2^i$.\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{Forbidden Set Routing (Faulty Edges are Known)}\\label{sec:routing-known}\nWe start by describing the routing scheme in the forbidden set setting, where the faulty edges $F$ are known to the source vertex $s$. We show the following.\n\n\\begin{theorem}\\label{thm:routing-known}[Forbidden-Set Routing]\nFor every integers $k,f$, there exists an $f$-sensitive compact routing scheme that given a message $M$ at the source vertex $s$, a label of the destination $t$, and labels of at most $f$ forbidden edges $F$ (known to $s$), routes $M$ from $s$ to $t$ in a distributed manner over a path of length at most $(8k-2)(f+1)\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$. The table size of each vertex is bounded by $\\widetilde{O}(n^{1\/k} \\log{(nW)})$. The header size of the messages is bounded by $\\widetilde{O}(f)$ bits. The labels of vertices and edges have size $\\widetilde{O}(n^{1\/k} \\log{(nW)})$.\n\\end{theorem}\n\n\\mtodo{I edited a bit the proof so it's more similar to the next section and uses Lemma \\ref{lem:succint_path_routing}. The routing tables here still use the distance labels. Also, added a bit more details on the routing process.}\n\n\\begin{proof}\nThe algorithm is based on the distance labels from Section \\ref{sec:ft-distance}. Recall that the distance labels are based on applying fault tolerant connectivity labels on different graphs $G_{i,j}$, we use the connectivity labels from Section \\ref{sec:ftconn-sketch}.\nThe routing table of each vertex $u$ consists of its distance label and the tree routing labels from Fact \\ref{fc:route-trees} for every tree $T_{i,j} \\in \\mathcal{T}$ containing $u$. That is,\n$$R_{route}(u)=\\mathsf{DistLabel}\\xspace(u) \\cup \\{ \\langle L_{T_{i,j}}(u), (i,j) \\rangle ~\\mid~ i \\in [1,K], j \\in \\{1,\\ldots, |\\mathsf{TC}\\xspace_{i}|\\}, u \\in T_{i,j}\\}~.$$\nSince each vertex $u$ participates in $O(k n^{1\/k} \\log(n W))$ trees of $\\mathcal{T}$, and since the distance labels are of size $O(k n^{1\/k} \\log(n W) \\cdot \\log^3{n})$, the table $R_{route}(u)$ has $O(k n^{1\/k} \\log(n W) \\cdot \\log^3{n})$ bits.\nThe label of a vertex $u$ is just $R_{route}(u)$, and the label of an edge $e$ is $\\mathsf{DistLabel}\\xspace(e)$. The labels have size $O(k n^{1\/k} \\log(n W) \\cdot \\log^3{n})$. \\mtodo{we can have shorter labels for vertices as in the next section (but then we need to \"open the box\" of the distance labels), as the labels of edges are anyway large I'm not sure if we want to do it in this section.}\n\n\\mtext{\nIn the routing algorithm, the vertex $s$ is given the label $R_{route}(t)$ of $t$, and the labels $\\{\\mathsf{DistLabel}\\xspace(e)\\}_{e \\in F}$, and it needs to route a message to $t$ in the graph $G \\setminus F$. First, from the distance labels of $s,t$ and $F$, $s$ can check using the algorithm from Section \\ref{sec:ft-distance} if $s$ and $t$ are connected in $G \\setminus F$, and if so get a succinct description of a path between them. In more detail, recall that the algorithm from Section \\ref{sec:ft-distance} works in $K$ phases, where in phase $i$ it checks if $s$ and $t$ are connected in the graph $G_{i,i^*(s)} \\setminus F$ that contains the $2^i$-ball around $s$. Let $i$ be the first iteration where $s$ and $t$ are connected in $G_{i,i^*(s)} \\setminus F$ according to the algorithm, and denote $G_i = G_{i,i^*(s)},T_i = T_{i,i^*(s)}$, and let $F_i = F \\cap G_i$. The algorithm can also give a succinct description of an $s$-$t$ path in $G_i \\setminus F_i$ following Lemma \\ref{lem:succint_path_routing}. For this, note that we indeed have all the required information. The distance labels of edges in $F$ in particular contain the labels $\\{\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)\\}_{e \\in F_i}$, and we can also tell which edges of $F$ are in $G_i$ from the labels. Also, the labels of $s,t$ contain the information $ID_{T_i}(s),ID_{T_i}(t)$ if they are both in $T_i$ (otherwise, they are not connected in level $i$).\n\nThe path $\\widehat{P}$ as described in Lemma \\ref{lem:succint_path_routing} is composed of $O(f)$ parts, where segment $(x,y)$ in the path corresponds either to an edge in $G_i$ or to a tree path in $T_i \\setminus F$, it also has the relevant port and tree routing information. Our goal is to route a message according to this path. For this we add to the header of the message the description of $\\widehat{P}$, the indexes $(i,i^*(s))$ of the tree we explore and an index $1 \\leq q \\leq 2f+1$ that represents the segment of $\\widehat{P}$ we currently explore, initially $q=1$. Overall, the header size is $\\widetilde{O}(f)$. To route a message according to the path, we work as follows. The header specifies the current segment in $\\widehat{P}$. If the current segment corresponds to an edge $(x,y) \\in G$, then $x$ uses the port information to route the message to $y$ and increases the index $q$. Otherwise, the current segment represents a tree path $(x,y) \\in T_i$ and a vertex $u$ in this path uses its routing label in $T_i$ and the routing label of $y$ in $T_i$ (that is part of the header) to route the message towards $y$. When the message reaches $y$, it increases the index $q$. This completes the description of the routing process. The length of the path described is at most $(8k-2)(f+1)\\cdot \\mbox{\\rm dist}_{G\\setminus F}(s,t)$, as shown in Section  \\ref{sec:ft-distance}. } \n\\end{proof}\n\n\n\n\\subsection{Fault Tolerant Routing (Faulty Edges are Unknown)}\\label{sec:route-unknown}\nWe now consider the more involved setting where the set of failed edges $F$ are unknown to $s$. In this case, an edge $(u,v) \\in F$ is detected only when the message arrives, during the routing procedure, to one of the endpoints of $e$. Note that the routing scheme should, by definition, be prepared to any set of faulty edges $F$. However, the space bound of our scheme is required to be bounded by $\\widetilde{O}(f n^{1+1\/k})$, which is possibly much smaller than the number of graph edges $m$. This in particular implies that we cannot store the FT-distance labels of all the graph edges. Nevertheless, we show that it is sufficient to explicitly store the labeling information for the tree edges in $\\mathcal{T}$. The required information for the failed non-tree edges would be revealed throughout the process, by applying the decoding algorithm of Lemma \\ref{lem:succint_path_routing}.\nOur routing scheme eventually routes the message along the $s$-$t$ path encoded by the FT-distance labels of $s,t$ and $F$. However, since the labels of $F$ are unknown in advance, the routing scheme will detect these edges in a trail and error fashion which induces to an extra factor of $f$ in the final multiplicative stretch. This extra $f$ factor is also shown to be essential, in the end of the section.\nWe proceed by describing the routing tables.  \n\n\\remove{\n\\paragraph{The routing tables.} For ease of presentation, we first describe a solution with a multiplicative stretch of $O(kf^2)$, and \\emph{global} space of $\\widetilde{O}(f K \\cdot n^{1+1\/k})$, \\mtodo{should it be $n^{1+2\/k}$?} but the individual tables of some of the vertices might be of linear space. \nRecall that $\\mathcal{T}=\\bigcup_i^{K} \\mathsf{TC}\\xspace_i$, for $K=O(\\log (nW))$ is a collection of tree covers in all $K=\\lceil \\log (nW) \\rceil$ distance scales, see Eq. (\\ref{eq:TC-i}). For every vertex $u$, let $\\deg_T(u)=\\sum_{T_{i,j} \\in \\mathcal{T}}\\deg(u,T_{i,j})$ be the sum of degrees of $u$ in the collection of trees  $\\mathcal{T}$.\nThe algorithm computes each vertex $u$ a table of $\\widetilde{O}(f \\cdot \\deg_T(u) \\cdot n^{1\/k})$ bits.\nSince the total number of tree edges in $\\mathcal{T}$ is bounded by $\\widetilde{O}(K \\cdot n^{1+1\/k})$, this provides a global space bound of $\\widetilde{O}(fK \\cdot n^{1+2\/k})$ bits. We later on improve the space of each table to $\\widetilde{O}(fK \\cdot n^{2\/k})$ bits. \\mtodo{maybe extra $k,K$ are needed in some places.}\n\nThe routing process has at most $f+1$ phases, and each phase uses an independent set of randomized FT-distance labels. Therefore, the preprocessing algorithm employs $f$ \\mtodo{$f$ or $f+1$?} independent applications of the $(f,(8k-2)(f+1))$ FT-distance labels scheme. Denote the output labels obtained by the $i^{th}$ application by $\\mathsf{DistLabel}\\xspace_i(w)$ for every $w \\in V(G)\\cup E(G)$. Let $E_T=\\bigcup_{T_{i,j} \\in \\mathcal{T}} E(T_{i,j})$ be the collection of tree edges. The \\emph{routing label} \\mtodo{maybe we want to have shorter labels, and have this information only in the routing tables?} of each $w \\in V\\cup E_T$ is given by \n$$L_{route}(w)=\\{\\mathsf{DistLabel}\\xspace_1(w),\\ldots, \\mathsf{DistLabel}\\xspace_{f+1}(w)\\}~.$$ \nThe routing label $L_{route}(e)$ of a non-tree edge $e \\in E \\setminus E_T$ is simply the extended identifier of $e$. For each vertex $u$, the routing table $R_{route}(u)$ consists of the following:\n\\begin{enumerate\n\\item Tree labels $L_{T_{i,j}}(u)$ for every tree $T_{i,j}$ in $\\mathcal{T}$ that contains $u$ (from Fact \\ref{fc:route-trees}). These labels are augmented with their corresponding tree index $(i,j)$.\n\\item The routing labels $L_{route}(u)$ and $\\{L_{route}(e=(u,v)) ~\\mid~ e \\in T_{i,j}, T_{i,j} \\in \\mathcal{T}\\}$.\n\\end{enumerate}\nThe header information initially contains the succinct path information\\footnote{\\textbf{MP: This is not essential, we can always make the header contain the labels of the currently detected faults.}} obtained by the decoding algorithm of the FT-distance labeling when given the labels $L_{route}$ of $s$ and $t$. \\mtodo{do we want to have $f$ paths? Or only one of them that is relevant for the current phase}  As we will see, throughout the routing procedure, the header information will be augmented with the labels of the currently detected faulty edges. \n}\n\n\n\\paragraph{The routing tables.} For ease of presentation, we first describe a solution with a multiplicative stretch of $O(kf^2)$, and \\emph{global} space of $\\widetilde{O}(f K \\cdot n^{1+1\/k})$, but the individual tables of some of the vertices might be large. We later on improve the space of each table to $\\widetilde{O}(fK \\cdot n^{1\/k})$ bits.\n\nRecall that $\\mathcal{T}=\\bigcup_i^{K} \\mathsf{TC}\\xspace_i$, for $K=O(\\log (nW))$ is a collection of tree covers in all $K=\\lceil \\log (nW) \\rceil$ distance scales, see Eq. (\\ref{eq:TC-i}). For every vertex $v$, let $\\deg_T(v)=\\sum_{T_{i,j} \\in \\mathcal{T}}\\deg(u,T_{i,j})$ be the sum of degrees of $u$ in the collection of trees  $\\mathcal{T}$. Recall that $G_{i,j}=G[V(T_{i,j})]$.\n\\mtext{For the routing we apply the FT-connectivity labels on the graphs $G_{i,j}$, similarly to Section \\ref{sec:ft-distance}.\nThe routing process uses $f$ independent applications of randomized FT-connectivity labels from Section \\ref{sec:ftconn-sketch}, applied on each one of the graphs $G_{i,j}$. \\mtodo{There is some delicate issue here. In my understanding we do want to have the extended ids fixed in different applications, and only change the randomness of sketches, as I try to explain here.} In more detail, when we apply the labeling scheme on the graph $G_{i,j}$ with spanning tree $T_{i,j}$, we use $f$ independent random seeds $\\mathcal{S}_h$ to determine the randomness of the sketches. However, the seed $\\mathcal{S}_{ID}$ used to determine the extended ids of edges in $G_{i,j}$ is fixed in the $f$ applications, hence the ids are fixed and we only use fresh randomness to compute the sketch information. This process is done independently on each one of the graphs $G_{i,j}$. }  \nDenote the output labels obtained by the ${\\ell}^{th}$ application of the scheme on the graph $G_{i,j}$ by $\\mathsf{ConnLabel}\\xspace^{\\ell}_{G_{i,j},T_{i,j}}(w)$ for every $w \\in V(G_{i,j})\\cup E(G_{i,j})$, and we denote by $$\\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(w)=(\\mathsf{ConnLabel}\\xspace^1_{G_{i,j},T_{i,j}}(w),...,\\mathsf{ConnLabel}\\xspace^f_{G_{i,j},T_{i,j}}(w)).$$\nSince in each application the labels are of size $O(\\log^3{n})$, the total size of each label is $O(f \\log^3{n})$. \\mtext{Recall that for a non-tree edge $e \\in G_{i,j}$, we have that  $\\mathsf{ConnLabel}\\xspace^{\\ell}_{G_{i,j},T_{i,j}}(e)=\\operatorname{ID}_{T_{i,j}}(e)$. This label does not depend on the sketch information, and hence is the same in all $f$ applications, we will exploit this later in the algorithm.}\n\n\\mtodo{I changed a bit the definition of routing tables and labels. The labels currently are of polylog size (no extra $f$ or $n^{1\/k}$).}\n\\mtext{\nThe routing table $R_{route}(v)$ of a vertex $v$ has the following information for any tree $T_{i,j}$ such that $v \\in T_{i,j}$:\n\\begin{enumerate}\n\\item The values $(ID_{T_{i,j}}(v),i,j)$.\n\\item The connectivity labels $\\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(e)$ for any tree edge $e \\in T_{i,j}$ adjacent to $v$. \n\\end{enumerate}\n\nSince the connectivity labels are of size $\\widetilde{O}(f)$, and since $v$ appears in $\\deg_T(v)$ trees, the size of the table is $\\widetilde{O}(f \\deg_T(v)).$ Since the total number of tree edges in $\\mathcal{T}$ is bounded by $\\widetilde{O}(K \\cdot n^{1+1\/k})$, this provides a global space bound of $\\widetilde{O}(fK \\cdot n^{1+1\/k})$ bits.\n\nRecall that for a vertex $v$ and index $1 \\leq i \\leq K$, we denote by $i^*(v)$ an index such that the $2^i$-ball around $v$ is contained in $T_{i,i^*(v)}$.\nThe routing label $L_{route}(v)$ of $v$ has the id of $v$ in the trees $T_{i,i^*(v)}$.\n$$L_{route}(v) = \\{(i^*(v),ID_{T_{i,i^*(v)}}(v)) | i \\in [1,K]\\}$$\n\nThe size of the label is $O(K\\log{n})=O(\\log{n} \\log{nW})$.}\n\n\\mtodo{In the routing algorithm the main difference is that we have log iterations for the different trees, in each tree the algorithm is similar to before.}\n\n\\paragraph{The routing algorithm.} \\mtext{In the routing algorithm, the source vertex $s$ gets the routing label $L_{route}(t)$ of the destination $t$ and its goal is to find a graph $G_{i,j}$ such that $s$ and $t$ are connected in $G_{i,j} \\setminus F$, and use it for routing. As the set $F$ is not known in advance, the algorithm works in $K= O(\\log{nW})$ iterations, where in iteration $i$ it tries to route a message in $G_{i,i^*(t)}$. If $s$ and $t$ are connected in $G_{i,i^*(t)} \\setminus F$ the algorithm succeeds, and otherwise we proceed to the next iteration. \nWe next describe the algorithm for one iteration $i$, we denote $G_i = G_{i,i^*(t)}, T_i = T_{i,i^*(t)}$. Note that $s$ learns the index $i^*(t)$ from the label of $t$, and it can check if $s \\in T_i$ using its routing table. If $s \\not \\in T_i$, we proceed to the next iteration.}\n\nIf $s \\in T_i$, the routing procedure for iteration $i$ has at most $f+1$ phases. Each phase will start at the source vertex $s$. We will maintain the following invariant in the beginning of each phase $\\ell \\in \\{1,\\ldots, f+1\\}$: (i) the algorithm has already detected a subset of $\\ell-1$ faulty edges $F_\\ell \\subseteq F$, and (ii) the header contains the labels $\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)$ of all the edges $e \\in F_\\ell$. Each phase $\\ell \\leq f+1$ will terminate either at the destination vertex $t$, or at the source vertex $s$. In addition, w.h.p., if $s$ and $t$ are connected in $G_i \\setminus F$, phase $f+1$ will terminate at $t$. The invariant holds vacuously for phase $1$.\n\nWe now describe the $\\ell^{th}$ phase of the routing procedure given the invariant. The source vertex $s$ considers the $\\ell^{th}$ copy of the FT-connectivity labels, $\\mathsf{ConnLabel}\\xspace^{\\ell}_{G_i,T_i}(e)$ for every $e \\in F_\\ell$. \nUsing the connectivity labels, that are part of the header, and the ids $ID_{T_i}(s),ID_{T_i}(t)$ that are part of the routing table of $s$ and routing label of $t$, the algorithm first checks if $s$ and $t$ are connected in $G_i \\setminus F_\\ell$ using Lemma \\ref{lem:succint_path_routing}. If the answer is no, the algorithm moves to the next iteration $i+1$.\nOtherwise, by applying the decoding algorithm of Lemma \\ref{lem:succint_path_routing}, it computes the succinct path $\\widehat{P}_\\ell$. The path $\\widehat{P}_\\ell$ encodes an $s$-$t$ path in $G_i \\setminus F_\\ell$, that includes the relevant port and tree routing information of its vertices. The header of the message $H_\\ell$ then contains \n$$H_\\ell=\\langle \\widehat{P}_\\ell, i, i^*(t), \\{\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)\\}_{e \\in F_{\\ell}}, q \\rangle~,$$ where $q = O(f)$ is an index indicating the current segment of $\\widehat{P}_\\ell$ we explore. \\mertodo{$H_\\ell$ should contain $\\mathsf{ConnLabel}\\xspace^{\\ell}_{G_i,T_i}(e)$, right?}\n\n\n\\mtext{The size of the header is $\\widetilde{O}(f^2)$, as the description of the path has size $\\widetilde{O}(f)$, and additionally we have at most $f$ faulty edges with labels of size $\\widetilde{O}(f)$.} \\mtodo{in some sense we don't really need to have all this $O(f^2)$ information on the header at once because it's enough that $s$ learns in each phase on one of the labels and then stores it locally, but this I guess doesn't work with the formal definition of routing.}\nLet $P_\\ell$ be the $G$-path encoded by the path $\\mathcal{P}_\\ell$. The algorithm then routes the message along $P_\\ell$ in the same manner as in Sec. \\ref{sec:routing-known}. In the case where $P_{\\ell}\\cap F=\\emptyset$, the phase successfully terminates at the destination vertex $t$. From now on, we consider the case that $P_{\\ell}$ contains at least one faulty edge. \n\nLet $e=(u,v)$ be the first edge (closest to $s$) on the path $P_\\ell$ that belongs to $F$. Since $P_\\ell \\cap F_\\ell=\\emptyset$, it holds that $e \\in F \\setminus F_\\ell$. Without loss of generality, assume that $u$ is closer to $s$ on $P_\\ell$. Thus the faulty edge $e$ is detected upon arriving to the vertex $u$. \nIn the case where $e$ is a \\emph{non-tree edge}, then it must be a $G$-edge on $\\widehat{P}_\\ell$. Since this path has the ids $ID_{T_i}(e)$ of its $G$-edges, and since the connectivity label of a non-tree edge $e$ is $ID_{T_i}(e)$ in all the $f$ applications of the scheme on $G_i$, $u$ can add $\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)$ to the header of the message. \\mtodo{here we use the fact that the extended ids are the same in all $f$ phases (of the same tree)} Assume now that $e$ is a tree edge in $T_i$. The vertex $u$ then adds the label $\\mathsf{ConnLabel}\\xspace_{G_i,T_i}(e)$ to the header of the message, as $e$ is a tree edge adjacent to $u$ it has this information in its routing table. Finally, it marks the header with the sign $R$, indicating that the message should now be routed in the reverse direction, until arriving $s$ again. This completes the description of phase $\\ell$. It is easy to see that the invariant is maintained. If $s$ and $t$ are connected in $G_i \\setminus F$, after at most $f$ phases all faulty edges are detected. In the last phase, the path computed based on the labeling information is free from faulty edges, and the routing is completed (in the same manner as in Sec. \\ref{sec:routing-known}) at the destination $t$. We next bound the multiplicative stretch of the routing.\n\n\n\n\n\\mtodo{updated the stretch analysis.}\n\n\\mtext{\n\\begin{claim}\\label{cl:route-length}\nFix a set of faulty edges $F$, and let $s,t$ be vertices that are connected in $G \\setminus F$. Then, the message is routed from $s$ to $t$ within $32k (f+1)^2 \\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$ steps, w.h.p.\n\\end{claim}\n\\begin{proof}\nFirst note that since each phase and each graph $G_{i}$ uses an independent set of FT-connectivity labels, then in each iteration and each phase the decoding algorithm succeeds w.h.p. and outputs an $s$-$t$ path $\\widehat{P}_\\ell$ if exists. \n\nAssume that $\\mbox{\\rm dist}_{G \\setminus F}(s,t) \\in (2^{i-1},2^i]$. Then, $s$ and $t$ are connected in $G_i \\setminus F$, as $T_i = T_{i,i^*(t)}$ contains the $2^i$-ball around $t$. We show that the algorithm terminates at $t$ in iteration $i$ or before it, and that in any iteration $j \\leq i$, the routing algorithm traverses a path of length at most $2(4k-1)(f+1)^2 \\cdot 2^j$.\n\nLet $j \\leq i$. In the $\\ell$'th phase of iteration $j$, the algorithm first checks if $s$ and $t$ are connected in $G_j \\setminus F_{\\ell}$, where $F_{\\ell}$ is the set of currently detected faults. If the answer is no, the algorithm proceeds to the next iteration. Otherwise, it tries to route a message from $s$ to $t$ on the path encoded by $\\widehat{P}_{\\ell}$. The length of the path is bounded by $(4k-1)(f+1)\\cdot 2^j$ from Lemma \\ref{lem:succint_path_routing}. The algorithm either succeeds, or finds a faulty edge on the way in which case it returns to $s$ by traversing the same path on the reverse direction. Overall, the algorithm traverses a path of length at most $2(4k-1)(f+1)\\cdot 2^j$, in this phase. In all $f+1$ phases of iteration $j$, the length of the path explored is at most $2(4k-1)(f+1)^2 \\cdot 2^j$. Summing over all iterations $j \\leq i$, the stretch is bounded by $$\\sum_{j=1}^{i} 2(4k-1)(f+1)^2 \\cdot 2^j = 2(4k-1)(f+1)^2 \\sum_{j=1}^i 2^j \\leq 2^{i+2} (4k-1)(f+1)^2 \\leq 32k (f+1)^2 \\mbox{\\rm dist}_{G \\setminus F}(s,t).$$ The last inequality uses the fact that $2^{i-1} \\leq \\mbox{\\rm dist}_{G \\setminus F}(s,t).$ \n\nIn the $i$'th iteration, since $s$ and $t$ are connected in $G_i \\setminus F$, then for any $F_{\\ell} \\subseteq F$, $s$ and $t$ are connected in $G_i \\setminus F_{\\ell}$, hence the algorithm always finds a path $\\widehat{P}_{\\ell}$. Hence, it either succeeds in routing the message to $t$ in one of the phases (or one of the previous iterations), or learns about all the failures $F$. In the latter case, in phase $f+1$ it learns about a failure-free path $\\widehat{P}_{f+1}$, and the routing terminates at $t$. This completes the proof.\n\\end{proof}\n}\n\n\\remove{\n\\begin{claim}\\label{cl:route-length}\nFix a set of faulty edges $F$. Then, the message is routed from $s$ to $t$ within $(16k-4)(f+1) f\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$ steps, w.h.p.\n\\end{claim}\n\\begin{proof}\nFirst note that since each phase and each graph $G_{i}$ uses an independent set of FT-connectivity labels, w.h.p., the decoding algorithm of phase $\\ell$ applied on the label set $\\{\\mathsf{DistLabel}\\xspace(w), w \\in \\{s,t\\} \\cup F_\\ell\\}$ succeeds w.h.p. and outputs an $s$-$t$ path $\\widehat{P}_\\ell$. \n\nLet $P_{\\ell}$ be the $s$-$t$ path in $G \\setminus F_\\ell$ encoded by $\\widehat{P}_\\ell$. By Lemma \\ref{lem:approx-dist-recovery}, the length of $P_\\ell$ is bounded by $(8k-2)(f+1)\\cdot \\mbox{\\rm dist}_{G \\setminus F_\\ell}(s,t)$. Since in phase $\\ell$ the message is routed along a (possibly) partial path $P_\\ell$ and back to $s$, we have $2|P_\\ell|$ routing hops \\mtodo{change to capture weighted graphs?} in that phase. Overall, we have at most $2\\sum_{i=1} |P_\\ell|$ routing hops. Since \n$F_\\ell \\subseteq F$ for every $\\ell$, it holds that $\\mbox{\\rm dist}_{G \\setminus F_\\ell}(s,t)\\leq \\mbox{\\rm dist}_{G \\setminus F}(s,t)$, and consequently $(16k-4)(f+1) f \\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$ routing steps over all. The claim follows. \n\\end{proof}\n}\n\nTo conclude, we have the following.\n\n\\begin{theorem}\nFor every integers $k,f$, there exists an $f$-sensitive compact routing scheme that given a message $M$ the source vertex $s$ and a label $L_{route}(t)$ of the destination $t$, in the presence of at most $f$ faulty edges $F$ (unknown to $s$) routes $M$ from $s$ to $t$ in a distributed manner over a path of length at most $32k (f+1)^2\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$. \nThe global table size is $\\widetilde{O}(f \\cdot n^{1+1\/k} \\log{(nW)})$.\nThe header size of the messages is bounded by $\\widetilde{O}(f^2)$ bits, and the label size of vertices is $O(\\log{(nW)} \\log{n})$. \n\\end{theorem}\n\n\n\n\n\n\\mtodo{I didn't change the next section, as some of the definitions of routing tables\/labels changed, there are probably also changes here.}\n\n\\paragraph{Improving the size of the routing tables.} \nSo far, we have described a routing scheme that consumes a total space of $\\widetilde{O}(f\\cdot n^{1+1\/k}\\log (nW))$ bits, and multiplicative stretch\n$32(f+1)^2 k$. We now explain the required modifications needed to providing routing tables with $\\widetilde{O}(f^2\\cdot n^{1\/k})$ bits per vertex. The most space consuming information for a vertex $u$ is the connectivity labeling\ninformation of the edges incident to $u$ in each of the trees $T_{i,j} \\in \\mathcal{T}$. As the degree of $u$ in some of the trees might be $\\Theta(n)$, it leads to tables of linear size. To reduce the space of the individual tables, we apply a load balancing idea which distributes the labeling information incident to \\emph{high}-degree vertices among their neighbors. \n\nInstead of storing the labeling information of $e=(u,v)$ at the routing tables of $u$ and $v$, we define \nfor every tree $T \\in \\mathcal{T}$ and an edge $e=(u,v) \\in T$, a subset $\\Gamma_T(e)$ of vertices that store the connectivity labeling information of $e$ in $T$. We will make sure that the information on some vertex in $\\Gamma_T(e)$ can be easily extracted in the routing procedure, and that each vertex stores the information for a small number of edges in each tree. Consider an edge $e=(u,v)$ in a tree $T$, and assume, without loss of generality, that $u$ is the parent of $v$ in the tree $T$. In the case where $\\deg(u,T)\\leq f+1$, we simply let $\\Gamma_T(e)=\\{u,v\\}$. That is, the label of $e$ is stored by both endpoints of $e$ (as before). The interesting case is where $\\deg(u,T)\\geq f+2$, in which case, $u$ might not be able to store the label of $e$, and will assist its other children as follows.  Let $Child(u,T)=[v_1,\\ldots, v_\\ell]$ be the lexicographically ordered list of the children of $u$ in $T$.  The algorithm partitions $Child(u,T)$ into consecutive blocks of size $f+1$ (the last block might have $2f+1$ vertices). Letting $[v_{q,1}, \\ldots, v_{q,f+1}] \\subseteq Child(u,T)$ be the block containing $v$, define\n$$\\Gamma_T(e)=\\{v_{q,1}, \\ldots, v_{q,f+1}\\}~.$$\nNote that in particular, $v \\in \\Gamma_T(e)$. Thus, the label of $e$ is stored by $v$ and $\\ell \\in [f,2f-1]$ additional children on $u$ in $T$. \n\nWe then modify the tree labels from Fact \\ref{fc:route-trees} to contain the port information of $\\Gamma_T(e)$. \nTo do that, we re-define the identifier of an edge $e$ to include the port information of $\\Gamma_T(e)$. Thus, the identifier of an edge has $O(f\\log n)$ bits. Using Fact \\ref{fc:route-trees}, we then have tree labels of $O(f\\log n)$ such that given the labels of $u$ and $t$, the algorithm can compute the port information $\\Gamma_T(e)$ where $e$ is the edge incident to $u$ on the $u$-$t$ tree path in $T$. We have:\n\\begin{claim}\\label{cl:tree-label-port}\n\n\n\\end{claim}\n\nIn addition, we modify the routing tables of each vertex $v$, so that its includes the connectivity labels $\\mathsf{ConnLabel}\\xspace_{G_{i,j},T_{i,j}}(e)$ for any tree edge $e \\in T_{i,j}$ such that $v \\in \\Gamma_{T_{i,j}}(e)$ for every tree $T_{i,j} \\in \\mathcal{T}$. This should be compared with the solution of the previous subsection, where $v$'s table contained the information for all its incident edges in $T_{i,j}$. We therefore have:\n\\begin{claim}\\label{cl:route-balance-table}\nThe size of each routing table $R_{route}(v)$ is bounded by $\\widetilde{O}(f K n^{1\/k})$ bits.\n\\end{claim}\n\\begin{proof}\nFor every tree $T_{i,j}$ containing $v$, $v$ stores that connectivity label of $\\widetilde{O}(f)$ bits of all edges in the set $E(v,T_{i,j})=\\{e \\in T_{i,j} ~\\mid~ v \\Gamma_{T_{i,j}}(e)\\}$. Note that $|E(v,T_{i,j})|\\leq f$ as each vertex stores the label of its parent in the tree, $O(f)$ child edges, and $O(f)$ child edges of its parent in the tree. Since each $v$ participates in $\\widetilde{O}(K n^{1\/k})$ trees, overall its routing table has $\\widetilde{O}(f K n^{1\/k})$ bits, as required. \n\\end{proof}\n\nIt remains to explain the required modifications for the routing procedure. Upon arriving to a vertex $u$ incident to a faulty \\emph{tree} edge $e=(u,v)$ the procedure is as follows. The interesting case is where $e$ is a tree edge in some path $\\pi(x,y,T_{i,j})$ where $(x,y)$ is an edge in the succinct path $\\widehat{P}$ indicated on the header of the message. \nIn this case, the tree routing algorithm applied at $u$ is given the tree labels $L_{T_{i,j}}(u)$ and $L_{T_{i,j}}(y)$ and output the port information of $\\Gamma_{T_{i,j}}(e)$ for the edge $e=(u,v)$ that lies on the tree path $\\pi(u,y, T_{i,j})$. Since there are at most $f$ edge faults, and $\\Gamma_{T_{i,j}}(e)$ contains information on at least $f+1$ ports of $u$'s neighbors that contain the label of $e$, the vertex $u$ can access a non-faulty neighbor, say $w$, that has the label information of $e$. That vertex can then add the labeling information of $e$ to the header of the message, and the routing algorithm proceeds as before. \n\n\n\nWe now bound the size of each routing table. \n\\begin{claim}\nThe routing table of each vertex has $\\widetilde{O}(f^2 n^{2\/k} \\log (nW))$ bits.\n\\end{claim}\n\\begin{proof}\nEach vertex appears in $O(k n^{1\/k} \\log (nW))$ trees in $\\mathcal{T}$. For each such tree, it might store the routing labels of at $O(f)$ edges. Since each label $L_{route}(e)$ has $\\widetilde{O}(f n^{1\/k} \\log (nW))$ bits, over all it stores $\\widetilde{O}(f^2 \\cdot n^{2\/k}\\log (nW))$ bits. \n\\end{proof}\n\n\\mtodo{how is the stretch affected from the process?}\n\nWe have:\n\\begin{theorem}\\label{thm:routing-unknown}[Fault-Tolerant Routing]\nFor every integers $k,f$, there exists an $f$-sensitive compact routing scheme that given a message $M$ at the source vertex $s$ and a destination $t$, in the presence of at most $f$ faulty edges $F$ (unknown to $s$) routes $M$ from $s$ to $t$ in a distributed manner over a path of length at most $32k f^2\\cdot \\mbox{\\rm dist}_{G \\setminus F}(s,t)$. The table size of each vertex is $\\widetilde{O}(f^2 \\cdot n^{1\/k} \\log(nW))$. The header size of the messages is bounded by $\\widetilde{O}(f^2)$ bits. \n\\end{theorem}\n\n\\paragraph{Lower Bound.} Finally, we show that the price of not knowing the set of faulty edges $F$ in advance might indeed incur a multiplicative stretch of $\\Omega(f)$. \n\n\\begin{theorem}[Stretch Lower-Bound for FT-Routing]\\label{thm:lb-routing}\nAny FT-routing randomized scheme resilient to $f$ faults induces an expected stretch of $\\Omega(f)$ regardless of the size of the routing tables and labels. In particular, this holds even if each routing table contains a complete information on the graph. \n\\end{theorem}\n\\begin{proof}\nConsider a graph that consists of $f+1$ vertex disjoint $s$-$t$ paths, each of length $L=\\Theta(n\/f)$. The last edge of each of the paths, except for one, is faulty. Assume that the non-faulty path is chosen uniformly at random. Since the routing scheme is oblivious to the faulty edges, it can discover a faulty edge only upon sending the message to one of the edge endpoints. The expected length of the routing is given by:\n$$\\frac{L}{f+1} +2L \\cdot \\left(1-\\frac{1}{f+1} \\right)\\cdot \\frac{1}{f} + \\ldots+ \\left(f+1\\right)L\\cdot \\prod_{i=0}^{f-1} \\left(1-\\frac{1}{f+1-i}\\right)=\\Omega(f L)~.$$ \nSince the $s$-$t$ shortest path under these faults is $L$, the proof follows. See Fig. \\ref{fig:LB-stretch} for an illustration.\n\\end{proof}\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.40]{lb.pdf}\n\\caption{\\sf Illustration for a stretch lower bound for any FT-routing schemes. The $s$-$t$ pair are connected by $f+1$ vertex disjoint paths. Since the faulty-edge is the last edge of the path, the routing requires $\\Omega(L)$ steps to discover a single faulty edge. As the non-faulty path is chosen uniformly at random, in expectation, the routing requires $\\Omega(fL)$ steps.  \\label{fig:LB-stretch}\n}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Our Techniques} \\label{sec:techniques}\n\nFor our FT labeling schemes, we present two constructions based on different techniques. \nThe first construction uses the \\emph{cycle-space sampling} technique of Pritchard and Thurimella \\cite{pritchard2011fast} to determine if $s$ and $t$ are disconnected by a set of failures $F$. This technique has been applied in the past mainly in the context of computing small cuts in the distributed setting. \nThe second construction uses the tool of \\emph{linear sketches} by Ahn et al. \\cite{ahn2012analyzing} to try to find a path that connects $s$ and $t$ in $G \\setminus F$. This scheme is also useful for routing.\nWe next give an overview of the two approaches, and the applications for routing. Throughout, we assume that the graph $G$ is originally connected, otherwise the scheme can be applied to each connected component of $G$, which can be indicated in the label of the vertex.\n\n\\paragraph{Connectivity Labels Based on Cycle Space Sampling.} The cycle space sampling technique, introduced by Pritchard and Thurimella \\cite{pritchard2011fast}, allows one to detect cuts in a graph by exploiting the interesting connection between cuts and cycles in a graph. This technique was used in \\cite{pritchard2011fast} to design distributed algorithms for identifying small cuts in a graph. In more details, the technique is based on the relation between \\emph{induced edge cuts} and \\emph{binary circulations}, defined as follows. For a subset of vertices $S$, we denote by $\\delta(S)$ the set of edges with exactly one endpoint in $S$. An \\emph{induced edge cut} is a set of edges of the form $\\delta(S)$ for some $S$. A \\emph{binary circulation} is a set of edges in which every vertex has an even degree. For example, a cycle is a binary circulation. Note that if $F$ is an induced edge cut, and $\\phi$ is a cycle, the number of edges in the intersection $|F \\cap \\phi|$ is even, as the cycle crosses the cut even number of times. This is also true for any binary circulation $\\phi$. The cycle space technique extends this observation and shows that if $\\phi$ is a random binary circulation and $F \\subseteq E$, then\n$$Pr[|F \\cap \\phi| \\ is \\ even] = \\left\\{\n                \\begin{array}{ll}\n                  1,\\ if\\ F\\ is\\ an\\ induced\\ edge\\ cut\\\\\n                  1\/2,\\ otherwise\n                \\end{array}\n              \\right. $$ \nHence, by choosing a \\emph{random} binary circulation, one can detect if a set of edges $F$ is an induced edge cut with probability $1\/2$.  To increase the success probability, we can choose $b$ random binary circulations. \nBased on these ideas, \\cite{pritchard2011fast} showed how to assign the edges of the graph $b$-bit labels with the following property.  See Appendix \\ref{sec:cycle_space_overview} for an overview.\n\n\\begin{restatable}{lemma}{cycle} \\label{cycle_space_lemma}\nThere is an algorithm that assigns the edges of a graph $G=(V,E)$, $b$-bit labels $\\phi(e)$ such that given a subset of edges $F \\subseteq E$, we have:\n$$Pr[\\Moplus_{e \\in F} \\phi(e) = 0] = \\left\\{\n                \\begin{array}{ll}\n                  1,\\ if\\ F\\ is\\ an\\ induced\\ edge\\ cut\\\\\n                  2^{-b},\\ otherwise\n                \\end{array}\n              \\right. $$ \nWhere $0$ is the all-zero vector. The time complexity for assigning the labels is $O((m+n)b)$.\n\\end{restatable}\n\n\\noindent\\textbf{The connectivity labels.} We next explain how to use this technique to build FT connectivity labels. Our goal is to assign labels to the vertices and edges of the graph, such that given the labels of two vertices $s,t$ and a set of failures $F$, we can check if $s$ and $t$ are disconnected by $F$. It is easy to show that $s$ and $t$ are disconnected by $F$ iff there is an \\emph{induced edge cut} $F' \\subseteq F$ that disconnects $s$ and $t$. While we can use the cycle space labels to check if a subset of edges $F' \\subseteq F$ is an induced edge cut, this is still not enough to solve FT connectivity. To do so, we should check if an induced edge cut $F'$ \\emph{disconnects} the vertices $s$ and $t$. To check this, we bring to our construction \\emph{ancestry labels} in trees, and show that we can determine if $s$ and $t$ are in the same side of cut (induced by $F'$) based on the ancestry labels of $s,t$ and $F'$. The key observation is that a spanning tree $T$ of the graph is disconnected to at most $|F'|+1$ connected components, upon removing $F'$, where for any $e \\in F'$ both its endpoints reside on two different sides of the induced edge cut defined by $F'$.\nWe can use this to identify which components of $T \\setminus F'$ are on the same side of the induced edge cut. Moreover, we show that the ancestry labels allow us to determine the connected components of $s$ and $t$ in $T \\setminus F'$. A brute-force implementation of this approach leads to a decoding time that is \\emph{exponential} in $|F|$. I.e., the algorithm should check for any subset $F' \\subseteq F$ if $F'$ is an induced edge cut. To overcome it, we show an efficient way to find $F' \\subseteq F$ that disconnects $s$ and $t$ if exists, by translating our problem to a system of linear equations. This results in a decoding time polynomial in $|F|$ and $\\log{n}$. The size of the labels is $O(f+\\log{n})$, to guarantee that the cycle space labels are correct for any $F' \\subseteq F$ w.h.p.      \n\n\\paragraph{Connectivity Labels Based on Graph Sketches.} We next provide some flavor of our labels based graph sketches. The length of the labels obtained in this technique is $O(\\log^3{n})$ bits, which is dominated by the sketching information. A \\emph{graph sketch} of a vertex $v$ is a randomized string of $\\widetilde{O}(1)$ bits that compresses $v$'s edges. The linearity of these sketches allows one to infer, given the sketches of subset of vertices $S$, an outgoing cut edge $(S, V \\setminus S)$. Graph sketches have numerous applications in the context of connectivity computation under various computational settings, e.g., \\cite{kapron2013dynamic,kapralov2014spanners,GibbKKT15,DBLP:conf\/podc\/KingKT15,DBLP:conf\/wdag\/MashreghiK18,GhaffariP16,DuanConnectivitySODA17}. More concretely, our sketch-based labels are inspired by the centralized connectivity sensitivity oracles of Duan and Pettie \\cite{DuanConnectivitySODA17}.   A common approach for deducing the graph connectivity merely from the sketches of the individual vertices is based on the well-known Boruvka algorithm \\cite{Boruvka}. This algorithm works in $O(\\log n)$ phases, where in each phase, from each growable component an outgoing edge is selected. All these outgoing\nedges are added to the forest, while ignoring cycles. Each such phase reduces the number of\ngrowable components by a $2$ factor, thus within $O(\\log n)$ phases, a maximal forest is computed. Since this algorithm only requires the computation of outgoing edges it can simulated using $O(\\log n)$ independent sketches for each of the vertices. \n\nOur high level approach for determining the $s$-$t$ connectivity in $G \\setminus F$ mimics this above mentioned procedure. For simplicity assume that $G$ is connected and let $T$ be some spanning tree in $G$. Using ancestry labels, one can infer the components of $T \\setminus F$. Moreover, by augmenting the labels with graph sketching information, one can also deduce the sketch of each component in $T \\setminus F$. \nNote however that these sketches are in $G$ and therefore might encode outgoing edges that belong to $F$. To overcome this technicality, our sketching scheme allows us to cancel out the effect of the faulty edges $F$ from the sketching information. Consequently, we obtain the sketches of each $T \\setminus F$ component in the surviving graph $G \\setminus F$. We can then apply the Boruvka's algorithm on the components of $T \\setminus F$, and infer the $s$-$t$ connectivity in $G \\setminus F$. The actual implementation of this labeling scheme is somewhat more delicate. We note that some of these technicalities are for the sake of our later extension of these labels into compact routing schemes.\n\n\n\n\\remove{\n\\mertodo{This text can be omitted now, see if want to move elsewhere. We start by illustrating the underlying intuition for sketch. For a vertex $v$ and a subset of edges $E' \\subseteq E$, let $\\mathsf{Sketch}\\xspace_{E'}(v)$ be the bitwise XOR of all the IDs of $E'$ edges adjacent to $v$.  \nFor a subset of vertices $S$, define $\\mathsf{Sketch}\\xspace_{E'}(S) = \\oplus_{v \\in S} \\mathsf{Sketch}\\xspace_{E'}(v)$. The useful property of sketches is that all edges of $E'$ that have both endpoints in $S$ are cancelled, and thus $\\mathsf{Sketch}\\xspace_{E'}(S)$ corresponds to the XOR of the identifiers of the $E'$ edges outgoing from $S$. If there is only one such edge, then\nits ID corresponds to the value of $\\mathsf{Sketch}\\xspace_{E'}(S)$. By combining this idea with a basic sampling trick one can \nidentify one outgoing edge from any subset $S$. But here we should also require special edge IDs in order to distinguish between an illegal ID, obtained by XORing IDs of several edges, and a true ID of a single edge.\nIntuitively, we first define $O(\\log{m})$ sets of edges $E_j$, where $E_j$ is obtained from $E$ by sampling each edge with probability $1\/2^j$. Next, we define $\\mathsf{Sketch}\\xspace(v) = (\\mathsf{Sketch}\\xspace_{E_0}(v),...,\\mathsf{Sketch}\\xspace_{E_{\\log{m}}}(v))$, and $\\mathsf{Sketch}\\xspace(S) = (\\mathsf{Sketch}\\xspace_{E_0}(S),...,\\mathsf{Sketch}\\xspace_{E_{\\log{m}}}(S))$. These $O(\\log^2{n})$-bit sketch units have the property that given $\\mathsf{Sketch}\\xspace(S)$, with constant probability there is a sketch unit that holds the identifier of exactly one outgoing edge of $S$. In our algorithm, we use $\\Theta(\\log{n})$ sketch units (each time with different sampled sets $E_j$) to be able to eventually find outgoing edges w.h.p. \nA crucial point in this regard is to be able to distinguish between sketch units that hold the XOR of at least two edge identifiers vs. units that store the identifier of \\emph{exactly} one edge (i.e., an outing edge). For that purpose we employ the computation of edge identifiers by \\cite{GhaffariP16}, that have the property that the XOR of any two edge identifiers is not a \\emph{legal} edge identifier of a single edge. We also show that identification of the legal edge can be done efficiently, with no global information.\n\n\\noindent\\textbf{FT Connectivity from graph sketches.} We use the graph sketches together with the well-known Boruvka algorithm \\cite{Boruvka} to identify the connected components of the graph $G \\setminus F$. This eventually allows us to check if $s$ and $t$ are connected in $G \\setminus F$ based on their components. We start by describing our general approach, and then explain how to simulate it based only on labels of $s,t$ and $F$. Let $T$ be a spanning tree of the graph $G$. If the edges $F$ are removed from $G$, it breaks the tree $T$ to at most $|F|+1$ connected components. To figure out the connected components in $G \\setminus F$, we apply Boruvka algorithm on the connected components of $T \\setminus F$. In this algorithm, at each iteration we have a set of components, and our goal is to find an outgoing edge from each connected component, and then merge components connected by an edge. To find outgoing edges, we use the sketches of the components. After repeating the process for $O(\\log{n})$ iterations, we find the connected components of $G \\setminus F$ w.h.p. The vertices $s$ and $t$ are connected in $G \\setminus F$ iff they are in the same component. \n\\noindent\\textbf{The connectivity labels.} At a high-level, to simulate the algorithm using only the information provided by the connectivity labels, we include in the labels of vertices and edges ancestry labels in a spanning tree $T$. In addition, for any tree edge $e=\\{u,v\\} \\in T$ we include in its label the sketch information of $T_v$ and $T_u$, as well as the sketch information of $T$, where $T_v,T_u$ are the subtrees of $T$ rooted at $v$ and $u$, respectively. \nWe show that this information allows us to identify the connected components in $T \\setminus F$, and compute the sketch information of each one of the components. Moreover, the ancestry labels give us an efficient way to identify the connected component of any vertex $v$. This is useful both for identifying the connected components of $s$ and $t$, and to find the connected components of any outgoing edge that the algorithm finds. When we merge components, the sketch information of the new component can be obtained by XORing the sketches of the components we merge. One delicate point in the algorithm is that we originally compute sketches in the original graph $G$, where our algorithm works in the graph $G \\setminus F$, and so we need the sketch information in $G \\setminus F$. To obtain this, we cancel the information about edges from $F$ in the sketches by XORing their IDs in the relevant places. This can be done efficiently by using pairwise independence hash functions to generate the sketches. \n}\n}\n\n\n\\paragraph{Applications for Routing Schemes.} The starting point to our routing scheme is given by our (sketch-based) labeling scheme. These labels allows one to deduce also a succinct description of an $s-t$ path in $G \\setminus F$ if exists, by following the component merging procedure of the Boruvka algorithm. This description is composed of $O(f)$ path segments, where each segment $\\{u,v\\}$ either corresponds to an outgoing (non-tree) edge found in the algorithm using the sketch information, or to a tree path between two vertices $u$ and $v$ in the same connected component in $T \\setminus F$. Given the connectivity labels of $s,t$ and $F$, we can find this description, and use it for routing. Routing across an edge $\\{u,v\\}$ just requires sending a message over the edge, while routing on a tree path between $u$ and $v$ can be done using a routing scheme for trees. \nWhile this approach allows to send a message from $s$ to $t$, there is no bound on the length of the path traversed. Additionally, this approach assumes that the set of failures $F$ is known in advance. We next explain how to overcome these issues.\n\\\\\n\\noindent\\textbf{Bounding the stretch.} To route messages on low-stretch paths we use the notion of \\emph{tree covers}, following the approach in \\cite{chechik2012f}. This approach also allows us to translate our connectivity labels to approximate distance labels as we discuss in Section \\ref{sec:ft-distance}. Here, instead of applying our connectivity scheme on just one graph $G$, we apply it on many subgraphs $G_{i,j}$ of $G$ with the following properties. \n\\begin{enumerate}\n\\item Each vertex $v$ is contained in $\\widetilde{O}(n^{1\/k})$ subgraphs.\n\\item For any $1 \\leq i \\leq \\log(nW)$, and any vertex $v$, there is a subgraph $G_{i,i^*(v)}$ that contains all the vertices in the $2^i$-neighborhood of $v$.\n\\item If $v$ and $u$ are connected in the graph $G_{i,i^*(v)} \\setminus F$, then there is a path between them of length at most $O(k|F| 2^i)$ in the graph $G_{i,i^*(v)} \\setminus F$.\\label{prop_path}\n\\end{enumerate}\nBy applying our connectivity scheme on each one of the subgraphs $G_{i,j}$, we can route a message from $s$ to $t$ on a path of stretch $O(k|F|)$. The size of the labels and routing tables of vertices is $\\widetilde{O}(n^{1\/k})$ as each vertex and edge participate in $\\widetilde{O}(n^{1\/k})$ subgraphs.\n\n\\\\\n\\noindent\\textbf{Faulty edges are unknown.} The scheme we described assumes that the routing algorithm knows the labels of $s,t$ and $F$ in advance, we next explain how to avoid this assumption. Our general approach is to work in phases, where in each phase we try to route a message from $s$ to $t$ according to the currently set of known faults. We either succeed, or learn about the label of a new faulty edge $e \\in F$ and try again. The stretch of the scheme increases to $O(k|F|^2)$ because of the $|F|+1$ phases. Direct application of this approach may require large routing tables, as each vertex may need to know the labels of all edges adjacent to it, to be able to learn the labels of faulty edges found in the algorithm. To overcome it we use the following ingredients. \n\nFirst, recall that in our connectivity labeling scheme we use a spanning tree $T$. In the routing scheme, these are the trees of the tree cover. We show that it is enough for each vertex to store labels only of its adjacent \\emph{tree} edges. \nConsequently, the total size of all routing tables can be bounded by $\\widetilde{O}(fn^{1+1\/k})$.\\footnote{The $f$ term in the size comes from the fact we apply the connectivity labels $f+1$ times to support the $|F|+1$ phases.}\nHowever, this alone is not enough to bound the size of individual routing tables of vertices, as the degree of a vertex in a tree may be linear. To overcome this, we show a clever way to load balance the labels' information between $v$ and its children in the tree. This results in tables of size $\\widetilde{O}(f^3 n^{1\/k})$ per vertex, while keeping the same stretch of the scheme. The increase in the total size of tables comes from the fact we now duplicate labels $f+1$ times, to be able to recover them in the presence of $f$ failures. ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s_intro}\n\\addtocounter{footnote}{9}\n\n\nPolycyclic aromatic hydrocarbons (PAHs) are large carbon molecules that can be thought of as the transition between the gaseous and solid phases of the interstellar medium (ISM).  They are commonly identified as the source of emission for multiple broad emission features in the mid-infrared, including those at 3.3, 6.2, 7.7, 11.3, 12.7~$\\mu$m \\citep{2008tielens}.  While these complex molecules can be excited by optical and infrared photons \\citep{2002li}, excitation by ultraviolet photons is more efficient \\citep{2008tielens}.  \n\n\nMany authors have sought to calibrate PAH emission as an extragalactic star formation tracer, mainly because it is relatively unaffected by dust extinction and because PAHs emit at shorter wavelengths than hot dust and can therefore be imaged with better angular resolutions.  While early analyses with the Infrared Space Observatory \\citep{1996kessler} found some evidence for a relation between PAH emission and other star formation tracers \\citep{2001roussel,2004forster}, later work, including work with the {\\it Spitzer} Space Telescope \\citep{2004werner}, demonstrated that the PAH emission was actually poorly correlated with other star formation tracers.  PAH emission appeared suppressed relative to other star formation tracers in star forming regions or overluminous in diffuse regions \\citep{2004boselli,2004helou,2005calzetti,2007prescott,2008bendo}.  The ratio of PAH to hot dust continuum emission was also found to decrease as metallicity decreased \\citep{2005engelbracht,2006madden,2007calzetti,2008engelbracht,2008galliano,2008gordon,2013galametz}. The common explanations are either that low metallicity regions contain fewer PAHs or that the PAHs are exposed to harder ultraviolet radiation in low-metallicity environments with less dust attenuation.  None the less, globally-integrated PAH emission has been shown to be correlated with other globally-integrated star formation tracers, and methods have been developed for calculating extinction-corrected star formation rates using a combination of H$\\alpha$ and PAH emission in the 8~$\\mu$m {\\it Spitzer} band \\citep{2008zhu,2009kennicutt}, although \\citet{2007calzetti} warns that even global measurements could be affected by metallicity effects.\n\n\nIn contrast, a few authors have found that PAH emission was associated with emission at $>100~\\mu$m that will primarily trace $\\ltsim30$~K large dust grains.  One of the first groups to draw attention to this was \\citet{2002haas}, who demonstrated that PAH emission was better correlated with 850~$\\mu$m from cold dust than 15~$\\mu$m emission from hot dust, although this analysis was limited to regions with high infrared surface brightnesses.  The ISO-based analysis by \\citet{2004boselli} also implied that PAHs were associated with diffuse dust rather than star forming regions.  Later work by \\citet{2006bendo} and \\cite{2008bendo} demonstrated that the PAH emission showed a strong correlation with 160~$\\mu$m emission and that the 8\/160~$\\mu$m surface brightness ratio was dependent upon the 160~$\\mu$m surface brightness.  These results combined with the breakdown in the relation between the PAH and hot dust emission implied that the PAHs were primarily associated with dust in the diffuse ISM or in cold molecular clouds near star forming regions and that both the PAHs and the large dust grains were heated by the same radiation field.\n\n\nData from the {\\it Herschel} Space Observatory \\citep{2010pilbratt} can be used to further study the relation between PAHs and large dust grains.  The telescope is able to resolve emission at 160 and 250~$\\mu$m on $<18$~arcsec scales, which is a marked improvement in comparison to the 38~arcsec scales that could be resolved by {\\it Spitzer} at 160~$\\mu$m.  Moreover, \\citet{2010bendo}, \\citet{2011boquien}, and \\citet{2012bendo} have demonstrated that $\\leq160$ and $\\geq250$~$\\mu$m emission from nearby galaxies may originate from dust heated by different sources.  The 70\/160 and 160\/250~$\\mu$m surface brightness ratios were typically correlated with star formation tracers such as ultraviolet, H$\\alpha$, and 24~$\\mu$m emission and peaked in locations with strong star formation, suggesting that the dust seen at $\\leq160$~$\\mu$m is primarily heated locally by star forming regions.  Meanwhile, the 250\/350 and 350\/500~$\\mu$m ratios were more strongly correlated with near-infrared emission and generally varied radially in the same way as the emission from the total stellar population (including both young, intermediate-aged, and evolved stars), demosntrating that the dust was primarily heated by the diffuse interstellar radiation field (ISRF) from these stars.  PAH emission can be compared to dust emission observed by {\\it Herschel} to determine which of these two dust components are more closely associated with PAH emission, which would ultimately lead to a better understanding of how the PAHs are excited and how they survive in certain environments in the ISM.\n\n\nSo far, the relation between PAH and dust emission has been investigated using {\\it Herschel} data for only two galaxies.  \\cite{2014calapa} have shown that 8~$\\mu$m emission from PAHs in M33 is well correlated with 250~$\\mu$m emission.  They go on to further demonstrate the 8\/250~$\\mu$m ratio is correlated with the 3.6~$\\mu$m band tracing the total stellar population, implying that the PAHs are excited by the diffuse ISRF.  \\citet{2014lu} present an alternative analysis with M81 in which they divide the PAH emission into components heated by two sources: a component heated by star forming regions traced by H$\\alpha$ emission and a component heated by the diffuse ISRF traced by the cold dust emission at 500~$\\mu$m emission.  The results show that most ($\\sim85$\\%) of the 8~$\\mu$m emission from diffuse regions is associated with the cold dust emission, while in star forming regions, most ($\\sim60$\\%) of the 8~$\\mu$m emission is excited by young stars.\n\n\nThe goal of this paper, which is a continuation of the work by \\citet{2013jones}, is to further study the relationship between PAH emission at 8~$\\mu$m and far-infrared emission from large dust grains using {\\it Herschel} Space Observatory \\citep{2010pilbratt} far-infrared images of NGC 2403 and M83.  These are two of the fourteen nearby galaxies within the Very Nearby Galaxies Survey (VNGS; PI: C. Wilson), a {\\it Herschel}-SPIRE Local Galaxies Guaranteed Time Program.  The VNGS was meant to sample galaxies with multiple morphological and active galactic nucleus types, and includes several well-studied galaxies including the Antennae Galaxies, Arp 220, Centaurus A, M51, and NGC 1068.  These two specific galaxies were selected because they are non-interacting nearby ($<10$~Mpc) spiral galaxies with an inclination from face on $\\leq~60^{o}$ and major axes $>10$~arcmin\\footnote{M81 is also in the VNGS, but the analysis of PAH emission from that galaxy is covered by \\citet{2014lu}.}  The basic properties of these galaxies are given in Table \\ref{t_galaxies}.\n\n\nNGC 2403 is an SAB(s)cd galaxy \\citep{1991devaucouleurs} with no clear bulge and flocculent spiral structure \\citep{1987elmegreen}.  Since the brightest star forming regions are found well outside the centre of the galaxy, it is easy to differentiate between effects related to star forming regions and either effects related to the evolved stellar population (which peaks in the centre of the galaxy) or effects tied to galactocentric radius.  This has been exploited previously to illustrate how PAH emission is inhibited relative to hot dust emission in star forming regions \\citep{2008bendo} and to differentiate between different heating sources for the dust seen at 70-500~$\\mu$m \\citep{2012bendo}.  M83 (NGC 5236) is an SAB(s)c galaxy \\citep{1991devaucouleurs} with a bright starburst nucleus \\citep{1983bohlin} and two strongly defined grand design spiral arms \\citep{1998elmegreen}.  Since we can resolve the spiral structure with {\\it Herschel}, we can compare the properties of arm and interarm regions quite effectively.  Both galaxies are at similar distances; we can resolve structures of $<400$~pc in the {\\it Herschel} data.  Although both of these galaxies are late-type spiral galaxies, they have the potential to yield different information on how PAHs relate to the far-infrared emission from large dust grains.\n\n\nWe focus our analysis on the {\\it Spitzer} 24~$\\mu$m data, which trace emission from very small grains and hot dust heated locally by star forming regions, and {\\it Herschel} 160 and 250~$\\mu$m data, which trace emission from large dust grains.  The prior analysis by \\citet{2008bendo} had shown an association between the 8~$\\mu$m and 160~$\\mu$m emission, but as stated above, the 160~$\\mu$m band may contain significant emission from large dust grains heated by star forming regions, while the 250~$\\mu$m band, at least for NGC~2403 and M83, originates more from dust heated by the diffuse ISRF \\citep{2012bendo} and could be better associated with PAH emission if PAHs are destroyed in star forming regions.  The next shortest waveband for which we have data for these two galaxies is at 70~$\\mu$m, but the available 70~$\\mu$m data have a lower signal to noise ratio, and the {\\it Spitzer} data are strongly affected by latent image artefacts.  Moreover, the 70~$\\mu$m emission may include emission from the same sources as the 24~$\\mu$m band.  The available 350 and 500~$\\mu$m data trace the same thermal component of dust seen at 250~$\\mu$m, but because the resolution of those data are coarser compared to the 250~$\\mu$m waveband, using the data would provide no additional benefit. \n\n\nFor this analysis, we use the techniques developed by \\citet{2008bendo} and \\citet{2012bendo} based upon qualitative analyses of surface brightness ratio maps based on images with matching point spread functions (PSFs) and quantitative analyses of the surface brightnesses and surface brightness rations measured in rebinned versions of these images.  Section \\ref{s_data} introduces the data and the data preparation steps.  We then present the analysis in Section~\\ref{s_analysis_ratios} and then use these results to identify the PAH excitation sources in Section~\\ref{s_pahexcitation}.  Following this, we discuss the implications of these results in Section~\\ref{s_discussion} and provide a summary in Section \\ref{s_conclusions}. \n\n\n\\begin{table*}\n\\centering\n\\begin{minipage}{94mm}\n\\caption{Properties of the sample galaxies$^a$.}\n\\label{t_galaxies}\n\\begin{tabular}{@{}lccccc@{}}\n\\hline\nName \t&\n    RA & \n    Dec &\n    Hubble &\n    Distance &\n    Size of Optical \\\\ \n&\n    (J2000) & \n    (J2000) &\n    Type &\n    (Mpc)$^b$ &\n    Disc (arcmin) \\\\ \n\\hline\nNGC 2403 & \n    07 36 54.5 &\n   +65 35 58 &\n   SAB(s)cd &\n   3.2 $\\pm$0.3 &\n   $22.0 \\times 12.3$ \\\\\nM83 &\n    13 37 00.3 &\n    -29 52 04 &\n    SAB(s)c &\n    4.5$\\pm$ 0.2 &\n    $12.9 \\times 11.3$ \\\\\n\\hline\n\\end{tabular}\n$^{a}$ Data are taken from \\cite{1991devaucouleurs} unless otherwise specified.\\\\\n$^{b}$ Distances are taken from \\cite{2001freedman}.\\\\ \n\\end{minipage}\n\\end{table*}\n\n\n\\section{Data}\n\\label{s_data}       \n\n\nThe 3.6, 4.5, 5.8 and 8.0~$\\mu$m data for NGC~2403 were observed with the Infrared Array Camera \\citep[IRAC; ][]{2004fazio} on {\\it Spitzer} as part of the {\\it Spitzer} Infrared Nearby Galaxies Survey \\citep[SINGS; ][]{2003kennicutt}, and the 3.6-8.0~$\\mu$m images for M83 were observed with IRAC by the Local Volume Legacy (LVL) Survey \\citep{2009dale}.  Both groups used similar drizzle techniques to mosaic basic calibrated data to produce final images with 0.75~arcsec pixels. The full-width at half maxima (FWHMs) of the PSFs are listed in Table~\\ref{t_IRAC}.  We also applied correction factors that optimise the data for photometry of extended source emission as suggested by the IRAC Instrument Handbook; these correction factors are listed in Table~\\ref{t_IRAC}.  The calibration uncertainty of the data is 3\\% \\citep{2013irac}.\n\n\n\\begin{table}\n\\caption{Properties of the IRAC instrument$^a$.}\n\\label{t_IRAC}\n\\begin{center}\n\\begin{tabular}{lcc}\n\\hline\nChannel &\n  FWHM  &\n  Correction Factors$^b$ \\\\\n\\hline\n3.6~$\\mu$m &\n  1.7 &\n  0.91 \\\\\n4.5~$\\mu$m &\n  1.7 &\n  0.94 \\\\\n5.8~$\\mu$m &\n  1.9 &\n  0.66 \\\\\n8.0~$\\mu$m &\n  2.0  &\n  0.74 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n$^a$ These data are given by the IRAC Instrument Handbook \\citep{2013irac}\\footnote{http:\/\/irsa.ipac.caltech.edu\/data\/SPITZER\/docs\/irac\\\\ \/iracinstrumenthandbook\/IRAC\\_Instrument\\_Handbook.pdf}.\\\\\n$^b$ These correction factors are for extended, diffuse emission. \n\\end{table}\n\n\nThe 24~$\\mu$m data were acquired using the Multiband Imaging Photometer for Spitzer \\citep[MIPS; ][]{2004rieke} on {\\it Spitzer} and were reprocessed by \\citet{2012bendomips} using the MIPS Data Analysis Tools \\citep{2005gordon} along with multiple modifications.  The final images have pixel scales of 1.5~arcsec, PSF with FWHM of 6~arcsec \\citep{2007engelbracht}, and calibration uncertainties of 4\\% \\citep{2007engelbracht}.\n\n\nThe 160~$\\mu$m data are updated versions of the 160~$\\mu$m data published by \\citet{2012bendo} and \\citet{2012foyle}.  The galaxies were observed at 160~$\\mu$m with the Photodetector Array Camera and Spectrometer \\citep[PACS; ][]{2010poglitsch} on {\\it Herschel} in four pairs of orthogonal scans performed at the 20 arcsec s$^{-1}$ rate.  The observations of NGC~2403 covered a $40\\times40$ arcmin region, while the observations of M83 covered a $25\\times25$ arcmin region.  The data were processed using the {\\it Herschel} Interactive Processing Environment \\citep[\\small{HIPE}; ][]{2010ott} version 11.1.  We used the standard data processing pipeline, which includes cosmic ray removal and cross-talk corrections, for the individual data frames.  We then remapped the data using {\\small SCANAMORPHOS} version 23 \\citep{2013roussel}, which also removes additional noise in the data and drift in the background signal.  We applied a colour correction of $1.01 \\pm\n0.07$, which has a mean value and uncertainty appropriate for emission from a modified blackbody with a temperature between 15 and 40~K and an emissivity function that scales as $\\lambda^{-\\beta}$ where $\\beta$ is between 1 and 2 \\citep{2011muller}\\footnote{http:\/\/herschel.esac.esa.int\/twiki\/pub\/Public\/PacsCalibrationWeb\\\\ \/cc\\_report\\_v1.pdf}.  The FWHM of the PSF is $\\sim12$~arcsec \\citep{2012lutz}\\footnote{\n  https:\/\/herschel.esac.esa.int\/twiki\/pub\/Public\/PacsCalibrationWeb\\\\ \/bolopsf\\_20.pdf}, and the flux calibration uncertainty is 5\\%\n\\citep{2013altieri}\\footnote{http:\/\/herschel.esac.esa.int\/Docs\/PACS\/pdf\/pacs\\_om.pdf}.\n\n\nThe 250~$\\mu$m images, produced using data from the Spectral and Photometric Imaging REceiver \\citep[SPIRE; ][]{2010griffin} on {\\it Herschel}, are also updated versions of the 250~$\\mu$m images originally published by \\citet{2012bendo} and \\citet{2012foyle}.  The observations consisted of one pair of orthogonal scans using the 30 arcsec s$^{-1}$ scan rate and nominal bias voltage settings.  The maps cover a $30\\times30$ arcmin region around NGC~2403 and $40\\times40$~arcmin region around M83.  The data were reprocessed using HIPE version 12.1 through a pipeline that includes the standard signal jump correction, cosmic ray removal, low pass filter correction, and bolometer time response corrections, but we used the BRIght Galaxy ADaptive Element method \\citep[][Smith et al., in preparation]{2012smith, 2013auld} to remove drift in the background signal and to destripe the data.  The final maps were produced using the naive mapmaker in HIPE and have pixel scales of 6~arcsec.  The FWHM of the PSF is specified by the SPIRE Handbook \\citep{2014valtchanov}\\footnote{herschel.esac.esa.int\/Docs\/SPIRE\/spire\\_handbook.pdf} as 18.1~arcsec, and the calibration uncertainty is 4\\% \\citep{2013bendo}.  To optimise the data for extended source photometry, we multiplied the data by the point source to extended source conversion factor of 91.289 MJy sr$^{-1}$ (Jy beam$^{-1}$)$^{-1}$ \\citep{2014valtchanov} and then applied a colour correction of $0.997 \\pm 0.029$, which should be appropriate for a modified blackbody with a temperature between 10 and 40~K and a $\\beta$ between 1.5 and 2 \\citep{2014valtchanov}.\n\n\nFor a discussion on the spiral density waves in M83 in Section~\\ref{s_m83pah}, we also included 0.23~$\\mu$m data from the Galaxy Evolution Explorer \\citep[GALEX; ][]{2005martin} produced by \\citet[][ see also \\citealt{2011lee}]{2009dale}.  The images have pixel scales of 1.5~arcsec, PSF FWHM of $\\sim6$~arcsec \\citep{2005martin}, and calibration uncertainties of $<1$\\% \\citep{2007morrissey}.  We applied a foreground extinction correction based on $A_{0.23\\mu\\text{m}}=0.56$ given by \\citet{2011lee} based on the \\citet{1989cardelli} extinction law with $R_V=3.1$.\n\n\nFor measuring quantitative star formation rates (so that we could identify locations that are strongly influenced by star formation using quantitative criteria), we included H$\\alpha$ data for these two galaxies in our analysis.  The H$\\alpha$ image for NGC~2403 was originally produced by \\citet{2002boselli} using data from the 1.20~m Newton Telescope at the Observatoire de Haute Provence.  The H$\\alpha$ image for M83 was produced by \\citet{2006meurer} using observations from the Cerro Tololo 1.5 Meter Telescope taken as part of the Survey for Ionization in Neutral Gas Galaxies.  We applied extinction corrections for dust attenuation within the Milky Way using calculations performed by the NASA\/IPAC Extragalactic Database\\footnote{http:\/\/ned.ipac.caltech.edu\/} based on data from \\citet{1998schlegel}, and we also use data from the literature to correct for [N{\\small II}] emission falling within the wavebands covered by the H$\\alpha$ filters.  Details on the data are given in Table~\\ref{t_hadata}.\n\n\\begin{table}\n\\caption{Properties of and corrections for the H$\\alpha$ images}\n\\label{t_hadata}\n\\begin{center}\n\\begin{tabular}{lp{1.8cm}p{1.8cm}}\n\\hline\nGalaxy &                              NGC~2403 &                        M83 \\\\ \\hline\nSource &                              \\citet{2002boselli} &             \\citet{2006meurer} \\\\\nPixel Scale  (arcsec ~pixel$^{-1}$) & 0.69 &                            0.43 \\\\\nPSF FWHM (arcsec) &                   3 &                               1.6 \\\\\nCalibration uncertainty &             5\\% &                             4\\% \\\\\nForground extinction ($A_R$) &        0.87 &                            0.144 \\\\ \\relax\n[N{\\small II}] \/ H$\\alpha$ ratio &      $0.28 \\pm 0.05^a$ &               $0.40 \\pm 0.13^b$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n$^a$ Both the 6548 and 6583 {\\AA} [N{\\small II}] lines fall within the band covered by the H$\\alpha$ filter used in the NGC~2403 observations.  This number represents the ratio of emission from both lines to H$\\alpha$ emission measured in the radial strip data from \\citet{2010moustakas}.\\\\\n$^b$ The H$\\alpha$ filter used in the M83 observations includes emission from only the [N{\\small II}] 6583 {\\AA} line.  This ratio is based on the ratio of ony that line to H$\\alpha$ emission and is calculated using data from \\citet{2005boissier}.\n\\end{table}\n\n\n\n\n\n\\subsection{Data preparation}\n\\label{s_data_prep}\n\n\nTo study the relation of PAH emission to far-infrared emission from large dust grains, we perform analyses using maps in which the PSFs have been matched to the PSF of the 250~$\\mu$m data (with a FWHM of 18~arcsec), and we plot data from images with matching PSFs that have been resampled into 18~arcsec bins that represent individual resolution elements within the maps.  The data from these bins should be statistically independent.  See \\citet{2008bendo} and \\citet{2012bendo} for additional discussion on this topic.\n\n\nIn the first step of preparing the data, foreground stars were identified by eye and removed from the H$\\alpha$, 3.6, 4.5, 5.8, 8 and 24~$\\mu$m data; these were typically sources that appeared unresolved and that had 3.6\/24~$\\mu$m flux density ratios $~\\rlap{$>$}{\\lower 1.0ex\\hbox{$\\sim$}}$10.  Next, the data were convolved with kernels from \\citet{2011aniano}\\footnote{Available from http:\/\/www.astro.princeton.edu\/$\\sim$ganiano\/Kernels.html .  Note that the kernels are created using circularised versions of instrumental PSFs.  In the case of the IRAC data, the circularised PSFs have FWHM ranging from 1.9 to 2.8~arcsec, which is larger than the original PSFs.} to match the PSFs of the H$\\alpha$, 3.6, 4.5, 5.8, 8, 24 and 160~$\\mu$m data to the 18~arcsec PSF of the 250~$\\mu$m data. This was done to preserve the colour variations across the data when it was rebinned, and it eliminated the need to perform additional aperture corrections.  The median background was then measured outside of the optical disc of each galaxy in each waveband and subtracted from the data.  For the qualitative map-based analyses, the 3.6 and 8 $\\mu$m images were shifted to match the world coordinate systems of the 24, 160 and 250~$\\mu$m maps so that we could create 8\/24, 8\/160, and 8\/250~$\\mu$m surface brightness ratio maps; the pixel size of each ratio map is set to the pixel size of the image for the longer-wavelength data used in the ratios.  For the analyses on binned data and for producing the profiles in Section~\\ref{s_m83pah}, the images were all shifted to match the world coordinate system of the 250~$\\mu$m data and then rebinned into 18~arcsec pixels to match the size of the PSF of the 250~$\\mu$m data.  The rebinning was done so that the centre of each galaxy was located at the centre of an 18~arcsec bin.  \n\n\nThe emission observed in the 4.5-24~$\\mu$m bands contains stellar continuum emission.  We remove this stellar emission by subtracting a rescaled version of the IRAC 3.6~$\\mu$m image \\citep[e.g. ]{2004helou, 2010marble, 2014ciesla}.  The IRAC 3.6~$\\mu$m band is suitable for this step because it generally contains unobscured stellar emission \\citep{2003lu}.  While hot dust emission may produce 3.6~$\\mu$m emission \\citep{2009mentuch, 2010mentuch} and while emission from PAHs at 3.3~$\\mu$m also falls within the IRAC 3.6~$\\mu$m band, the comparison of 3.6~$\\mu$m emission to H-band emission by Bendo et al. (2014, submitted) suggests that, on the spatial scales of our data, local enhancements in hot dust and 3.3~$\\mu$m PAH emission have a very minor effect on the total 3.6~$\\mu$m emission.  The continuum subtraction equations derived by \\citet{2004helou} were based on using an earlier version of Starburst99 \\citep{1999leitherer} to simulate the infrared stellar SED of a stellar population with a Salpeter initial mass function \\citep[IMF; ][]{1955salpeter} and two different metallicities.  From this analysis, Helou et al. derived mean 3.6\/8 and 3.6\/24~$\\mu$m stellar surface brightness ratios that could be used to rescale the 3.6~$\\mu$m emission and subtract it from the 8, 4.5, 5.8 and 24~$\\mu$m data.  We re-derived these values using a newer version of Starburst99 (version 6.0.3) to simulate a solar metallicity stellar population with a Kroupa IMF \\citep{2001kroupa}, which is now becoming more popular to use than the Salpeter IMF.  We also examined the differences resulting from using both the Geneva and Padova stellar evolutionary tracks and found that the selection of one set of tracks over the other did not significantly affect the results.  From these tests, we derive the following equations to subtract the stellar continuum from the 4.5-24~$\\mu$m data:\n\\begin{equation}\n\\begin{multlined}\nI_{\\nu}(4.5 \\mu \\mbox{m}~\\mbox{(SCS)} ) \\\\\n  = I_\\nu(4.5 \\mu \\mbox{m}) - (0.60 \\pm 0.02) I_\\nu(3.6 \\mu \\mbox{m}) \n\\end{multlined}\n\\end{equation}\n\n\n\\begin{equation}\n\\begin{multlined}\nI_{\\nu}(5.8 \\mu \\mbox{m}~\\mbox{(SCS)} ) \\\\\n  = I_\\nu(5.8 \\mu \\mbox{m}) - (0.40 \\pm 0.03) I_\\nu(3.6 \\mu \\mbox{m}) \n\\end{multlined}\n\\end{equation}\n\n\n\\begin{equation}\n\\begin{multlined}\nI_{\\nu}(8 \\mu \\mbox{m}~\\mbox{(SCS)} ) \\\\\n  = I_\\nu(8 \\mu \\mbox{m}) - (0.246 \\pm 0.015) I_\\nu(3.6 \\mu \\mbox{m}) \n\\end{multlined}\n\\end{equation}\n\n\n\\begin{equation}\n\\begin{multlined}\nI_\\nu(24 \\mu \\mbox{m} ~\\mbox{(SCS)} ) \\\\\n  = I_\\nu(24 \\mu \\mbox{m}) - (0.033 \\pm 0.003) I_\\nu(3.6 \\mu \\mbox{m}). \n\\end{multlined}\n\\end{equation}\nIn these equations, \"SCS\" stands for stellar continuum subtracted.  The scaling terms are based on calculations performed at time intervals equally spaced in logarithm space between $10^{7}$ and $10^{10}$ yr.  The values of the scaling terms are based on the mean of the results from using the Geneva and Padova tracks.  The uncertainties are the greater of either the difference in the mean values measured between the results for the two tracks or the larger of the standard deviations measured in the scaling terms derived for the separate tracks. Changing the metallicities to $Z=0.008$ changed the factors by $\\ltsim1\\sigma$.  The uncertainties in the coefficients translate to a $\\ltsim1$\\% uncertainty in the corrected 8 and 24~$\\mu$m maps, which is negligible compared to the calibration uncertainties.  The 4.5-8.0~$\\mu$m coefficients derived here are typically within $1\\sigma$ of equivalent coefficients derived in other studies \\citep[e.g.][]{2004helou, 2010marble, 2014ciesla}.  The coefficients for the 24~$\\mu$m data may disagree with coefficients from other papers by up to 0.012 or $4\\sigma$, although the values derived in these other papers differ among each other by 0.018.  However, this correction is so small for the 24~$\\mu$m data (typically $\\sim1$\\% in NGC~2403 and M83) that the relatively high disagreement among the values should not have a major impact on our analysis or on other analyses relying upon this type of stellar continuum subtraction.\n\n\nThe 8~$\\mu$m band still contains continuum emission from very hot grains.  In most solar-metallicity galaxies, this continuum emission may constitute $\\sim20$\\% of the total stellar-continuum-subtracted 8~$\\mu$m emission \\citep[e.g.][]{2007smith}, although in locations with very weak PAH emission, such as star-forming regions or metal-poor dwarf galaxies, a much higher percentage of the 8~$\\mu$m emission may be thermal continuum emission \\citep[e.g.][]{2005engelbracht, 2006cannon, 2008engelbracht, 2008gordon}.   To remove the excess dust continuum emission, we use the following equation derived in an empirical analysis of photometric and spectroscopic data by \\citet{2010marble}\\footnote{The equation given by \\citet{2010marble} also includes a term that integrates the emission in frequency and converts the data into units of erg s$^{-1}$ cm$^{-2}$.  Since we are comparing the PAH emission in the 8~$\\mu$m band to continuum emission in other bands that is measured in Jy arcsec$^{-2}$, it is easier to keep the 8~$\\mu$m data in units of Jy arcsec$^{-2}$, so we do not include the unit conversion term in this equation.}:\n\\begin{equation}\n\\label{e_pahdustsub}\n\\begin{multlined}\nI_{\\nu}(8 \\mu \\mbox{m}~{\\mbox{(PAH)})} =  (I_{\\nu}(8 \\mu \\mbox{m}~{\\mbox{(SCS)})}\\\\\n  - (0.091 + .314 I_{\\nu}(8 \\mu \\mbox{m})\/I_{\\nu}(24 \\mu \\mbox{m}))  \\\\\n  \\times (I_{\\nu}(4.5 \\mu \\mbox{m}~{\\mbox{(SCS)})} + I_{\\nu}(5.8 \\mu \\mbox{m}~{\\mbox{(SCS)})})^{0.718} \\\\\n  \\times I_{\\nu}(24 \\mu \\mbox{m}~{\\mbox{(SCS)})}^{0.282}) \n\\end{multlined}\n\\end{equation}\nWhen this equation is applied to our data, the 8~$\\mu$m surface brightnesses typically decrease by $15-20$\\%.  Based on the analysis from \\citet{2010marble}, the percentage difference between the 8~$\\mu$m PAH fluxes calculated using this equation and the fluxes of the spectral features measured spectroscopically is 6\\%. Throughout the rest of this paper, when we refer to 8~$\\mu$m emission, we are referring to the 8~$\\mu$m emission calculated using Equation~\\ref{e_pahdustsub}.\n\n\nFor the binned analysis, we wanted to illustrate which bins were more strongly influenced by emission from star forming regions and which regions tend to trace emission from dust predominantly heated by evolved stars. To do this, we created specific star formation rate (SSFR) maps.  We first applied an intrinsic extinction correction to the H$\\alpha$ intensities (measured in erg cm$^{-2}$ s$^{-1}$ arcsec$^{-2}$) using \n\\begin{equation}\n\\label{e_hacorr}\n\\begin{multlined}\nI(\\mbox{H$\\alpha$ (corrected)}) = I(\\mbox{H$\\alpha$ (observed)})\\\\\n\t+ 2.0 \\times 10^{-25} (12.5~\\mbox{THz})I_{\\nu}(24 \\mu \\mbox{m}) \\left( \\frac{\\mbox{erg cm$^{-2}$ s$^{-1}$}}{\\mbox{Jy}} \\right),\n\\end{multlined}\n\\end{equation}\nwhich is a variant of the correction equation given by \\citet{2009kennicutt}.  Since 24~$\\mu$m emission has been shown to be associated with H$\\alpha$ emission and other star formation tracers \\citep[e.g.][]{2005calzetti, 2007calzetti, 2007prescott, 2014bendo}, it is the best band to use when correcting H$\\alpha$ emission for intrinsic dust extinction.  The 24~$\\mu$m band may also contain emission from diffuse dust heated by the radiation field from evolved stars \\citep{2009kennicutt}, which we would expect to affect low surface brightness regions in these galaxies, so low star formation rates derived using Equation~\\ref{e_hacorr} should be treated cautiously.\n\n\nAfter converting the corrected H$\\alpha$ intensities to units of erg s$^{-1}$ pc$^{-2}$ (written as $L(\\mbox{H}\\alpha)\/A$, with $A$ representing the area per pixel in pc$^2$), we used \n\\begin{equation}\n\\Sigma(\\mbox{SFR}) = 7.9 \\times 10^{-42}\\left(\\frac{L(\\mbox{H}\\alpha)}{A}\\right)\\left(\\frac{\\mbox{erg s$^{-1}$}}{\\mbox{M$_\\odot$ yr$^{-1}$}}\\right)\n\\end{equation}\nfrom \\citet{1998kennicutt} to calculate star formation rate surface densities $\\Sigma(\\mbox{SFR})$.  To produce maps of the total stellar surface mass density $\\Sigma(\\mbox{M$_{\\star}$})$, we used\n\\begin{equation}\n\\label{e_totalstellar}\n\\begin{multlined}\n\\Sigma(\\mbox{M$_{\\star}$})\n   =10^{5.65}\n   \\left(\\frac{I_{\\nu}(3.6 \\mu \\mbox{m})^{2.85}I_{\\nu}(4.5 \\mu \\mbox{m})^{-1.85}\\Omega}\n   {A}\\right)\\\\  \n   \\left(\\frac{D}{0.05}\\right)^{2}\n   \\left(\\frac{\\mbox{M$_{\\odot}$ arcsec$^{2}$}}\n   {\\mbox{Jy Mpc$^2$ pc$^2$}}\\right)\n\\end{multlined}\n\\end{equation}\nbased on the equation from \\citet{2012eskew}.  In this equation, $\\Omega$ is the angular area of the bin in the map, and $D$ distance to the source.  We then divided $\\Sigma(\\mbox{SFR})$ by $\\Sigma(\\mbox{M$_{\\star}$})$ to calculate the SSFR.\n\n\n\n\n\\section{Analysis of 8\/24, 8\/160, and 8\/250~$\\mu$\\lowercase{m} ratios}\n\\label{s_analysis_ratios}\n\n\n\\subsection{Map-based analysis}\n\\label{s_analysis_maps}\n\n\n\\begin{figure*}\n\\begin{center}\n\\epsfig{file=jonesa_fig_01.ps}\n\\caption{The 21 $\\times$ 18~arcmin images of NGC 2403 used in the analysis. North is up and east is to the left in each image. The 3.6~$\\mu$m maps trace the intermediate-age and older stars. The 8~$\\mu$m image mainly shows the PAH 7.7~$\\mu$m emission feature but may also contain small amounts of emission from hot dust and stellar sources.  The 24~$\\mu$m band traces emission from hot (100~K) dust, and the 160 and 250~$\\mu$m trace emission from colder (15-30~K) dust. The FWHM for each image is shown as a green circle in the lower left corner of each panel, and the light blue ellipse in the 3.6~$\\mu$m image outlines the optical disk of the galaxy.}\n\\label{f_ngc2403_maps}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\epsfig{file=jonesa_fig_02.ps}\n\\caption{The 20 $\\times$ 20~arcmin images of M83 used in the analysis.  See Figure \\ref{f_ngc2403_maps} for additional information on the image format.}\n\\label{f_m83_maps}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure}\n\\epsfig{file=jonesa_fig_03.ps}\n\\caption{The 8\/24, 8\/160 and 8\/250~$\\mu$m surface brightness ratio maps for NGC 2403.  These images are based on data where the PSFs are matched to the PSF of the 250~$\\mu$m data. The FWHM for the 250~$\\mu$m PSF is shown by the green circle in the lower left corner of each panel.  The colour scales in the images have been adjusted to show the structure in the surface brightness ratios; some of the red or purple pixels may be outside the range of values shown in the colour bars.  Data not detected at the $5\\sigma$ level in either band is left blank.  The 8\/24~$\\mu$m ratio is low in star forming regions where the 24~$\\mu$m emission is brightest.  The 8\/160 and 8\/250~$\\mu$m images are similar in that the ratios generally decrease with radius.  However, the 8\/160~$\\mu$m ratio shows more structure, while the 8\/250~$\\mu$m map is generally smoother.}\n\\label{f_ngc2403_ratio}\n\\end{figure}\n\n\n\\begin{figure}\n\\epsfig{file=jonesa_fig_04.ps}\n\\caption{The 8\/24, 8\/160 and 8\/250~$\\mu$m surface brightness ratio maps for M83.  These images are based on data where the PSFs are matched to the PSF of the 250~$\\mu$m data.   The maps are formatted in the same way as the maps in Figure~\\ref{f_ngc2403_ratio}.  We see spiral arm structure in all images.  The arm structure in the 8\/24~$\\mu$m image is traced by a series of red point-like sources where the ratio decreases in star forming regions.  However, the filamentary spiral structures in the 8\/160 and 8\/250~$\\mu$m maps are locations offset from the 160 and 250~$\\mu$m emission in Figure~\\ref{f_m83_maps} where PAH emission is enhanced relative to cold dust emission.}\n\\label{f_m83_ratio}\n\\end{figure}\n\n\\begin{figure}\n\\epsfig{file=jonesa_fig_05a.ps}\n\\epsfig{file=jonesa_fig_05b.ps}\n\\caption{ Maps showing the specific star formation rate (SSFR) for NGC~2403 and M83. The data are formatted the same as the corresponding ratio maps for each galaxy in Figures \\ref{f_ngc2403_ratio} and \\ref{f_m83_ratio}}\n\\label{f_ssfr}\n\\end{figure}\n\nFigures \\ref{f_ngc2403_maps} and \\ref{f_m83_maps} show the 3.6, 8, 24, 160 and 250~$\\mu$m images used in the analysis (before the application of the convolution or rebinning steps described in Section~\\ref{s_data_prep}).  Figures \\ref{f_ngc2403_ratio} and \\ref{f_m83_ratio} show the 8\/24, 8\/160 and 8\/250~$\\mu$m surface brightness ratios of the data after the convolution step but before rebinning.  The 8, 24, 160, and 250~$\\mu$m images all look very similar, demonstrating that the PAHs, hot dust, and cold dust are found in the same large-scale structures.  However, the ratio maps demonstrate how the PAH emission varies with respect to the dust traced by the other infrared bands. For comparison to these figures, we also show maps of the SSFR in Figure \\ref{f_ssfr}.\n\n\nIn both galaxies, we see a decrease in the 8\/24~$\\mu$m ratio in locations where the 24~$\\mu$m emission peaks. If the PAH emission was tracing star formation in the same way as the hot dust emission, we would see little variation across the 8\/24~$\\mu$m ratio maps. The disparity indicates either that the 24~$\\mu$m emission is enhanced in regions with high SSFR, that the PAH emission is inhibited where the hot dust emission peaks in the centres of regions with high SSFR, or that both effects occur within the star forming regions. In NGC 2403, we see the 8\/24~$\\mu$m ratio is higher in the diffuse regions outside the regions with high SSFR, particularly in the southern half of the galaxy.  In M83, we see the enhancement of PAHs relative to the 24~$\\mu$m emission in not only the interarm regions but also between high SSFR regions in the spiral arms.  The 8\/24~$\\mu$m ratio is also very low in the starburst nucleus of M83, as is also seen by Wu et al. (2014, submitted). \n\n\nThe 8\/160 and 8\/250~$\\mu$m ratio maps for NGC~2403 and M83 present different results for each galaxy.  In NGC~2403, the 8\/160 and 8\/250~$\\mu$m ratios peak near the centre and decrease with radius, although the 8\/160~$\\mu$m map looks more noisy than the 8\/250~$\\mu$m map.  Instead of seeing the PAH emission decrease relative to the cold dust emission in individual regions with high SSFR, as was the case in the 8\/24~$\\mu$m ratio maps, we see the PAH emission enhanced relative to the 160 and 250~$\\mu$m emission at the location of the infrared-brightest star forming region in the northeast side of the disc. The 8\/160 and 8\/250~$\\mu$m ratios generally do not change significantly near most other star forming regions.  With the exception of the infrared-brightest star forming region, the radial gradients in the 8\/160 and 8\/250~$\\mu$m ratios look similar to the radial gradients in the 3.6~$\\mu$m image seen in Figure~\\ref{f_ngc2403_maps}.  In NGC~2403, \\citet{2012bendo} found that the 160\/250~$\\mu$m surface brightness ratios were correlated with H$\\alpha$ emission and peaked in locations with strong star formation, while the 250\/350~$\\mu$m ratios were more strongly correlated with near-infrared emission and generally varied radially in the same way as the older stellar populations.  These results demonstrated that the 160~$\\mu$m emission is dominated by dust heated locally in star forming regions but the dust seen at 250~$\\mu$m is heated by the diffuse ISRF.  The similarity between the 8\/160, 8\/250, 250\/350, and 3.6~$\\mu$m radial gradients suggests that the PAHs are intermixed with the cold large dust grains and that the enhancement of PAH emission relative to the large dust grains depends on the surface brightness of the evolved stellar population.  If this is the case, the 8\/160~$\\mu$m map may looks noisy compared to the 8\/250~$\\mu$m map because emission in the 8 and 160~$\\mu$m bands is affected by different stellar populations while emission in the 8 and 250~$\\mu$m bands is affected by mainly the evolved stellar population.\n\n\nThe dust emission for the different wavebands peak in slightly different places in these profiles of the M83's spiral arms.  The 250~$\\mu$m emission, which trace most of the dust mass in the spiral arms, peak on the downstream (or inner) side of the spiral arm.  The profile of the 24~$\\mu$m emission tends to appear narrower and peaks $0-7$~arcsec further towards the upstream (or outer) side of the spiral arms (although this is small relative to the 18~arcsec resolution of the data used to create these plots).  This is consistent with the classical description of star formation within spiral arms \\citep[e.g.]{1969roberts, 1979elmegreen}.  First the gas flows into the spiral arms, then the gas is shocked by the spiral density waves and collapses into stars, and finally young stars emerge on the upstream side of the spiral arms.  \n\n\nThe 8~$\\mu$m emission peaks slightly further downstream from the 250~$\\mu$m emission, and the profiles of the 8~$\\mu$m emission on the downstream side of the spiral arms is broader than the profiles on the upstream side.  This is particularly pronounced for profiles A, C, and F.  While the 8\/24~$\\mu$m ratio drops sharply near the star forming regions as expected, the 8\/250~$\\mu$m ratio peak 10-30~arcsec (or $\\sim$200-650 pc) downstream from the dust lane, as is also seen in the countour overlays in Figure~\\ref{f_m83_overlay}. This demonstrates that the PAHs emission is enhanced relative to the cold dust on the downstream side of the spiral arms well outside the dust lanes.  The ultraviolet emission also peaks downstream from the 250~$\\mu$m emission in many of these profiles and that the profiles of the ultraviolet emission in B and C look broader on the downstream side.  The possible connection of these profiles to the ultraviolet emission and to the 160\/250~$\\mu$m ratios is discussed further in Section~\\ref{s_m83pah}.\n\n\nThe offset enhancements in PAH emission in the arms of M83 could appear because of astrometry problems, but we have checked the astrometry among the images using foreground and background sources outside the optical disc of the galaxy and found no significant offsets greater than $\\sim1$~arcsec in the sources between images.  It is also possible that the broader PAH emission could result from issues related to the PSF matching step, but usually these types of artefacts will appear symmetric around bright sources, whereas the enhanced PAH emisson appears asymmetric.  It is more likely that the phenomenon is real and has been difficult to detect before because of limitations in the angular resolution of far-infrared data.\n\n\n\\begin{figure}\n\\epsfig{file=jonesa_fig_06.ps}\n\\caption{The 8~$\\mu$m image of M83 (after the PSF has been matched to the PSF of the 250~$\\mu$m data) showing locations where we produced additional plots to illustrate the offset between the 8\/250~$\\mu$m ratio and the dust emission from the spiral arms.  The blue boxes show the locations in Figure~\\ref{f_m83_overlay} where we overlay contours of the 8\/250~$\\mu$m and 160\/250~$\\mu$m ratios on the 250~$\\mu$m data.  The cyan lines show the locations of the surface brightness profiles plotted in Figure~\\ref{f_m83_line}.  The image is formatted in the same way as Figure~\\ref{f_m83_maps}.}\n\\label{f_m83_line_map}\n\\end{figure}\n\n\n\\begin{figure*}\n\\begin{center}\n\\epsfig{file=jonesa_fig_07a.ps,height=7.1cm}\n\\epsfig{file=jonesa_fig_07b.ps,height=7.1cm}\n\\epsfig{file=jonesa_fig_07c.ps,height=7.1cm}\n\\epsfig{file=jonesa_fig_07d.ps,height=7.1cm}\n\\caption{The 250~$\\mu$m images of the two spiral arms in M83 with the 8\/250~$\\mu$m ratio and 160\/250~$\\mu$m ratio overlaid as contours.  The contours for the 8\/250~$\\mu$m ratio start at 0.07 and increase upwards in increments of 0.01.  The contours for the 160\/250~$\\mu$m ratio start at 2.4 and increase upwards in increments of 0.2.  The images are formatted in the same way as Figure~\\ref{f_m83_maps}.  The 8\/250~$\\mu$m ratios themselves are discussed in Section~\\ref{s_analysis_maps}, while both the 8\/250 and 160\/250~$\\mu$m ratios are discussed in Section~\\ref{s_m83pah}.}\n\\label{f_m83_overlay}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\epsfig{file=jonesa_fig_08.ps}\n\\caption{Profiles of the H$\\alpha$, 0.23, 8, 24, 160, and 250~$\\mu$m surface brightnesses and 8\/24, {8\/250, and 160\/250}~$\\mu$m surface brightness ratios measured in 18~arcsec wide regions along the locations shown in Figure~\\ref{f_m83_line_map}.  The x-axis shows the distance from the peak of the 250~$\\mu$m emission; negative numbers are for the upstream side of the arms, while positive materials are for the downstream side.  The profiles were measured in 18~arcsec wide regions in images where the PSF had been matched to the PSF of the 250~$\\mu$m data, which has a FWHM of 18~arcsec (or $\\sim400$~pc), but the data are supersampled at 1~arcsec resolutions to produce smooth curves.  All surface brightnesses are normalised so that the peak values are 1, and all ratios are normalised so that they range between 0 and 1.  The uncertainties in the normalised surface brightnesses are $~\\rlap{$<$}{\\lower 1.0ex\\hbox{$\\sim$}} 1$\\%, and the uncertainties in the normalised ratios are $~\\rlap{$<$}{\\lower 1.0ex\\hbox{$\\sim$}} 5$\\%.  The H$\\alpha$ data shown here are corrected for foreground dust attenuation but not corrected for dust extinction within M83.}\n\\label{f_m83_line}\n\\end{center}\n\\end{figure*}\n\n\n\n\n\\subsection{Analysis of binned data}\n\\label{s_binned_analysis}\n\nIn Figures \\ref{f_8-24}-\\ref{f_8-250}, we plot the relations between the PAH emission at 8~$\\mu$m and either the hot dust emission at 24~$\\mu$m or the cold dust emission at 160 and 250~$\\mu$m.  Pearson correlation coefficients for these relations are given in Table \\ref{t_correlation}.  To first order, the 8~$\\mu$m surface brightness is well correlated with the 24, 160 and 250~$\\mu$m surface brightnesses.  However, the 8\/24, 8\/160 and 8\/250~$\\mu$m ratios reveal the presence of both scatter in the relations between the 8~$\\mu$m emission and emission in other bands as well as systematic variations in these relations. \n\n\nThe variations in the 8\/24~$\\mu$m ratio in Figure~\\ref{f_8-24} show that the 8\/24~$\\mu$m ratio decreases in areas where the 24~$\\mu$m emission is strongest in both galaxies.  However, the relations between the 8\/24~$\\mu$m ratio and the 24~$\\mu$m emission differ somewhat between the two galaxies.  In NGC~2403, we found that we could see different trends in the data when we separated the 18~arcsec binned data into two subsets where the SSFR was either $\\geq1\\times10^{-10}$~yr$^{-1}$ or $<1\\times10^{-10}$~yr$^{-1}$.  The data with low SSFR follow a relation in which $\\log (I_\\nu(8\\mu\\mbox{m})\/I_\\nu(24\\mu\\mbox{m}))$ increases slightly from $\\sim 0.0$ to $\\sim 0.2$ as $\\log (I_\\nu(24\\mu\\mbox{m}))$ increases from -6 to -4.  The emission from these regions may orginate mainly from locations in the diffuse ISM with relatively soft radiation fields where the PAH emission is well-correlated with emission from hot, diffuse dust heated by the diffuse ISRF.  As both the PAHs and the hot, diffuse dust are stochastically heated, the ratio of PAH to hot dust emission is expected to be roughly constant.  The slight decrease in the 8\/24~$\\mu$m ratio as the diffuse 24~$\\mu$m surface brightness decreases is potentially a result of an increase in the hardness of the radiation field as the 24~$\\mu$m surface brightness decreases (possibly as a result of changes in metallicity with radius as found by multiple authors \\citep[e.g. ][]{1994zaritsky,2010moustakas}), which could lead to either PAH emission being suppressed or 24~$\\mu$m emission being enhanced.  Data points tracing locations with high SSFR fall below the relation between the 8\/24~$\\mu$m ratio and 24~$\\mu$m surface brightness.  These locations would be expected to have harder radiation fields that may enhance the 24~$\\mu$m emission or suppress the PAH emission.  While we are able to empirically separate data into regions with high and low 8\/24~$\\mu$m ratios using a SSFR cutoff value of $1\\times10^{-10}$~yr$^{-1}$, additional work in modelling the stellar populations, the PAH excitation, and the dust heating is needed to understand the details of how the SSFR of the stellar populations affects the variations in the 8\/24~$\\mu$m ratio within NGC~2403.\n\n\nIn M83, the relationship between the 8\/24~$\\mu$m ratio and 24~$\\mu$m emission is close to linear at $\\log (I_\\nu(24\\mu\\mbox{m}))>-4$, but it flattens at $\\log (I_\\nu(24\\mu\\mbox{m})) < -4$. Some of the lowest 8\/24~$\\mu$m ratios corresponds to regions with high SSFR in the nucleus and spiral arms where, again, the 24~$\\mu$m emission may be strongly enhanced or the PAH emission is suppressed.  The relation between 24~$\\mu$m emission and the 8\/24~$\\mu$m ratio at $\\log (I_\\nu(24\\mu\\mbox{m}))>-4$ in M83 is similar to the relation seen in NGC~2403.  Unlike NGC~2403, however, we found that we could not readily separate data in the plots of $\\log (I_\\nu(8\\mu\\mbox{m})\/I_\\nu(24\\mu\\mbox{m}))$ versus $\\log (I_\\nu(24\\mu\\mbox{m}))$ for M83 simply by selecting data by SSFR, as some regions with low SSFR have low 8\/24~$\\mu$m ratios.  These regions are mostly locations within radii of 1.5~kpc.  The exact reason why we see this is unclear, although it is possible that hard ultraviolet photons from the starburst nucleus leak into the diffuse ISM in this region and destroy the PAHs in the diffuse ISM.\n\n\nFigure \\ref{f_8-160} shows good correlations between the 8~$\\mu$m and 160~$\\mu$m surface brightnesses.  The plot of the 8\/160~$\\mu$m vs 160~$\\mu$m for M83 shows that the 8\/160~$\\mu$m ratio is close to constant over a range of infrared surface brightnesses that vary by a factor of 100, indicating that the relation between 8 and 160~$\\mu$m emission is very close to a one-to-one relationship.  Some scatter is seen in the 8\/160~$\\mu$m ratio at high 160~$\\mu$m surface brightnesses.  Some of these data points are for locations around the infrared-bright centre of M83 where the outer regions of the PSF were not matched perfectly in the convolution step, while other data points sample regions along the spiral arms where the enhancement in the 8\/160~$\\mu$m ratio is offset from the 160~$\\mu$m surface brightness as discussed in Section~\\ref{s_analysis_maps}.  In NGC 2403, the 8\/160~$\\mu$m ratio increases with 160~$\\mu$m surface brightness, and the relation exhibits more scatter, indicating that the relation of 8~$\\mu$m emission to 160~$\\mu$m emission in NGC~2403 is different from the relation for M83. \n\n\nThe relations between the 8 and 250~$\\mu$m emission in Figure \\ref{f_8-250} are similar to the relations between the 8 to 160~$\\mu$m emission.  For both galaxies, the 8\/250~$\\mu$m ratio increases with the 250~$\\mu$m surface brightness, and the correlation coefficients are relatively strong.  In NGC~2403, the correlation coefficient between the 8\/250~$\\mu$m ratio and the 250~$\\mu$m surface brightness is 0.83, which is much higher than the correlation coefficient of 0.66 for the relation between the 8\/160~$\\mu$m ratio and the 160~$\\mu$m surface brightness.  Given that the square of the Pearson correlation coefficient indicates the fraction of variance in one quantity that depends upon the other quantity, the difference in the correlation coefficients is equivalent to a $\\sim25$\\% difference in being able to describe the variance in the relations.  This suggests that the PAHs are more strongly associated with the colder dust seen at 250~$\\mu$m than the warmer dust seen at 160~$\\mu$m.  In M83, the relation between the 8\/250~$\\mu$m ratio and 250~$\\mu$m surface brightness is sloped and also shows significant scatter at high surfaces brightnesses in the same way as the relationship between the 8\/160~$\\mu$m ratio and the 8~$\\mu$m emission.\n\n\nBecause M83 is at a distance $\\sim$1.5 further than NGC 2403, the 18~arcsec bins used in this analysis will cover regions with different spatial scales.  In Appendix~\\ref{a_binsizecheck}, we examined how the results for the analysis on NGC~2403 would change if we used 27~arcsec bins, which cover approiximately the same spatial scales as the 18~arcsec bins used for the M83 data. We see no noteable difference in the results using the 27~arcsec bins compared to the 18~arcsec bins; most correlation coefficients change by $\\leq0.05$.  Hence, adjusting the bin sizes for the two galaxies to similar spatial scales is unimportant.  We will therefore use data measured in the smaller bins as it takes full advantage of the capabilities of the {\\it Herschel} data that we are using and as it allows us to illustrate how the relations are still found in smaller structure in NGC 2403.\n\n\n\\begin{table}\n\\caption{Pearson correlation coefficients for the binned data.}\n\\label{t_correlation}\n\\begin{tabular}{p{5.6cm}cc}\n\\hline\n       & \n   \tNGC &\n\tM83 \n\t\\\\ \n       & \n   \t2403 &\n\t\\\\ \\hline\n$\\log (I_\\nu(8\\mu\\mbox{m}))$ vs $\\log(I_\\nu(24\\mu\\mbox{m}))$ &\n   \t0.96\t&\n\t0.97\t\\\\\n$\\log (I_\\nu(8\\mu\\mbox{m})\/I_\\nu(24\\mu\\mbox{m}))$ vs $\\log(I_\\nu(24\\mu\\mbox{m}))$ &\n\t-0.13\t&\n\t-0.74\t\\\\\n$\\log (I_\\nu(8\\mu\\mbox{m}))$ vs $\\log(I_\\nu(160\\mu\\mbox{m}))$ &\n\t0.98\t&\n\t0.98\t\\\\ \n$\\log (I_\\nu(8\\mu\\mbox{m})\/I_\\nu(160\\mu\\mbox{m}))$ vs $\\log(I_\\nu(160\\mu\\mbox{m}))$ &\t\n\t0.66\t&\n\t0.17\t\\\\\n$\\log (I_\\nu(8\\mu\\mbox{m}))$ vs $\\log(I_\\nu(250\\mu\\mbox{m}))$ &\n\t0.98\t&\t\n\t0.97\t\\\\ \n$\\log (I_\\nu(8\\mu\\mbox{m})\/I_\\nu(250\\mu\\mbox{m}))$ vs $\\log(I_\\nu(250\\mu\\mbox{m}))$ &\n\t0.83\t&\n\t0.44\t\\\\\t\t\n\t\\hline\n\\end{tabular}\t\n\\end{table}\n\n\n\\begin{figure*}\n\\begin{center}\n\\epsfig{file=jonesa_fig_09.ps}\n\\caption{The 8~$\\mu$m surface brightness and the 8\/24~$\\mu$m ratios as a function of 24~$\\mu$m emission for the 18~arcsec binned data for both galaxies.  Only data detected at the $5\\sigma$ level are displayed.  The best fitting linear functions between the surface brightnesses (weighted by the errors in both quantities) are shown as black lines in the top panels. The blue points are locations with high SSFR, and the red points and error bars are locations which are predominantly heated by the diffuse ISRF; see Section~\\ref{s_data_prep} for more details. In M83 we highlight locations within a 1.5~kpc radius of the centre in black. }\n\\label{f_8-24}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\epsfig{file=jonesa_fig_10.ps}\n\\caption{The 8~$\\mu$m surface brightness and the 8\/160~$\\mu$m ratios as a function of 160~$\\mu$m emission for the 18~arcsec binned data for both galaxies.  The data are formatted in the same way as in Figure~\\ref{f_8-24}.}\n\\label{f_8-160}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\epsfig{file=jonesa_fig_11.ps}\n\\caption{The 8~$\\mu$m surface brightness and the 8\/250~$\\mu$m ratios as a function of 250~$\\mu$m emission for the 18~arcsec binned data for both galaxies.  The data are formatted in the same way as in Figure~\\ref{f_8-24}.}\n\\label{f_8-250}\n\\end{center}\n\\end{figure*}\n\n\n\\section{Identification of PAH excitation sources}\n\\label{s_pahexcitation}\n\n\n\\subsection{PAH excitation in NGC 2403}\n\\label{s_ngc2403pah}\n\nWe can conclude from our analysis that the 8~$\\mu$m emission that we observe from NGC~2403 does not originate from PAHs excited locally within the centres of star forming regions, as the relationship between 8 and 24~$\\mu$m emission shows that PAH 8~$\\mu$m emission decreases relative to 24~$\\mu$m emission within regions with high SSFR.  This is partly because the 24~$\\mu$m emission is very sensitive to dust heating and increases significantly within star forming regions \\citep[e.g.][]{2001dale, 2002dale}.  However, the strong ultraviolet radiation from these massive young stars probably photodissociates the PAHs in the centres of these regions including species that produce features other than the 7.7~$\\mu$m feature, as has been seen spectroscopically in other galactic and extragalactic star forming regions \\citep{2007berne, 2007lebouteiller, 2007povich, 2008gordon}.  PAH emission has been observed in the outer regions of photodissociation regions \\citep{2007berne, 2007lebouteiller, 2007povich}; the photons that excite the PAHs in these locations can also heat the very small grains that produce the 24~$\\mu$m emission.  In our data, the emission from the inner and outer regions of these regions will be blended.  Integrating over the centres of these regions, the 8\/24~$\\mu$m ratio will still appear low comapred to diffuse regions outside these regions because the PAH emission is suppressed in parts of the regions while the 24~$\\mu$m emission is not, a result also obtained by \\citet{2005calzetti}.  The dust emitting at 160~$\\mu$m is also heated by light from star forming regions \\citep{2012bendo}.  While the 8~$\\mu$m emission is better correlated with the 160~$\\mu$m band than with the 24~$\\mu$m band, the 8\/160~$\\mu$m ratio still shows significant scatter as a function of 160~$\\mu$m surface brightness, possibly because the PAH emission is still inhibited in the regions from which the 160~$\\mu$m emission is originating.\n\n\nHowever, we see a strong correlation between the 8 and 250~$\\mu$m surface brightnesses, and the relation between the 8\/250~$\\mu$m ratio and the 250~$\\mu$m surface brightness shows that the residuals in the relation between the 8 and 250~$\\mu$m are very small, especially compared to the equivalent residuals for the relations between the 8~$\\mu$m emission and emission in either the 24~$\\mu$m or 160~$\\mu$m bands. This is particularly evident when comparing the correlation coefficients for the 8\/24~$\\mu$m ratio versus 24~$\\mu$m emission, the 8\/160~$\\mu$m versus 160~$\\mu$m emission, and the 8\/250~$\\mu$m ratio versus 250~$\\mu$m emission in Table~\\ref{t_correlation}.  This indicates that the PAH emission is much more strongly tied to the dust emitting in the 250~$\\mu$m band.  \\citet{2012bendo} demonstrated that the dust emitting at $\\geq 250$~$\\mu$m in NGC~2403 was heated mainly by the diffuse ISRF from the total stellar population.  This implies that the PAHs in NGC~2403 are also mainly excited by the diffuse ISRF.  Moreover, the map of the 8\/250~$\\mu$m ratio in Figure~\\ref{f_ngc2403_ratio} looks very similar to both the 3.6~$\\mu$m map in Figure~\\ref{f_ngc2403_maps} that traces the light from the total stellar population and the 250\/350~$\\mu$m ratio map from \\citet{2012bendo} that shows the variations in the colour temperatures of the large dust grains heated by the ISRF from these stars.\n\n\nTo examine this relation further, we plot the 8\/250~$\\mu$m ratio versus the 3.6~$\\mu$m surface brightness for NGC 2403 in Figure \\ref{f_2403-8-250}. We find a strong correlation between 8\/250~$\\mu$m ratio and the 3.6~$\\mu$m surface brightness; the Pearson correlation coefficient for the relation between these data in logarithmic space is 0.89.  This shows that the enhancement of PAH emission relative to cold dust emission scales with the stellar surface brightness, which implies that the PAHs are primarily mixed in with the large dust grains in the diffuse ISM and that the PAHs are predominantly heated by the diffuse ISRF from the total stellar population.  \n\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=jonesa_fig_12a.ps}\n\\epsfig{file=jonesa_fig_12b.ps}\n\\caption{The 8\/250$~\\mu$m surface brightness ratio plotted as a function of the 3.6~$\\mu$m surface brightness and galactocentric radius for the 18~arcsec binned data for NGC~2403. The data are formatted in the same way as in Figure~\\ref{f_8-24}.  The radii are based on using an inclination of $62.9\\deg$ from \\citet{2008deblok}.}\n\\label{f_2403-8-250}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=jonesa_fig_13a.ps}\n\\epsfig{file=jonesa_fig_13b.ps}\n\\caption{To examine whether the relation between the 3.6~$\\mu$m emission and the 8\/250~$\\mu$m ratio is related to hot dust or PAH emission in the 3.6~$\\mu$m band, we first show the relation between 3.6 and 8~$\\mu$m emission.  The relation in the top panel could be the result of either the 3.6 and 8~$\\mu$m bands tracing emission from similar sources or the 3.6~$\\mu$m band tracing starlight exciting the PAHs seen in the 8~$\\mu$m band.  Next, we plot the 4.5\/3.6$~\\mu$m ratio as a function of the 24$~\\mu$m emission in the bottom panel to examine whether the slope of the SED at 3.6-4.5~$\\mu$m is influenced by hot dust emission.  The absence of such a relation as well as the relative invariance of the 4.5\/3.6~$\\mu$m ratio implies that the 3.6 and 4.5~$\\mu$m bands are relatively unaffected by non-stellar emission.  The data are formatted in the same way as in Figure~\\ref{f_8-24}.}\n\\label{f_2403_stars}\n\\end{center}\n\\end{figure}\n\n\nIt is also possible that, because the correlation coefficient for the relation between the logarithm of the 3.6~$\\mu$m surface brightness and radius is -0.95, the 8\/250~$\\mu$m ratio actually depends on radius rather than 3.6~$\\mu$m surface brightness.  Any radial dependence would be expected to be related to metallicity, which decreases with radius, and this could influence the PAHs.  However, the results from \\citet{2008engelbracht} and \\citet{2008gordon} indicated that the apparent dependence of PAH emission on metallicity is really the result of changes in the radiation field illuminating the ISM in lower metallicity environments.  The radiation field would be harder in low metallicity systems first because of increases in the stellar temperatures of O and B stars \\citep{2004massey, 2005massey, 2007trundle} and second because of decreases of extinction related to a decrease in the gas to dust ratio. This in turn leads to harder interstellar radiation fields in low-metallicity environments that potentially destroy PAHs.  To examine this, we plot the 8\/250~$\\mu$m ratio as a function of radius in the bottom panel of Figure~\\ref{f_2403-8-250}.  As expected, the 8\/250~$\\mu$m ratio decreases as radius increases.  The relation between the logarithm of the 8\/250~$\\mu$m ratio and radius has a Pearson correlation coefficient of -0.82, which has an absolute value that is similar to the value of 0.89 for the relation between the lorarithms of the 8\/250~$\\mu$m ratio and the 3.6~$\\mu$m emission.  However, at the resolution of these data, we do see non-axisymmetric substructures in the 8\/250~$\\mu$m image in Figure~\\ref{f_ngc2403_ratio} that correspond to similar substructure in the 3.6~$\\mu$m images, but these structures are mainly visible at radii of $<10$ kpc.  When we look at data within this inner 10~kpc region, we get a correlation coefficient of -0.77 for the relation with radius and 0.91 for the relation with 3.6~$\\mu$m surface brightness.  This implies that the total stellar surface brightness is more influentual on PAH excitation than any effects related to radius.\n\n\nAlthough it is unlikely, the correlation between the 8\/250~$\\mu$m ratio and 3.6~$\\mu$m emission could be the result of a correlation between emission in the 3.6 and 8~$\\mu$m bands themselves.  Both bands may contain thermal continuum dust emission (although the thermal continuum emission should have been removed from the 8~$\\mu$m data when we applied Equation~\\ref{e_pahdustsub}), and the 3.6~$\\mu$m band may also contain emission from the 3.3~$\\mu$m PAH emission feature, although previous work by \\citet{2003lu} indicated that emission at $<5$~$\\mu$m from nearby galaxies is dominated by stellar emission.  To investigate this further, we plot the relation between the 8~$\\mu$m PAH emission and 3.6~$\\mu$m stellar surface brightness in Figure~\\ref{f_2403_stars}.  The data are well correlated; the correlation coefficient for the relation in logarithm space is 0.94.  While this could indicate that the same emission sources are seen at 3.6 and 8~$\\mu$m, it is also possible that the 8~$\\mu$m emission is correlated with 3.6~$\\mu$m emission because the stars seen at 3.6~$\\mu$m excite the PAHs seen at 8~$\\mu$m.  Hence, the correlation between 3.6 and 8~$\\mu$m does not necessarily prove anything about the relation between the emission in these bands.  \\citet{2010mentuch} illustrated that it was possible to identify the influence of non-stellar emission at near-infrared wavenegths by examining the 4.5\/3.6~$\\mu$m surface brightness ratio.  This ratio would be relatively invariant for stellar emission because it traces emission from the Rayleigh-Jeans side of the stellar SED, but if hot dust emission influences the bands, the ratio should increase.  We plot the 4.5\/3.6~$\\mu$m ratio versus 24~$\\mu$m emission in the lower panel of Figure \\ref{f_2403_stars}. This relationship is almost flat.  Most of the data points have log($I_\\nu$(4.5~$\\mu$m)\/$I_\\nu$(3.6~$\\mu$m)) values that lie within a range of -0.14 to -0.23, which would be consistent with what was observed for evolved stellar populations by \\citet{2010mentuch}.  The absence of significant variations in the 4.5\/3.6~$\\mu$m ratio with 24~$\\mu$m implies that the 3.6 and 4.5~$\\mu$m bands are largely uninfluenced by hot dust emission.  We do see a few data points with values of log($I_\\nu$(4.5~$\\mu$m)\/$I_\\nu$(3.6~$\\mu$m))$>-0.14$ where the 3.6 and 4.5~$\\mu$m may be more strongly influenced by non-stellar emission, but these data only weakly influence our results.  If we exclude these data, the correlation coefficient for the relation between the 3.6~$\\mu$m data and the 8\/250~$\\mu$m data changes by $<0.01$.  This shows that the 3.6~$\\mu$m emission in NGC~2403 is largely dominated by the stellar population and is relatively unaffected by hot dust or PAH emission. Therefore, the most likely explanation for the correlation between the 3.6 and 8~$\\mu$m emission as well as the correlation between the 3.6~$\\mu$m emission and the 8\/250~$\\mu$m ratio is that the PAHs are excited by the stellar population seen at 3.6~$\\mu$m.\n\n\n\\subsection{PAH excitation in M83}\n\\label{s_m83pah}\n\n\nThe results from the 8~$\\mu$m to 24~$\\mu$m relationship in M83 are similar to NGC 2403.  We see the 8~$\\mu$m PAH emission is low in regions where the 24~$\\mu$m emission peaks.  Again, PAHs are probably being destroyed locally in regions with high SSFR.  However, we find that 8~$\\mu$m emission is more strongly related to the 160 and 250~$\\mu$m emission.  We also see offsets between the 8\/250~$\\mu$m ratios and the dust mass (as traced by the 250~$\\mu$m band).  \\citet{2012bendo} and \\citet{2014bendo} also found that the 160\/250~$\\mu$m colours appeared offset relative to the star forming regions in the spiral arms, and \\citet{2012foyle} found a related offset in the dust colour temperatures.  This implies that the enhancement in PAH emission relative to cold dust emission is related to the enhancement of the temperature of the dust seen at 160~$\\mu$m.  To examine this relationship further, we map the 250~$\\mu$m emission from the spiral arms overlaid with contours showing the 160\/250~$\\mu$m ratio in Figure~\\ref{f_m83_overlay}, and we show profiles of the 160\/250~$\\mu$m ratio across the spiral arms in Figure~\\ref{f_m83_line}.  These plots show that the 8\/250 and 160\/250~$\\mu$m ratios trace similar structures offset from the dust mass as well as the H$\\alpha$ and 24~$\\mu$m emission associated with star formation.  To check how well the 8\/250 and 160\/250~$\\mu$m ratios are correlated, we plot the two ratios in Figure~\\ref{f_8-250-160-250}.  The relation has a Pearson correlation coefficient of 0.65, implying that the excitation of PAH emission and the heating of the dust seen at 160~$\\mu$m are, to some degree, linked.\n\n\nIn spiral density waves, as mentioned before, large quantities of gas and dust are expected where the ISM is shocked on the upsteam sides of the spiral arms, star forming regions would be found immediately downstream of the shocks, and older stars would be expected further downstream \\citep[e.g.][]{1969roberts, 1979elmegreen, 2008tamburro, 2009martinezgarcia, 2011sanchezgil}.  This could cause offset enhancement of PAH emission relative to spiral arm dust lanes, as seen in Figure~\\ref{f_m83_line} and as also implied by the relation in Figure~\\ref{f_8-250-160-250}, in two possible ways.  \n\n\nOne possible explanation is that the dense dust lanes on the upstream edge of the spiral arms severely attenuate the starlight escaping from the photoionising stars within star forming regions, but light easily escapes across the downstream side of the spiral arms where the dust density is lower.  Such a geometrical arrangement of the star forming regions relative to the dust would produce the slight offsets between the 24~$\\mu$m emission (tracing obscured star formation) and H$\\alpha$ emission (tracing unobscured star formation) seen in most of the profiles in Figure~\\ref{f_m83_line} and may also explain the downstream areas with enhanced H$\\alpha$ emission in profiles C and F.  If photoionising light is primarily travelling asymmetrically from the star forming regions and if the 160~$\\mu$m band traces dust heated by the light escaping from the star forming regions into the diffuse ISM, the 160~$\\mu$m emission would appear enhanced relative to 250~$\\mu$m emission along the downstream side of the spiral arms.  Similarly, PAHs mixed in with the dust emitting at 160~$\\mu$m would be excited by the ultraviolet light escaping from star forming regions and appear enhanced relative to the dust emission in the same locations, although the total PAH emission itself will peak along the spiral arms where the total mass of the PAHs is greater (which is also true for dust emission observed in any single band).\n\n\nThe other possible explanation is that the PAHs and the dust seen at 160~$\\mu$m are locally heated by a young, non-ionising population of stars (stars with ages older than 4 Myr) that have left the dusty star forming regions in the spiral arms.  As shown by \\citet{1999leitherer}, such a population would still produce a substantial amount of ultraviolet and blue light that could strongly enhance the PAH emission and the temperature of the large dust grains on the downstream side of the arms, but the radiation from these stars may not include higher energy photons that destroy PAHs.  In the profiles in Figure~\\ref{f_m83_line}, the ultraviolet emission either peaks downstream of the dust mass or has a profile on the downstream side that is broader than the dust emission profile.  Additionally, the ultraviolet emission appears to stronger relative to the H$\\alpha$ emission in most downstream locations.  This provides additional support for the possibility that the 8~$\\mu$m emission observed in M83 originates from PAHs excited locally by young, non-ionising stars, although additional analysis would be needed to confirm this.\n\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=jonesa_fig_14.ps}\n\\caption{The 8\/250$~\\mu$m surface brightness ratio plotted as a function of the 160\/250~$\\mu$m surface brightness ratio for the 18~arcsec binned data. The data are formatted in the same way as in Figure~\\ref{f_8-24}.}\n\\label{f_8-250-160-250}\n\\end{center}\n\\end{figure}\n\n\nIt is also worth briefly noting that we do not see variations in the 8\/160 or 8\/250~$\\mu$m ratios in M83 that imply a dependence upon radius and hence a dependence upon the metallicity, which decreases with radius \\citep{1994zaritsky}.  The variations in the PAH emission with respect to the cold dust emission are driven mainly by local excitation of the PAHs.\n\n\n\n \n\\section{Discussion}\n\\label{s_discussion}\n\n\nThe results show that the 8\/24~$\\mu$m ratio decreases in many regions with high SSFR, in agreement with previous findings from \\citet{2004helou}, \\citet{2005calzetti}, \\citet{2006bendo}, \\citet{2006madden}, \\citet{2007berne}, \\citet{2007lebouteiller}, \\citet{2007povich}, \\citet{2008bendo}, \\citet{2008gordon}, and \\citet{2014calapa}.  Even though our analysis is mainly focused on the 7.7~$\\mu$m PAH feature that falls within IRAC channel 4, the results from \\citet{2007lebouteiller}, \\citet{2007povich}, and \\citet{2008gordon} suggest that other PAH emission features may also decrease relative to hot dust emission within star forming regions.\n\n\nOur results showing the correlation of the 8~$\\mu$m emission from PAHs with the far-infrared emission from large dust grains is largely in agreement with the results from \\citet{2002haas}, \\cite{2008bendo}, and \\citet{2014calapa}.  However, the general conclusion from these papers had been that the PAHs are mixed with large grains heated by the diffuse ISRF.  While NGC~2403 certainly fits that scenario, M83 does not.  Instead, PAH emission in M83 is more strongly associated with large dust grains heated either by light escaping from star forming regions and travelling hundreds of pc away from the spiral arms or locally by young stars that produce substantial non-ionising ultraviolet radiation, which was unexpected.  Further study would be needed to determine whether such variations in PAH excitation are seen among other nearby galaxies as well.\n\n\nPAH may be excited by different radiation fields in NGC 2403 and M83 because of the differences in the spiral structure in the two galaxies.  Because NGC~2403 is a flocculent spiral galaxy, star formation is expected to be triggered by clouds collapsing in local gravitational instabilities \\citep[e.g.][]{2003tosaki,2010dobbs}.  In such a scenario, inflowing dust on scales of tens or hundreds of parsecs may be roughly symmetrically distributed around star forming regions, although more modelling work on cloud collapse in flocculent galaxies is needed to confirm this.  If the dust is distributed this way, dust near the centres of these shells would absorb the ultraviolet and blue light from the star forming regions inside, and any PAHs within these central regions would be destroyed.  Meanwhile, dust and PAHs in the outer shells would be shielded from the light from the star forming regions and would instead be heated by the diffuse ISRF.  Also note that the stars in spiral arm filaments in flocculent spiral galaxies are not expected to exhibit any age gradients like grand-design spiral galaxies \\citep{2010dobbs}.  If young, non-photoionising stars contribute significantly to PAH excitation, then the relatively homogeneous distribution of these stars may result in the PAHs appearing enhanced over broad areas rather than appearing enhanced near the spiral filaments.  In contrast to NGC~2403, M83 is a grand design spiral galaxy in which cold, dusty gas flows into star forming sites mainly from one side of the spiral arms and star forming regions emerge from the other side \\citep[e.g. ][]{1979elmegreen, 1993garciaburillo, 2008tamburro, 2009egusa, 2013vlahakis}.  Hence, dust will preferentially be located upstream of individual star forming regions within M83.  As described in Section~\\ref{s_m83pah}, PAHs are destroyed within the centres of these star forming regions, but PAHs in the diffuse ISM could be excited by starlight diffusing out of the optically-thin side of the star forming regions.  Additionally, multiple studies \\citep[e.g. ][]{2009martinezgarcia, 2011sanchezgil} have found gradients in the ages of the stellar populations downstream of spiral arms in grand design spiral galaxies.  PAH emission could appear enhanced downstream of star forming regions if the PAHs are destroyed in photoionising regions but strongly excited locally in regions with soft ultraviolet emission from young, non-photoionising stars.  If these geometrical descriptions for the relation between PAHs and excitation sources is accurate, then we should find that PAHs are excited by the diffuse ISRF in other flocculent late-type spiral galaxies while PAHs are excited in regions offset from star forming regions in other grand design spiral galaxies.\n\n\nSome dust emission models \\citep[e.g. ][]{2007draine} and radiative transfer models \\citep[e.g. ][]{2011popescu} show PAHs as excited by the radiation fields from all stellar populations regardless of the hardness or intensity of the fields\\footnote{The version of the \\citet{2007draine} model typically applied to infrared SEDs is usually based on dust heated by a radiation field with the same spectral shape as the local ISRF as specified by \\cite{1983mathis}.  When applying the dust model to data, only the amplitude of the radiation field is treated as a free parameter.  However, \\citet{2014draine} includes an example of SED fitting with the \\citet{2007draine} model in which the spectral shape of the illuminating radiation field is also allowed to vary.}.  Our results show that this approach is an oversimplification of PAH excitation.  New refinements in dust emission and radiative transfer models are needed to replicate how PAHs are excited by radiation fields from different stellar populations within different galaxies and how the PAH\/dust mass ratio may change with variations in the hardness of the illuminating radiation field.  For example, \\citet{2013crocker} used stellar population synthesis and simplified models of dust and PAH absorption to predict the contributions of different stellar populations to PAH excitation in NGC 628 and found that $\\sim$40\\% of the PAHs are excited by stars $<10$~Myr in age, $\\sim20$\\% are excited by stars with ages of 10-100~Myr, and the remainder are excited by stars $>$100~Myr in age.  It is also apparent that PAH excitation changes across spiral density waves (either because of details in the geometry of the star forming regions or because of variations in the stellar populations on either side of the waves), and it would be appropriate to make improvements to radiative transfer models so that they can replicate these effects.\n\n\nThese results have multiple implications for using PAH emission as a proxy for other quantities.  While PAH emission cannot be used on sub-kpc scales to measure accurate star formation rates, groups such as \\citet{2008zhu} and \\citet{2009kennicutt} have suggested using globally-integrated PAH emission to estimate extinction corrections for optical star formation tracers such as H$\\alpha$ emission, thus producing extinction corrected global star formation metrics.  When PAHs are excited by star forming regions, globally-integrated PAH emission should more accurately represent the light attenuated by dust in star forming regions and should provide fairly accurate star formation rates.  When PAHs are excited by the diffuse ISRF, however, the connection between star formation and PAH emission is less clear, and star formation rates calculated using PAH emission could be less reliable.\n\n\nPrevious results showing a relation between PAH emission and far-infrared emission from large dust grains had implied that PAHs could be used as a proxy of dust mass \\citep[e.g.][]{2008bendo}.  In cases where the PAHs are associated with dust heated by the diffuse ISRF, this should still be appropriate, although metallicity-related effects would still need to be taken into account.  In cases where the PAHs are heated by diffuse light from star forming regions or from young, non-photoionising stars that have emerged from star forming regions, the PAH emission will still scale approximately with dust mass but will also vary depending upon the radiation field from the young stars.  In this situation, using PAH emission to trace dust mass may be less reliable.  \n\n\nMultiple authors have identified an empirical relation between either radially-averaged or globally-integrated PAH and CO emission \\citep{2006regan,2010bendoco, 2013tan, 2013vlahakis}, implying that the PAHs are, to some degree, correlated with molecular gas.  This would be expected if the PAHs also trace the cold dust that is found associated with the molecular gas.  However, the relation between PAH and CO emission breaks down on small spatial scales, including in NGC~2403 (\\citealt{2010bendoco}, but also see \\citealt{2013tan}).  In M51, \\citet{2013vlahakis} found an offset between PAH and CO emission in the spiral arms, with the molecular gas associated with the cold dust in the locations where material is entering the spiral arms and the PAH emission appearing enhanced further downstream where it is excited by light from young stars.  Our results imply that, in future work, we may be able to measure a similar offset between PAH and CO emission in M83 as well as other grand design spiral galaxies.  While PAH emission was already shown to be a poor tracer of molecular gas on sub-kpc scales, the phenomenology of PAHs excitation in spiral density waves causes even more problems with using it as a proxy for molecular gas.\n\n\n\\section{Conclusions}\n\\label{s_conclusions}\n\n\nWe identified different relations between PAH emission and far-infrared emission from large dust grains in the two galaxies we examined.  For NGC 2403, we find the 8~$\\mu$m emission is most strongly associated with emission from cold dust at 250~$\\mu$m.  In particular, we find that the 8\/250~$\\mu$m ratio shows a very strong dependence upon the 3.6~$\\mu$m emission from the total stellar population, indicating that the PAHs are mixed in with the diffuse dust and heated by the diffuse ISRF from the total stellar population.  Star forming regions play a much less significant role in the excitation of the PAHs observed in the 8~$\\mu$m band. In contrast, we see in M83 that the PAH emission is more strongly associated with the 160~$\\mu$m emission from large grains heated by star forming regions as implied by the strong correlation between the 160\/250~$\\mu$m and 8\/250~$\\mu$m ratios. This illustrates that PAHs in M83 are excited either by starlight escaping asymmetrically from star forming region so that locations towards the downstream edges of the spiral arms show enhancement in 8~$\\mu$m emission compared to the dust mass or that the PAHs are excited locally by young, non-photoionising stars that have migrated downstream from the spiral arms.  \n\n\nMany dust emission and radiative transfer models currently treat PAHs as though they are excited by all radiation fields of all intensities from all stellar populations within the galaxies to which they are applied, much in the same way that emission from silicate and large carbonaceous dust grains is modelled.  The results from just these two galaxies show that this assumption is not universally applicable.  These dust models need to be adjusted to account for the observational results showing that PAHs are sometimes excited by the diffuse ISRF from the total stellar population and sometimes excited either by young, non-photoionising stars that have emerged from star forming regions or light escaping from these regions and travelling hundreds of pc away (although the PAH emission may be inhibited within the star forming regions themselves).  Additionally, some models rely upon using a single SED shape (such as the SED of the local ISRF) for the radiation field illuminating PAHs and dust.  Such models cannot account for the possibility that PAH emission could be inhibited if the radiation fields are excessively hard.  To properly characterise the PAH excitation, it is necessary to model the PAHs as being illuminated by radiation fields with different spectral shapes.\n\n\nOur data here show differences between the PAH excitation within two galaxies with similar Hubble types.  We should expand the analysis to include galaxies with a wider range of Hubble types, including E, S0, and Sa galaxies where the evolved stellar populations may play a larger role in dust heating and therefore may be more responsible for PAH excitation.  We should also examine other grand-design spiral galaxies to determine whether PAH emission from the spiral arms in these galaxies is offset from the star forming regions in the same way that it is in M83. This research will lead to a better understanding of PAH excitation mechanisms as well as the relation of PAHs to star formation and large dust grains. \n\n\n\\section*{Acknowledgments}\n\n\nThis work has ben produced as part of a MSc Thesis for the University of Manchester. SPIRE has been developed by a consortium of institutes led by Cardiff Univ. (UK) and including: Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA (USA).  IDL is a postdoctoral researcher of the FWO-Vlaanderen (Belgium).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{1. Introduction}\nIn recent years the effect of the spin of electrons on the\ntransport properties of nanostructures have been studied\nintensively, both theoretically and experimentally. In the context\nof spin-based electronics (spintronics) the possibility to control\nelectrical currents by a weak external magnetic field using the\nZeeman and\/or the spin-orbit interaction is one of the main goals.\n\nMagnetic materials and especially half-metals are natural sources of\nspin-polarized electrons for spintronics. Transport of spin-polarized\nelectrons in nanostructures (quantum dots, suspended nanowires,\netc.) in external magnetic field results in new phenomena where spin,\ncharge and mechanical degrees of freedom are strongly inter-related.\nIn this new field of investigations (spintromechanics, see\nRef.~\\onlinecite{pulkin}) the presence of a mechanically ``soft\"\nsubsystem results both in a strong enhancement of spintronic effects\nand in magnetic control of the mechanical subsystem in the\nclassical as well as in the quantum transport regimes.\n\nVibrational effects are known to be important for the transport\nproperties of molecular transistors (see, e.g., the reviews in\nRefs.~\\onlinecite{galperin} and \\onlinecite{krive}). In single-molecule\ntransistors a strong electron-vibron coupling was observed\nin a $C_{60}$-based transistor with nonmagnetic (gold)\nleads \\cite{park}. The measured current-voltage characteristics in\nthis experiment revealed low-energy periodic step-like features.\nThey were interpreted as a signature of vibron-assisted electron\ntunneling via the fullerene molecule. Experimental $I-V$ curves were\ntheoretically explained \\cite{shekhter,flensberg} in the frames of a\nsimple model of a single-level quantum dot strongly coupled to a\nsingle vibrational mode and weakly coupled to the source and drain\nelectrodes.\n\nLater on $C_{60}$-based molecular transistors with magnetic (Ni)\nleads were fabricated \\cite{pasupathy}. In samples where the\ntunneling coupling to the ferromagnetic electrodes were\nrelatively strong ($\\sim$ tens of meV), Kondo-assisted tunneling via the\n$C_{60}$ molecule was observed. These measurements also proved the\npresence of a strong inhomogeneous magnetic field produced by the\nferromagnetic electrodes in the nano-gap between them. In samples with weak\ntunneling couplings the usual Coulomb blockade picture for a single-electron \ntransistor was observed.\n\nIn the present paper we formulate the conditions for the\nappearance of a vibrational instability of a fullerene molecule\nsuspended in the gap between two magnetic leads with opposite\nmagnetization. Electron shuttling of spin-polarized electrons\nproduced by magnetic (exchange) forces was\npredicted in Ref.~\\onlinecite{kul} for the case of 100\\%\npolarization of the leads. In this limit (realized for\nhalf-metals) the electric current is blocked (spin blockade) in\nthe absence of spin-flips induced by, e.g., an external magnetic\nfield. It was shown that in the absence of dissipation in the\nmechanical subsystem such a magnetic field triggers a shuttle\ninstability even for vanishingly small fields \\cite{kul}. In the\npresence of dissipation a threshold magnetic field is determined\nby the rate of dissipation and it is small for a weak dissipation.\n\nOne of\nour aims here is to develop a theory of\nmagnetic shuttling for conditions corresponding to the experimental\nset-up of Ref.~\\onlinecite{pasupathy}, where the electrons in the\nferromagnetic leads were partially polarized ($\\sim 30\\%$). The\nabsence of a spin blockade in this case qualitatively changes the criterion for\nelectron shuttling. We will show that even in the absence of mechanical\ndissipation, a shuttling regime of electron transport occurs in a\nfinite interval of external magnetic field strengths,\n$H_{\\text{min}}<H<H_{\\text{max}}$, where $H_{\\text{min}}$ is\ndetermined by the degree of spin polarization $\\eta$.\n\nIn particular, for a high degree of spin polarization\n$(\\eta\\rightarrow 1)$ and if $\\Gamma\\gg\\hbar\\omega$ (where $\\Gamma\/\\hbar$\nis the tunneling\nrate of majority spin electrons and $\\omega\/2\\pi$ is the\nmechanical vibration frequency of the fullerene) the threshold magnetic field for\nreaching the shuttling regime of electron transport reads:\n$H_{\\text{min}}\\sim\\sqrt{1-\\eta}\\,\\Gamma$. In the limit of ``hard\"\nvibrons, $\\Gamma\\ll\\hbar\\omega$, the threshold field is determined\nby the vibron energy, $H_{\\text{min}}\\sim\\sqrt{1-\\eta}\\,\\hbar\\omega$.\n\nThe calculations outlined below aim at determining the rate of change\n$r(H)$ (to be defined later) of the\ncenter-of-mass coordinate of the single-molecule shuttle, the sign of\nwhich then allows us to formulate the\nconditions required to observe shuttling of spin-polarized\nelectrons in $C_{60}$-based molecular transistors.\n\n\\section{2. Hamiltonian and Equations of Motion}\nThe Hamiltonian of a magnetically driven single electron shuttle (see\nRefs.~\\onlinecite{kul} -- \\onlinecite{parafilo}) consists of four terms,\n\\begin{equation}\\label{4}\n\\hat H=\\sum_{j=S,D}\\left(\\hat H_j+\\hat H_{t,j}\\right)+\\hat H_d+\\hat\nH_v,\n\\end{equation}\nwhere $\\hat H_j$ is the standard Hamiltonian of noninteracting\nelectrons in the source ($j=S$) and drain ($j=D$) electrodes, $\\hat\nH_{t,j}$ is a tunneling Hamiltonian with coordinate dependent\ntunneling amplitudes $t_j(\\hat x)$ and $\\hat H_d$ is the Hamiltonian\nof a single level ($\\varepsilon_0$) quantum dot (QD) magnetically coupled to\nleads of spin-polarized electrons by coordinate dependent exchange\ninteractions $J_j(\\hat x)$. It has been shown \\cite{krive} that the shuttling\nregime of single-electron transport can be realized in the presence\nof an external magnetic field $H_{\\text{ext}}$ for oppositely\nmagnetized source and drain electrodes. If\nthe external\nmagnetic field is directed perpendicular to the antiparallel polarization vectors\nof the leads, the QD Hamiltonian reads\n\\begin{eqnarray}\\label{5}\n&&\\hat H_d= \\left[\\varepsilon_0-\\frac{J(\\hat x)}{2} \\right]\na^\\dagger_\\uparrow a_\\uparrow+\\left[\\varepsilon_0+\\frac{J(\\hat\nx)}{2} \\right] a^\\dagger_\\downarrow\na_\\downarrow\\nonumber\\\\&&\\hspace{0.65cm}-\\frac{g\\mu\nH}{2}\\left(a^\\dagger_\\uparrow a_\\downarrow+a^\\dagger_\\downarrow\na_\\uparrow\\right)+Ua^\\dagger_\\uparrow a_\\uparrow\na^\\dagger_\\downarrow a_\\downarrow.\n\\end{eqnarray}\nHere $a^\\dag_\\sigma$ $(a_\\sigma)$ is the creation (annihilation)\noperator for an electron with spin projection $\\sigma=\\uparrow,\\,\n\\downarrow$ on the dot, $J(\\hat x)=J_S(\\hat x)-J_D(\\hat x)$, $\\mu$\nis the Bohr magneton ($g$ is the gyromagnetic ratio), and $U$ is the\nCoulomb repulsion energy. QD vibrations are described by the\nharmonic oscillator Hamiltonian\n\\begin{equation}\\label{6}\n\\hat H_v=\\frac{\\hat p^2}{2m}+\\frac{m\\omega^2\\hat x^2}{2},\n\\end{equation}\nwhere $\\hat x$ is the displacement operator, $\\hat p$ is the\ncanonical conjugated momentum ($\\left[\\hat x, \\hat p\n\\right]={i}\\hbar$), $m$ is the mass and $\\omega$ is the\n(angular) vibration frequency of the QD.\n\nThe aim of the present paper is to find the conditions under\nwhich magnetically driven shuttling in experiments with\nfullerene-based single-molecule transistors \\cite{park,pasupathy}\ncan be realized.\nIn this case the ferromagnetic leads are characterized by a certain\ndegree $0<\\eta<1$ of spin polarization ($\\sim 30\\%$ in the\nexperiment of Ref.~\\onlinecite{pasupathy}) and the characteristic\nvibron energy $\\hbar\\omega$ ($\\sim$ several meV,\nRef.~\\onlinecite{park}) is larger or of the order of the energy\nscale $\\Gamma$ that characterizes the tunneling coupling to the leads,\n$\\Gamma\\leq\\hbar\\omega$.\n\nIt follows from the above considerations that we are not in the adiabatic regime\nof mechanical motion  (which is said to be antiadiabatic when\n$\\Gamma\\ll \\hbar\\omega$) and the physical picture of ``magnetic\nshuttling\" developed in Ref.~\\onlinecite{kul} (see also\nRef.~\\onlinecite{parafilo}) for the adiabatic regime does not hold\nhere. As in Ref.~\\onlinecite{ilin} we will solve the problem using\nequations of motion for the reduced density operator. In the limit\n$eV\\gg\\Gamma,\\, \\hbar\\omega,\\, \\mu H,\\, k_BT$ ($T$ is the\ntemperature, $V$ is the bias voltage) the density operator can be\nfactorized into the product of a QD density operator and an\nequilibrium density matrix for the leads. In the Coulomb blockade\nregime, $eV,\\,T\\ll U$, the matrix elements of the QD density\noperator ($\\rho_0=\\langle 0\\vert\\hat \\rho_d\\vert0 \\rangle$,\n$\\rho_\\sigma=\\langle \\sigma\\vert \\hat \\rho_d\\vert\\sigma\\rangle$,\n$\\rho_{\\uparrow\\downarrow}=\n\\langle\\uparrow\\vert\\hat\\rho_d\\vert\\downarrow\\rangle$) in the\nHilbert space of a singly occupied dot level (they are still\noperators in the Hilbert space of a harmonic oscillator) are\ndetermined by the following set of equations \\cite{mis},\n\\begin{eqnarray}\\label{baseq}\n&&\\frac{\\partial \\rho_0}{\\partial t}=-i\\left[ H_v,\n\\rho_0\\right]-\\frac{1}{2}\\left\\{\\Gamma_S^\\uparrow(\\hat\nx)+\\Gamma_S^\\downarrow(\\hat x),\n\\rho_0\\right\\}\\nonumber\\\\\n&&\\hspace{0.7cm}\\label{baseq1} +\\sqrt{\\Gamma_D^\\uparrow(\\hat\nx)}\\rho_\\uparrow\\sqrt{\\Gamma_D^\\uparrow(\\hat x)}\n+\\sqrt{\\Gamma_D^\\downarrow(\\hat x)}\n\\rho_\\downarrow\\sqrt{\\Gamma_D^\\downarrow(\\hat x)},\\\\\n&&\\frac{\\partial \\rho_\\uparrow}{\\partial t}=-i\\left[\nH_v,\\rho_\\uparrow\\right]+\\frac{{i}}{2}\\left[J(\\hat x),\n\\rho_\\uparrow\\right]+ \\frac{{i} h}{2}\n\\left(\\rho_{\\uparrow\\downarrow}\n-\\rho^{\\dag}_{\\uparrow\\downarrow}\\right)\n\\nonumber\\\\&&\\hspace{0.7cm}+\\sqrt{\\Gamma_S^\\uparrow(\\hat x)}\\rho_0\n\\sqrt{\\Gamma_S^\\uparrow(\\hat x)}-\n\\frac{1}{2}\\left\\{\\Gamma_D^\\uparrow\n(\\hat x), \\rho_\\uparrow\\right\\},\\\\\n&& \\frac{\\partial \\rho_\\downarrow}{\\partial t}=-i\\left[\nH_v,\\rho_\\downarrow\\right]-\n\\frac{{i}}{2}\\left[J(x),\\rho_\\downarrow\\right]-\\frac{{i}\nh}{2}\\left(\\rho_{\\uparrow\\downarrow}-\\rho^\\dag_{\\uparrow\\downarrow}\n\\right)\\nonumber\\\\\n&&\\hspace{0.7cm}+\\sqrt{\\Gamma_S^\\downarrow(\\hat\nx)}\\rho_0\\sqrt{\\Gamma_S^\\downarrow(\\hat x)} -\\frac{1}{2}\n\\left\\{\\Gamma_D^\\downarrow(\\hat x),\\rho_\\downarrow\\right\\},\\\\\n&& \\frac{\\partial \\rho_{\\uparrow\\downarrow}}{\\partial t}= -i\\left[\nH_v, \\rho_{\\uparrow\\downarrow}\\right]+\\frac{{i}}{2}\\left\\{J(\\hat\nx),\\rho_{\\uparrow \\downarrow}\\right\\}+\\frac{{i} h}{2}\n\\left(\\rho_\\uparrow-\\rho_\\downarrow\\right)\\nonumber\\\\&&\\hspace{0.7cm}\n-\\frac{\\rho_{\\uparrow\\downarrow} }{2}\\Gamma_D^\\downarrow(\\hat x)\n-\\Gamma_D^\\uparrow(\\hat x)\\frac{\\rho_{\\uparrow\\downarrow}}{2}.\n\\label{baseq2}\n\\end{eqnarray}\nIn order to solve Eqs.~(\\ref{baseq1}) -- (\\ref{baseq2})\nit is convenient to introduce dimensionless variables\nfor time ($t\\omega\\rightarrow t$), dot displacement ($\\hat\nx\/x_0\\rightarrow \\hat x$, where $x_0=\\sqrt{\\hbar\/m\\omega}$ is the\nzero-point oscillation amplitude), momentum ($\\hat p\nx_0\/\\hbar\\rightarrow \\hat p$) and for various characteristic\nenergies ($\\hbar\\omega\\rightarrow 1$, $g\\mu\nH\/\\hbar\\omega\\rightarrow h$, $\\Gamma_j^\\sigma(\\hat\nx)\/\\hbar\\omega\\rightarrow\\Gamma_j^\\sigma (\\hat x)$, where\n$\\Gamma_j^\\sigma(\\hat x)=2\\pi\\nu\\vert t_{j,\\sigma}(\\hat x)\\vert^2$\nis the level width, $\\nu$ is the density of states).\n\n\\section{3. Influence of Spin Polarization on the Magnetic Shuttle\nInstability} We are interested in the conditions under which a\nmagnetic shuttle instability occurs. Therefore it is sufficient to\nexpand tunneling amplitudes and exchange energies up to linear\nterms in the coordinate operator $\\hat x$,\n\\begin{equation}\\label{7}\nJ(\\hat x)=J_0-\\alpha \\hat x,\\,\\Gamma_{S\/D}^\\sigma(\\hat\nx)=\\Gamma_{S\/D}^\\sigma\\left(1\\mp\\frac{\\hat x}{l}\\right),\n\\end{equation}\nwhere $l$ is the characteristic electron tunneling length (of the\norder of 1~{\\AA}).\n\nNote that the zero-point fluctuation amplitude of a fullerene molecule \nin the vicinity of the Lennard-Jones potential minimum is\nmuch smaller than the electron tunneling length, $l\\gg 1$. Since the\nspatial scale of the exchange interaction is also determined by the\ntunneling length $l$, we can treat the mechanical coordinate $\\hat x$ as\na classical variable. The equation of motion for the classical\ncoordinate $x_c$ takes the form\n\\begin{equation}\\label{8}\n\\frac{d^2 x_c}{d\nt^2}+x_c=-\\frac{\\alpha}{2}\\text{Tr}\\left(\\rho_\\uparrow-\n\\rho_\\downarrow\\right)\\,.\n\\end{equation}\nThe r.h.s. of Eq. (\\ref{8}) can readily be found from the set of\nlinear equations (\\ref{baseq1}) -- (\\ref{baseq2}) when the operator $\\hat\nx$ is replaced by $x_c$.\n\nTo simplify analytical calculations we consider a symmetric junction for which\n$J_0=0$, $\\Gamma_S^\\uparrow=\\Gamma_D^\\downarrow=\\Gamma$, and\n$\\Gamma_S^\\downarrow=\\Gamma_D^\\uparrow=\\gamma$. To first order in\nthe small dimensionless parameter\n$\\alpha\/l\\sim\\left(J\/\\hbar\\omega\\right) \\left(x_0\/l\\right)^2\\ll 1$\nEq.~(\\ref{8}) is reduced to a linear equation. For the time\ndependent coordinate displacement $x_1=x_1(t)$ this equation reads\n\\begin{equation}\\label{9}\n\\frac{d^2\nx_1}{dt^2}+x_1=-\\frac{\\alpha}{2l}\\int_{-\\infty}^tdt'\\langle e_0\\vert\ne^{\\hat A(t-t')}\\vert e\\rangle x_1(t').\n\\end{equation}\nHere the matrix $\\hat A$ takes the form\n\\begin{equation}\\label{90}\n\\hat A=\\frac{1}{2}\\left(\n\\begin{array}{ccc}\n-\\Gamma_1&-\\Gamma_2& -2h \\\\\n\\Gamma_2 &-3\\Gamma_1 & 0 \\\\\n2h & 0& -\\Gamma_1\n\\end{array}\\right),\n\\end{equation}\n$\\Gamma_{1,2}=\\Gamma\\pm\\gamma$, $\\vert e_0\\rangle=(1,0,0)^T$, and the\nvector $\\vert e\\rangle=(e_1,e_2,e_3)^T$ has the components\n\\begin{eqnarray}\n&&e_1= -\\frac{2\\Gamma_2}{\\Delta}(\\Gamma_1^2-\\Gamma_2^2),\\,\ne_2=-\\frac{2\\Gamma_1}\n{\\Delta}\\left(\\Gamma_1^2-\\Gamma_2^2+4h^2\\right) \\nonumber\\\\\n&&e_3=-\\frac{4h\\Gamma_1\\Gamma_2}{\\Delta},\\,\n\\Delta=3\\Gamma_1^2+\\Gamma_2^2+12h^2.\n\\end{eqnarray}\n\nA shuttle instability occurs when the oscillatory solution of\nEq.~(\\ref{9}), $x_1(t)\\sim\\text{exp} ({i}\\Omega t)$ becomes\nunstable, that is when $\\text{Im}\\,\\Omega<0$. The smallness of the\nr.h.s. of Eq.~(\\ref{9}) allows one to obtain the imaginary part of\n$\\Omega$ by perturbation theory. The values of the parameters\n$\\Gamma_1, \\Gamma_2$ and $h$ for which $\\text{Im}\\,\\Omega<0$,\ncorresponding to the\nshuttle regime of electron transport, satisfy the inequality\n\\begin{equation}\\label{10}\nh^4+A_1h^2+A_2<0,\n\\end{equation}\nwhere\n\\begin{eqnarray}\nA_1&=&-3\\left(\\Gamma_1^2+1\\right)-\\frac{\\Gamma_1^2\n-\\Gamma_2^2}{8\\Gamma_1^2}\\left(5\\Gamma_1^2+4\\right),\\\\\nA_2&=&\\frac{\\left(\\Gamma_1^2-\\Gamma_2^2\\right)\n\\left(\\Gamma_1^2+4\\right)}{8\\Gamma_1^2}\\left(\\Gamma_1^2+1+\n\\frac{\\Gamma_1^2-\\Gamma_2^2}{4}\\right).\n\\end{eqnarray}\nNote that in the absence of an external magnetic field the\nnanoelectromechanical coupling results in additional damping of the\nmechanical subsystem (since the coefficient $A_2>0$).\n\nIt is evident from Eq.~(\\ref{10}), which is biquadratic in the magnetic field ($h$),\nthat a shuttle instability occurs in a finite interval\nof magnetic fields, $h_{\\text{min}}<h<h_{\\text{max}}$. Now we\nintroduce the degree of spin polarization as\n\\begin{equation}\\label{100}\n\\eta=\\frac{\\vert N_\\uparrow-N_\\downarrow\\vert}{N_\\uparrow+\nN_\\downarrow}\\simeq\\frac{\\Gamma-\\gamma}{\\Gamma+\\gamma}\n\\end{equation}\nwhere $N_\\sigma$ is the number of particles with spin projection\n$\\sigma$. The last relation in Eq.~(\\ref{100}) comes from the\ndefinition of tunneling rates and our assumption that tunneling\namplitudes do not depend on spin projections. We analyze\nEq.~(\\ref{10}) in two limiting cases: (i) $\\Gamma\\gg 1$, $1-\\eta\\ll 1$,\nand (ii) $\\Gamma\\ll 1$, $1-\\eta\\ll 1$.\n\nNote that in the limit of weak polarization, $\\eta\\rightarrow 0$,\nmagnetic forces are small, $J\\rightarrow 0$, and electron shuttling\nis supported by Coulomb forces. This case was considered in\nRef.~\\onlinecite{ilin}, where  it was shown that in the limit of\nweak polarization a magnetic field ceases to influence single electron\nshuttling.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{Fig1_Ilinskaya.eps}\n\\caption {Upper critical magnetic field (upper set of curves) and\nlower critical field (lower set of curves) for the shuttle\ntransport regime plotted as a function of the normalized tunneling\nrate $\\Gamma\/\\hbar\\omega$ of majority spin electrons for different\nvalues of spin polarization $\\eta$. The solid curves were plotted\nfor $\\eta =1$ [100\\% spin polarization, here the lower curve\ncoincides with the $x$-axis], the dashed curves were plotted for\n$\\eta = 0.8$, and the short-dashed curves for $\\eta = 0.3$. The\nshuttle regime corresponds to the area between the lower and upper\ncritical fields (dark region for the case of $\\eta = 0.8$), while\noutside this area one is in the vibronic regime.}\n\\end{figure}\n\nIn the adiabatic limit, (i), one finds (reverting to using\nparameters with dimensions) that the\nshuttle instability region is defined by the double inequality\n\\begin{equation}\\label{11}\n\\frac{\\Gamma}{2}\\sqrt{\\frac{1-\\eta}{3}}<g\\mu H<\\sqrt{3}\\,\\Gamma.\n\\end{equation}\nIn the antiadiabatic limit, (ii), the critical magnetic fields are\ndetermined by the vibron energy $\\hbar\\omega$ rather than the\ntunneling rate $\\Gamma$, so that\n\\begin{equation}\\label{12}\n\\hbar\\omega\\sqrt{\\frac{1-\\eta}{3}}<g\\mu H<\\sqrt{3}\\,\\hbar\\omega.\n\\end{equation}\n\nFor the general case that part of the $\\Gamma, H$ parameter space\nwhich corresponds to a magnetic shuttle instability is shown in\nFig.~1 for different degrees of spin polarization.\nFor a given (normalized) tunneling coupling $\\Gamma\/\\hbar\\omega$ \nthe range of magnetic fields for which an instability occurs is shifted \ntowards higher fields as the spin polarization decreases.\nThe upper critical\nfield for high spin polarizations, $\\eta\\rightarrow 1$, depends\nlinearly on $\\eta$,\n\\begin{equation}\\label{13}\ng\\mu\nH_{\\text{max}}\\sim\\sqrt{3\\left(\\Gamma^2+(\\hbar\\omega)^2\\right)}\n\\left[1+K(1-\\eta)\\right]\\,,\n\\end{equation}\nwhere $K\\sim 1$ is a positive constant.\n\nThe dependence of the lower critical magnetic field on $\\eta$ is\nweaker. In the antiadiabatic regime one finds that\n\\begin{equation}\\label{14}\ng\\mu H_{\\text{min}}\\sim\n\\hbar\\omega\\sqrt{1-\\eta},\\quad \\Gamma\\ll\\hbar\\omega,\n\\end{equation}\nand\nhence $H_{\\text{min}}$ rapidly saturates to a constant value of\norder $\\hbar\\omega\/g\\mu$ with decreasing spin polarization (see\nFig.~2, short-dashed curve).\nIn the adiabatic regime,\n$\\Gamma\\gg\\hbar\\omega$, the\nlower critical field\ndecreases linearly with increasing spin polarization, except in\nthe close vicinity of complete spin polarization, where\n$H_{\\text{min}} \\propto \\sqrt{1-\\eta}$ (compare the solid curve in\nFig.~2).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{Fig2_Ilinskaya.eps}\n\\caption {Plots of the lower critical magnetic field\n$H_{\\text{min}}$, which defines the border between the vibronic\nregime ($H<H_{\\text{min}}$) and the shuttle transport regime\n($H>H_{\\text{min}}$; compare Fig.~1) vs. spin polarization $\\eta$\nfor different values of the normalized tunneling rate\n$\\Gamma\/\\hbar\\omega$ of majority spin electrons (solid curve:\n$\\Gamma\/\\hbar\\omega=10$;  dashed curve: $\\Gamma\/\\hbar\\omega=3$;\nshort-dashed curve: $\\Gamma\/\\hbar\\omega=0.1$).}\n\\end{figure}\n\nThe appearance of an upper and a lower critical magnetic field has a\nsimple physical explanation. When $\\mu H$ is the largest energy\nscale in our problem, $\\mu H\\gg \\Gamma,\\,\\hbar\\omega$, the fast\nprecession of the electron spin of the dot in a perpendicular external magnetic field\nnullifies the average spin and the magnetic shuttle instability\ndisappears. To estimate the upper field one may compare the\ncharacteristic spin precession frequency, $\\mu\nH_{\\text{max}}\/\\hbar$, with the electron tunneling rate,\n$\\Gamma\/\\hbar$, or the frequency of vibrations $\\omega$. That is\n$\\mu H_{\\text{max}}\\sim\\text{max}\\left(\\Gamma, \\hbar\\omega\\right)$.\nThe lower critical field can be readily estimated for\na high degree of spin polarization, $1-\\eta\\ll 1$.\nIn this case we have to compare the average time between spin\nflips, $\\tau_f$, induced by a constant magnetic field $H$ in the\npresence of an electron tunneling coupling $\\Gamma$ with the\ncharacteristic life-time of minority spin electrons on the dot,\n$\\sim \\hbar\/\\gamma$. The spin-flip rate $\\nu_f$ in weak magnetic\nfields $H$ can be estimated by perturbation theory\nwith the result that $\\hbar\\nu_f\\sim(\\mu H)^2\/\\text{max}(\\Gamma,\n\\hbar\\omega)$. Therefore the lower magnetic field is strongly\nsensitive to spin polarization,\n\\begin{equation}\\label{15}\n\\mu H_{\\text{min}}\\sim\n\\sqrt{\\Gamma\\gamma}\\,\\text{max}(\\Gamma,\\hbar\\omega)\\sim\n\\sqrt{1-\\eta}\\,\\text{max}(\\Gamma,\\hbar\\omega),\n\\end{equation}\nand disappears for 100\\% spin-polarized electrons ($\\eta=1)$.\n\nNext we estimate the maximum rate of (exponential) increase,\n$r_m=-\\text{Im}\\{\\Omega (H_{\\text{opt}})\\}$, of the QD oscillation\namplitude in the shuttle regime.\nIn the adiabatic limit, $\\Gamma\\gg \\hbar\\omega$,\none finds that $g\\mu H_{\\text{opt}}\\simeq 0.4\\, \\Gamma$ and that\n\\begin{equation}\\label{16}\nr_m\\simeq C\\frac{\\omega J}{\\Gamma}\\left(\\frac{x_0}{l}\\right)^2,\n\\end{equation}\nwhere $C\\sim 0.1$ is a small numerical factor. In the case\n$\\Gamma\\ll \\hbar\\omega$, which we are interested in here, the\nmaximum rate is realized when $g\\mu H_{\\text{opt}}\\simeq\n\\hbar\\omega$, corresponding to\n\\begin{equation}\\label{17}\nr_m\\simeq\n\\frac{\\Gamma}{\\hbar}\\frac{J}{\\hbar\\omega}\\left(\\frac{x_0}{l}\\right)^2\\,,\n\\end{equation}\nwhere we omit a numerical factor of the order of one.\n\nIn the presence of dissipation in the mechanical subsystem, which\ncan be described by adding a phenomenological friction term\n$\\gamma_d \\dot{x}_c(t)$ to the equation of motion (\\ref{8})\n($\\gamma_d=\\omega\/Q$, where $Q$ is the quality factor), the\nshuttling regime appears when $r_m>\\omega\/Q$. Therefore electron\nshuttling in a $C_{60}$-based molecular transistor with magnetic\nelectrodes could be realized if the quality factor $Q$ of the\nmechanical resonator obeys the inequality\n\\begin{equation}\n\\label{18}\nQ>Q_{\\text{opt}}=\\frac{\\left(\\hbar\\omega\\right)^2}{J\\Gamma}\n\\left(\\frac{l}{x_0}\\right)^2.\n\\end{equation}\n\nFor the experimental setup in Ref.~\\onlinecite{park}, where\nfullerene vibrations were observed, the factor\n$\\left(l\/x_0\\right)^2\\simeq 10^3$ and\n$\\Gamma\\ll\\hbar\\omega\\sim$~5~meV (one can estimate\n$\\Gamma\\sim$~0.1 -- 0.5~meV from the maximal current measured in\nRef.~\\onlinecite{park}). In the $C_{60}$-based transistor with\nmagnetic (Ni) leads $J\\sim\\Gamma\\sim$~10~meV (see\nRef.~\\onlinecite{pasupathy}). From Eq.~(\\ref{18}) one can estimate\nthat the required quality factor is $Q\\geq 10^3-10^4$. However the\noptimal external magnetic field in this case,\n$H_{\\text{opt}}\\simeq$~50~T, is too high. Instead, we therefore\nestimate $Q$ for magnetic fields in the vicinity of the lower\ncritical magnetic field $H\\geq H_{\\text{min}}$ where magnetic\nfields for a very high degree of electron spin polarization ($\\sim\n99\\%$) could be of the order of a few tesla. In this case\n($\\hbar\\omega\\gg \\Gamma$, $1-\\eta\\ll 1$)\n\\begin{equation}\\label{19}\nr(\\eta)\\simeq \\omega\\frac{J \\Gamma\n(1-\\eta)}{\\Gamma^2+4(1-\\eta)(\\hbar\\omega)^2}\\left(\\frac{l}{x_0}\\right)^2.\n\\end{equation}\nAssuming that $\\Gamma\\simeq\\sqrt{1-\\eta}\\,\\hbar\\omega$ we find\n$Q\\sim Q_{\\text{opt}}\/(1-\\eta)$.\n\n\\section{4. Conclusions}\nIn summary we have considered the feasibility of observing magnetically\ndriven single-electron shuttling under\nrealistic conditions\ncorresponding to an already experimentally realized $C_{60}$-based\nsingle-molecule transistor with magnetic leads. The main\nrequirement for magnetic shuttling is the presence of an external\nmagnetic field that induces electron\nspin flips. We have shown that the optimal magnetic field, defined\nas the field that maximizes the rate of increase of the shuttling\namplitude, is determined by the vibration frequency $\\omega$. For\nfullerene-based single-electron transistors this frequency could\nbe in the THz region \\cite{park}\nwith\ncorresponding\nmagnetic fields\nin the region of several tenths of teslas. For magnetic electrodes\nwith a very high degree of spin polarization one needs less strong\n(by an order of magnitude) magnetic fields. However, the quality\nfactor of the corresponding mechanical resonator has to be\nexceptionally high, $Q\\geq 10^5$.\n\n{\\bf Acknowledgements:} Financial support from the Leading Foreign\nResearch Institutes Recruitment Program (2009-00514) of NRF,\nKorea, and the Swedish Research Council (VR) is gratefully\nacknowledged. OI, IK and SK acknowledge financial support from\nNational Academy of Science of Ukraine, Grant No 4\/15--N. IK and\nSK thank the Department of Physics at the University of Gothenburg\nand the Department of Physics and Astronomy at Seoul National\nUniversity for their hospitality.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nComplex systems of a dispersed phase in a solid matrix can behave very differently from one of their components taken alone. Their broad range of properties explains that examples of dispersions such as composites~\\cite{matthews1999composite} or porous media~\\cite{coussy2011mechanics} are widespread in nature and in the industry. In all dispersions, interfacial forces can appear at the boundary between the dispersed phase and the continuous matrix. A coupling of surface tension forces to the bulk elasticity of a solid has been evidenced in soft systems like biological tissues~\\cite{clements1961pulmonary}, or through the deformation of soft substrates like polymers at the contact line with a drop resting on the solid~\\cite{pericet2008effect}. Capillary forces also affect the overall mechanical properties of nanoporous media~\\cite{duan2005size}. For larger pores, because of the hardness of the matrix in usual porous media, the influence of interfacial effects on the overall properties of the saturated material is negligible~\\cite{dormieux2006microporomechanics}. Dispersions in softer materials could allow for observable coupling of interfacial forces to the bulk elasticity of the solid at larger scales than the nanometer. Many dense suspensions~\\cite{coussot2005rheometry} of geological interest, like muds, or with industrial applications, like fresh concrete or emulsions, behave as soft elastic solids below a critical level of stress~\\cite{coussot2012rheophysique}. To study the role of surface tension forces in soft elastic materials, we investigate the elastic behaviour of dispersions of bubbles in concentrated emulsions. Those aerated emulsions, which have applications in the food~\\cite{vanAken2001333} and cosmetic~\\cite{balzer1991alkylpolyglucosides} industry, have been the subject of stability and rheology studies~\\cite{C1SM06537H, PhysRevLett.104.128301, kogan2013mixtures}. However, their overall elastic properties have not yet been studied in detail.\n\nIn dispersions of bubbles in a soft material, coupling between the elasticity of the matrix and capillary effects is expected to occur through bubble deformation. The elastic deformation of the matrix tends to deform the bubbles and surface tension forces will thus act to minimize the area of the bubble by maintaining a spherical shape. The limit case of negligible surface tension forces is a soft porous medium. Theoretical work shows that adding holes in a solid softens it~\\cite{dormieux2006microporomechanics}. In the limit case of predominant surface tension forces compared to the matrix elasticity, a bubble should no longer be deformable and should behave as a rigid inclusion with no shear stiffness. Experimental and theoretical work have shown that rigid beads in a soft solid strengthen the solid~\\cite{mahaut2008yield}. The case of rigid bubbles is similar except for the boundary condition, changed from no-slip for beads to full-slip for bubbles. Theoretical models in the dilute limit predict a strengthening of the dispersion when adding rigid bubbles~\\cite{dormieux2006microporomechanics}. Between those two limit cases, more work is needed to investigate the elastic response of the soft aerated solid. In this work, we restrain to the range of gas volume fraction $\\phi<50\\%$, so that we do not consider foams of those materials, in which the bubbles are deformed by geometrical constraints. We design model systems and appropriate experimental methods that allow us to measure the shear modulus of dispersions of monodisperse bubbles embedded in a medium of chosen elasticity. We compare our experimental results to estimates of the elastic modulus through a micro-mechanical approach. \n\\section*{Experimental aspects}\nThe dispersion matrices we choose are concentrated oil in water emulsions of shear moduli ranging from 100 to 1000Pa. Concentrated emulsions behave as soft elastic solids for stresses well below their yield stress~\\cite{mason1995elasticity}. In the experimental systems, unless otherwise indicated, the radius of the droplets is around 1 to 2$\\mathrm{\\mu m}$ (the poydispersity is around 20\\%), which, at the considered gas volume fractions, should ensure that there is scale separation between the drops and the bubbles, and consequently validate the use of the emulsion as an elastic continuous medium embedding the bubbles~\\cite{goyon2008spatial}. In all the systems, the yield stress of the emulsion is high enough to ensure that no bubble rise occurs at rest or during measurements~\\cite{Dubash2007123}. Most dispersions are prepared by gently mixing the emulsion with a separately produced monodisperse foam. The foams are obtained by blowing nitrogen plus a small amount of perfluorohexane ($\\mathrm{C_{6}F_{14}}$) through a porous glass frit or through needles: we are able to produce nearly monodisperse foams with average bubble radii $R_{b}$ ranging from 40$\\mathrm{\\mu m}$ to 800$\\mathrm{\\mu m}$. Coarsening is strongly reduced by the presence of $\\mathrm{C_{6}F_{14}}$~\\cite{gandolfo1997interbubble}, meaning that the bubble size is stable during measurements. The continuous phase of the foam is the same as the one in the emulsion, ensuring that mixing is easy and does not induce any chemical effect in the dispersions. The mixing with the foam adds a small amount of continuous phase to the emulsion. To ensure that for a series of experiments at different $\\phi$ in a given emulsion, the elastic modulus of the matrix in the dispersions remains the same, we add controlled amounts of pure continuous phase in order to reach the same the oil volume fraction in the emulsion~\\cite{PhysRevLett.104.128301, kogan2013mixtures}. An example of a dispersion of bubbles in an emulsion is shown on figure~\\ref{fig:photo}. The composition of all the tested emulsions is indicated in table~\\ref{tab:recap_systs}, and illustrates the variety of chemical compositions, surface tensions and elastic properties of the matrix that were used to perform the study.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.6]{graph1_leg1_2.png}\n\\caption{Microphotograph of a dispersion of monodisperse bubbles (R=200$\\mu$m) in emulsion (2). The emulsion is transparent, allowing for the visualisation of in-depth bubbles that thus do not have the same apparent radius. Inset: close-up of droplets of emulsion (2) at the interface with a bubble.\\label{fig:photo}}\n\\end{figure}\n\nThe shear modulus of the dispersions is measured on a control stress rheometer by imposing small amplitude oscillations at a frequency of typically 1Hz. The oscillatory stress is chosen to be well below the yield stress of the systems, so that the oscillations are performed in the linear elastic regime of each material. At this frequency, the loss modulus of the systems is negligible. The geometry used to perform the rheometrical measurements is chosen according to the bubble size : for $R_{b}\\le 50\\mathrm{\\mu m}$, the material is sheared between parallel  plates (radius $R$=25mm, gap $h$=2.5mm). The planes are serrated to prevent slippage of the dispersion~\\cite{coussot2005rheometry}. Dispersions containing bigger bubbles require a larger thickness of sheared material and are studied in Couette-like devices : for $50\\mathrm{\\mu m}< R_{b}< 800\\mathrm{\\mu m}$, we use a vane in cup (exceptionally a serrated bob in cup) geometry (inner radius $R_i$=12.5mm, outer radius $R_o$=18mm), and for $R_{b}\\ge 800\\mathrm{\\mu m}$, we use vane in cup geometries (either $R_i$=12.5mm and $R_o$=25mm or $R_i$=22.5mm and $R_o$=45mm). \n\\begin{center}\n\\begin{table*}\n{\\renewcommand{\\arraystretch}{1.5}\n\\renewcommand{\\tabcolsep}{0.2cm}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n& \\textbf{oil - vol. fraction} & \\textbf{continuous phase} & \\textbf{$\\mathbf{G'(0)}$ (Pa)} & \\textbf{$\\mathbf{\\gamma}$ (mN.$\\mathbf{m^{-1}}$)}\\\\\n\\hline\nemulsion (1a) & silicon (V20) - 75\\% & Forafac$\\textregistered$ ($\\mathrm{{DuPont}^{TM}}$) 4\\% w. in water & 230 &15.5 $\\pm$ 0.1 \\\\\n\\hline\nemulsion (1b) & silicon (V20) - 73\\% & Forafac$\\textregistered$ ($\\mathrm{{DuPont}^{TM}}$) 4\\% w. in water & 163 &15.5 $\\pm$ 0.1 \\\\\n\\hline\nemulsion (2) & silicon (V350) - 79\\% & TTAB 3\\% w. in water\/glycerol 50\/50 w\/w & 650 & 35.5 $\\pm$ 0.1 \\\\\n\\hline\nemulsion (3) & dodecane - 73\\% & SDS 2.7\\% w. in water & 285 & 36 $\\pm$ 1 \\\\\n\\hline\nemulsion (4) & silicon (V350) - 70\\% & TTAB 3\\% w. in water\/glycerol 36\/64 w\/w & 799 & 35 $\\pm$ 1 \\\\\n\\hline\n\\end{tabular}}\n\\caption{Synthetic description of all the emulsions used as matrices in the bubble dispersions: nature and volume fraction of the oil dispersed phase, composition of the aqueous continuous phase (including the surfactant) and relevant physical constants for the determination of the capillary number: elastic modulus of the matrix, and surface tension between air and the continuous phase. The composition given is the one of the matrix actually embedding the bubbles.\\label{tab:recap_systs}}\n\\end{table*}\n\\end{center}\n\\section*{Results}\nWe start by studying the influence of the bubble radius $R_b$. In this aim, we prepare dispersions of bubbles in emulsion (3) (see table~\\ref{tab:recap_systs} for details). In a first series of experiments, we add bubbles of $R_b=(50\\pm10)\\mathrm{\\mu m}$ ($10\\mathrm{\\mu m}$ being the width of the volume-weighed bubble radius distribution) at various gas volume fractions $\\phi$ in the emulsion. Those bubbles are slightly more polydisperse than is generally used for this study, because of the foam production technique. The shear modulus $G'(\\phi)$ of the dispersions is measured to be slightly decreasing with $\\phi$. This result is reported in dimensionless quantities $\\hat{G}(\\phi)=G'(\\phi)\/G'(0)$ as a function of $\\phi$ on figure~\\ref{fig:1}. We then prepare dispersions of larger bubbles in emulsion (3): a series with $R_b=(143\\pm17)\\mathrm{\\mu m}$ and another one with $R_b=(800\\pm40)\\mathrm{\\mu m}$. The results for $\\hat{G}(\\phi)$ are also reported on figure~\\ref{fig:1}. The measurements show that the larger the bubbles, the softer the dispersion. This result can be understood as a manifestation of a simple physical effect, already evidenced in~\\cite{kogan2013mixtures} (see also~\\cite{rust2002effects} and~\\cite{llewellin2002rheology} for the effect of bubble deformation on the viscosity of bubbly Newtonian fluids), that the interfacial energy to volume ratio is lower in larger bubbles, resulting in least bubble resistance to deformation. \n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.29]{graph1.pdf}\n\\caption{Dimensionless elastic modulus $\\hat{G}$ as a function of the gas volume fraction $\\phi$ for dispersions with three different bubble radii $R_b$ in emulsion (3) [see legend]. The full lines are the computed $\\hat{G}_{homog}(\\phi)$ for $Ca=0.23$ (dark blue), $Ca=0.57$ (green) and $Ca=3.2$ (pink); experimentally measured $Ca$: $0.23\\pm0.05$, $0.57\\pm0.08$, $3.2\\pm0.4$. Inset: $\\hat{G}$ as a function of $\\phi$ for dispersions of $R_b\\approx 150\\mathrm{\\mu m}$ in \\textcolor{spinach}{$\\bullet$} emulsion (3) and \\textcolor{orange}{$\\blacklozenge$} emulsion (4). The full lines are the computed $\\hat{G}_{homog}(\\phi)$ for $Ca=0.57$ (green) and $Ca=1.65$ (orange); experimentally measured $Ca$: $0.57\\pm0.08$, $1.65\\pm0.15$.\\label{fig:1} }\n\\end{figure}\nWe now keep the bubble size constant, and vary the elastic modulus of the matrix: we prepare dispersions of $R_b=143\\mathrm{\\mu m}$ bubbles in emulsion (3) and of $R_b=(150\\pm10)\\mathrm{\\mu m}$ bubbles in emulsion (4) (see table~\\ref{tab:recap_systs}). In the two series of experiments, the bubble sizes are close and the surface tension is similar, but $G'(0)$ is almost three times higher in emulsion (4). $\\hat{G}(\\phi)$ is plotted for both systems on the inset in figure~\\ref{fig:1}. As observed on the previous suspensions, $\\hat{G}(\\phi)$ is a decreasing function of $\\phi$, and this decrease is all the stronger as $G'(0)$ is high.\nTo quantify the competition between the matrix elasticity and the bubble's resistance to deformation, we introduce a capillary number \n\\begin{equation}\nCa=\\frac{G'(0)}{2\\gamma\/R_b}\n\\label{eq:Ca}\n\\end{equation} \nwhich compares the shear modulus of the dispersion medium to the interfacial stress scale, the capillary pressure in the bubbles. This capillary number is equally affected by an increase in $R_b$ or a decrease in $G'(0)$. To quantify the relevance of $Ca$ on the overall elastic response of the dispersion at a given $\\phi$, we perform two series of experiments with close $R_b$, but very different capillary pressure because of very different surface tension, and we adjust the elastic modulus in one of the emulsions so that $Ca$ is similar in both systems. The two experimental systems are as follow: the first one is dispersions of $R_b=143\\mathrm{\\mu m}$ of radius bubbles in emulsion (3), which leads to $Ca=0.57\\pm0.08$, and the second one dispersions of (129$\\pm$10)$\\mathrm{\\mu m}$ of radius bubbles in emulsion (1b) for which $Ca=0.70\\pm0.08$. We observe that the measured values of $\\hat{G}(\\phi, Ca)$ are very close, as can be seen on figure~\\ref{fig:2}. The value of $Ca$ unequivocally determines the elastic behaviour of the dispersion at a given $\\phi$.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.30]{graph2.pdf}\n\\caption{Effect of a change in the surface tension: dimensionless elastic modulus $\\hat{G}$ as a function of $\\phi$ for dispersions of \\textcolor{spinach}{$\\bullet$} $R_b=143\\mathrm{\\mu m}$ bubbles in emulsion (3) and \\textcolor{red}{$\\bigstar$} $R_b=129\\mathrm{\\mu m}$ bubbles in emulsion (1). The surface tension is much lower in emulsion (1b), but $G'(0)$ has been chosen to get close values for $Ca$ in both systems. This experimentally measured $Ca$ are $0.57\\pm0.08$ and $0.70\\pm0.08$. The full line is the computed $\\hat{G}_{homog}(\\phi, Ca)$ at $Ca=0.63$, which is compatible with both systems, given the uncertainty on the value of $Ca$.\\label{fig:2}}\n\\end{figure}\n\nWe now investigate the limit value of $Ca\\to \\infty$, for which surface tension forces are negligible and the bubbles can be assimilated to holes in the matrix. This is the case in usual porous materials. $\\hat{G}(\\phi, Ca\\to\\infty)$ can then be computed in the dilute limit~\\cite{dormieux2006microporomechanics}: $\\hat{G}(\\phi, Ca\\to\\infty)=1-\\frac{5}{3}\\phi$. To compare this prediction to experimental data, we design a system in which surface tension effects are bound to be poor: we include the biggest bubbles of this study, of radius (1 $\\pm$ 0.1) mm, in emulsion (2), which has a high elastic modulus (see table~\\ref{tab:recap_systs}). Note that for this system the bubbles are injected directly in the emulsion in a tee-junction in a milli-fluidic device. As before, we measure the elastic modulus of the dispersion at various $\\phi$. The experimental data points for the dimensionless modulus $\\hat{G}(\\phi)$ are compared to the dilute limit for dispersions of holes in an elastic medium on figure~\\ref{fig:3}. We observe that $\\hat{G}$ is a decreasing function of $\\phi$. The exact value of $Ca$ in this system is $9.0\\pm1.2$. The theoretical dilute limit for spherical holes in an elastic medium is already a good estimate of $\\hat{G}(\\phi\\to 0, Ca)$at $Ca\\sim10$.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.30]{graph3.pdf}\n\\caption{Remarkable values of $Ca$: $\\mathbf{Ca\\to \\infty}$: $\\hat{G}(\\phi)$ for dispersions of \\textcolor{lgtblue}{$\\blacktriangledown$} $R_b$=1mm bubbles in emulsion (2). The full line is the computed $\\hat{G}_{homog}(\\phi, Ca)$ at $Ca=9.0$ (light blue); experimentally measured $Ca$: $9.0\\pm1.2$. $\\mathbf{Ca\\approx 0.25}$: dimensionless elastic modulus $\\hat{G}$ for dispersions of \\textcolor{darkblue}{$\\bullet$} $R_b$=50$\\mu$m bubbles in emulsion (3) and \\textcolor{gray}{$\\bigstar$} $R_b$=41$\\mu$m in emulsion (1a). The full lines are the computed $\\hat{G}_{homog}(\\phi, Ca)$ at $Ca=0.23$ (dark blue) and $Ca=0.30$ (grey); experimentally measured $Ca$: $0.23\\pm0.05$, $0.30\\pm0.05$. The dashed lines are the dilute limits for rigid (top) and fully deformable (bottom) spheres, with a full slip boundary condition.\\label{fig:3} }\n\\end{figure}\n\nThe limit case of $Ca\\to 0$ also leads to a simplification: the bubbles are stiff compared to the matrix and the dispersion is made of rigid spheres with a full slip boundary condition in an elastic medium. The theoretical dilute limit can be computed as $\\hat{G}(\\phi, Ca=0)=1+\\phi$~\\cite{dormieux2006microporomechanics} and is plotted on figure~\\ref{fig:3}. An experimental validation of this limit with our systems may be biased, because increasing the capillary pressure would mean reducing $R_b$, and we might no longer assume scale separation between the bubbles and the oil droplets. We thus choose not to investigate this limit.\nFrom $Ca\\to 0$ to $Ca\\to \\infty$, $\\hat{G}(\\phi, Ca)$ turns from an increasing to a decreasing function of $\\phi$. Between these two extreme values, we have observed on the dispersion of the smallest bubbles in emulsion (3), already discussed above on figure~\\ref{fig:1}, that $G'(\\phi)$ has little variation with $\\phi$ and is comparable to $G'(0)$. The capillary number in this system is $Ca=0.23\\pm0.05$. To further check the peculiarity of this value of $Ca$, we prepare another dispersion of small bubbles $R_b$=(41$\\pm$5)$\\mathrm{\\mu m}$ in emulsion (1a) (see table~\\ref{tab:recap_systs}), with a close capillary number: $Ca=0.30\\pm0.05$. $\\hat{G}(\\phi, Ca)$ for both dispersions of small bubbles is plotted on figure~\\ref{fig:3}. We observe that in both systems, $\\hat{G}(\\phi, Ca)$ exhibit little dependence on the gas volume fraction, and is of order 1. The non-perturbative effect of bubble addition in the matrix can be seen as an experimental validation of previous micro-mechanical calculations~\\cite{palierne1991rheologica, doi:10.1061\/9780784412992.224} which have shown that a spherical bubble of radius $R_b$ and surface tension $\\gamma$ in an elastic medium can be described as an equivalent elastic sphere of radius $R_b$ and no surface tension. Indeed, the deformation of the bubble under a strain $\\epsilon$ leads to an increase in the bubble area that is proportional to $\\epsilon^2$. The stored interfacial energy scales as $\\gamma \\epsilon^2$, which is analogous to an elastic energy. If the equivalent elasticity of the sphere is equal to that of the matrix, the bubbles are non-perturbative and $\\hat{G}(\\phi)= 1$. The equivalent elasticity of a bubble in a matrix $G'(0)$ can be written as a function of $G'(0)$ and $Ca$~\\cite{palierne1991rheologica, doi:10.1061\/9780784412992.224}:\n\\begin{equation}\nG^{eq}=G'(0)\\frac{8}{3+20Ca}\n\\end{equation}\nwith $Ca$ defined in equation~\\ref{eq:Ca}. The expression of $G^{eq}$ shows that $Ca$ introduced above does not actually compare the equivalent elasticity of the bubble to that of the matrix. This explains why the overall elasticity of the dispersion is unperturbed by the presence of the bubbles for a somewhat unnatural value of $Ca$ around 0.2 to 0.3, which can be understood thanks to the computation of $G^{eq}$: $G^{eq}=G'(0)$ for $Ca=1\/4$. Relying on the equivalent elastic sphere model for a bubble, a micro-mechanical approach allows to compute the overall elastic properties of the dispersions at finite $Ca$. The overall elasticity of a composite material made of elastic spheres in a matrix of another elastic material in the semi-dilute limit can be computed as a function of $Ca$ and $\\phi$, in the framework of the Mori-Tanaka scheme~\\cite{doi:10.1061\/9780784412992.224, palierne1991rheologica}: \n\\begin{equation}\n\\hat{G}_{homog}(\\phi, Ca)=1-\\frac{\\phi(4Ca-1)}{1+\\frac{12}{5}Ca-\\frac{2}{5}\\phi(1-4Ca)}\n\\end{equation}\nNote that this expression is compatible with the previously discussed limits of $Ca \\to \\infty$, $Ca \\to 0$ and $Ca=1\/4$. Predictions of the model for $\\hat{G}(\\phi, Ca)$ at the experimentally measured $Ca$ are plotted in full coloured lines on figures~\\ref{fig:1} to~\\ref{fig:3}. A comparison of $\\hat{G}_{homog}(\\phi, Ca)$ to $\\hat{G}(\\phi, Ca)$ for all the systems we used, at all tested gas volume fractions is presented on figure~\\ref{fig:4}. Experimental measurements and computations are generally in good agreement all over the range of systems we investigated. \n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.3]{graph4.pdf}\n\\caption{Consistency between $G'$ and ${G'}_{homog}$ for all tested systems. $\\bigstar$: emulsion (1), $\\blacktriangledown$: emulsion (2), $\\bullet$: emulsion (3), $\\blacklozenge$: emulsion (4).\\label{fig:4} }\n\\end{figure}\n\nAs we have seen that the two dimensionless parameters $Ca$ and $\\phi$ are enough to understand and predict the elasticity of the dispersions, we now plot $\\hat{G}(\\phi, Ca)$ as a function of $Ca$, for 4 values of $\\phi$, on figure~\\ref{fig:5}. As can be noticed on the graphs~\\ref{fig:1} to~\\ref{fig:3}, the achieved values of $\\phi$ are different for all tested systems. To be able to plot $\\hat{G}(\\phi,Ca)$ at a given $\\phi$, we interpolate the experimental data at the exact values of $\\phi$ used for plotting on figure~\\ref{fig:5}. The full lines are computations of $\\hat{G}_{homog}(\\phi, Ca)$. As expected, $\\hat{G}(\\phi, Ca)$ is a decreasing function of $Ca$: higher values of $Ca$ correspond to more deformable bubbles that lower the overall elastic modulus of the dispersions. The non-perturbative effect of the bubbles for $Ca=1\/4$ is evidenced by the crossing of $\\hat{G}_{homog}(\\phi, Ca)$ at 1 for $Ca=0.25$, whatever the gas volume fraction. Below this value, the increase of $\\hat{G}_{homog}(\\phi, Ca)$ is consistent with previously discussed theoretical limits, but could not be investigated with our experimental systems. The series of data points at $Ca=0.23\\pm0.05$ does not fit in the increasing $\\hat{G}_{homog}(\\phi, Ca)$ regime, perhaps because of broader polydispersity: the uncertainty on the value of $Ca$ mainly arises from the width of the bubble radius distribution and the value of $Ca$ for the largest bubbles in the foam is for instance higher than 0.25. A model computing $\\hat{G}_{homog}(\\phi, Ca)$ as a function of the whole measured distribution of radii may better represent the experimental data.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.32]{graph5.pdf}\n\\caption{Dimensionless elastic modulus $\\hat{G}$ as a function of $Ca$ for four different values of the gas volume fraction. The dots are interpolated experimental data points, the full lines are $\\hat{G}_{homog}(\\phi, Ca)$. \\label{fig:5} }\n\\end{figure}\n\n\\section*{Conclusions}\nAs a conclusion, we have designed model systems in which precise control of the bubble stiffness and the matrix elasticity allows to experimentally determine the elastic modulus of dispersions of bubbles in a soft matrix. The results show that the addition of bubbles leads to a softening of the dispersion that is finely tuned by the capillary number. Those model systems enable us to compare our experimental results to estimates of the shear modulus through a micro-mechanical approach. Precise control of the capillary number provides experimental data validating the theoretical description of the bubble as an equivalent elastic sphere. The good agreement between theoretical and measured elastic moduli confirms the generality of the study, which demonstrates that $\\phi$ and $Ca$ entirely govern the overall response of the dispersions. More work remains to do in the limit of small capillary numbers ($Ca<1\/4$), for which the predicted regime of increasing $\\hat{G}(\\phi, Ca)$ could not be experimentally investigated. Emulsions may not be the most relevant matrices for that study and dedicated experiments on even softer media could be more appropriate. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nUltraluminous X-ray sources (ULXs) are off-nuclear X-ray sources in galaxies with an X-ray luminosity above the Eddington luminosity of a 10 \\msun~black hole, or $\\sim 10^{39}$ erg s$^{-1}$. Several scenarios have been proposed to explain their high luminosities. Geometrical \\citep{king01} or relativistic \\citep{kording02} beaming may allow for the observation of super-Eddington luminosities. In some sources there is evidence for a new state with truly super-Eddington accretion rates \\citep{gladstone09}. Recent investigations into the X-ray luminosity function (XLF) of ULXs suggest that the majority of ULXs are formed by the high-luminosity tail of X-ray binaries (XRBs) and contain stellar mass black holes (BHs) \\citep{swartz11, mineo12}. The best-fitting XLF exhibits a cut-off around $10^{40}$ erg s$^{-1}$, suggesting that this may be the effective upper limit for the luminosity of the most massive objects in the sample.\n\nHowever, \\citet{swartz11} argue that if they extrapolate their best-fitting XLF, based on a complete sample of ULXs within 14.5 Mpc, to larger distances, they can not explain the relatively large number of ULXs with luminosities above $10^{41}$ erg s$^{-1}$~that have been observed. Hence these ULXs may belong to a different class of objects. These sources would have to exceed the Eddington limit by more than a factor 100 if they contained stellar-mass black holes. They may be good candidates to host the predicted but thus far elusive intermediate-mass black holes (IMBHs). These IMBHs may form in the collapse of a dense stellar cluster \\citep{portegieszwart02}, the collapse of population III stars in the early universe \\citep{madau01} or the direct collapse of massive gas clouds \\citep{begelman06}. They may reside in globular clusters \\citep{maccarone07}, but conclusive evidence for their existence there has not yet been found. For a review on IMBHs and their formation mechanisms see \\citet{vandermarel04}. The best candidate for an IMBH to date is the extremely bright source HLX-1 in ESO 243-49, which reaches maximum X-ray luminosities of $\\sim 10^{42}$ erg s$^{-1}$ \\citep{farrell09}. The recent work by \\citet{sutton12} provides more evidence for extreme ULXs as IMBHs.\n\nMost ULX candidates are discovered by searching for off-nuclear X-ray point sources in galaxies (e.g. from the {\\it Chandra}~or XMM-{\\it Newton}~serendipitous source catalogs; see for example \\citealt{walton11b, liu11}). The ULX catalogs compiled in this way are contaminated with objects that also show up as off-nuclear, bright X-ray sources but are not accreting IMBHs or stellar-mass BHs. Background active galactic nuclei (AGN) and quasars are obvious examples, but some X-ray bright supernovae (most likely type IIn, \\citealt{immler03}) and active foreground stars may also contaminate the catalogs. One way to identify these contaminants if the ULX candidate has a bright optical counterpart is to take an optical spectrum. If emission or absorption lines are present the redshift to the source can be measured. In this way we can determine whether the source is associated with the galaxy or is a background or foreground object (compare e.g. \\citealt{gutierrez13}).\n\nIf the X-ray source is associated with the galaxy, optical spectra can give us additional information to classify the object. Some ULXs \\citep{pakull02, kaaret09} are surrounded by bubbles of ionized gas, which can act as calorimeters and as such tell us if the emission is strongly beamed or not. The intensity ratios of the emission lines from these regions provide information on the source of the ionizing radiation, e.g. whether they are shock ionized or X-ray photo-ionized (e.g. \\citealt{abolmasov07}).\n\nWe selected four high-luminosity (L$_X \\geq 10^{40}$ erg s$^{-1}$) ULX candidates from the catalog of \\citet{walton11b} with accurate positions that we measured using archival {\\it Chandra}~observations and optical counterparts that are sufficiently bright for optical spectroscopy. Two of the ULX candidates are situated in elliptical galaxies (NGC~533~and NGC~741). ULX candidates in elliptical galaxies have a higher chance to be background AGN (39\\%, compared to 24\\% for all sources in the catalog of \\citet{walton11b}). On the other hand, IMBHs may form in dense (globular) clusters \\citep{portegieszwart02, miller02} and the optical counterparts to these ULX candidates could well be just that, making them interesting targets for further investigation. AM~0644-741~is a ring galaxy with a ULX candidate situated in between the nucleus and the ring. ESO~306-003~is a spiral galaxy with a ULX in the outer edge of the disk, apparently associated with an extended optical source. We obtained optical spectra of these four sources with the FOcal Reducer and low dispersion Spectrograph (FORS2) mounted on the Very Large Telescope (VLT) \\citep{appenzeller98}. The observations and data reduction steps are described in Section 2; Section 3 contains the results. In Section 4 we discuss our findings.\n\n\\section{Observations and data reduction}\n\\subsection{X-ray observations}\nWe use archival {\\it Chandra}~observations to get exact source positions for the ULX candidates in NGC~533, NGC~741, AM~0644-741~and ESO~306-003. Table \\ref{chantab} lists the details of all observations. \n\n\\begin{table*}\n\\begin{minipage}{135mm}\n\\caption{The \\textit{Chandra} observations of the four ULX candidates.}\\label{chantab}\n \\begin{tabular}{lccccc}\n \\hline\n Galaxy & Observation ID & Exposure time & Source on CCD & Off-axis angle & Observation date \\\\\n & & (kiloseconds)& & (arcmin) & (UT)\\\\\n \\hline\nNGC~533~& 2880 & 38.1 & ACIS S3 & 0.85 & 2002-07-28\\\\\n NGC~741~& 2223 & 30.74 & ACIS S3 & 2.74 & 2001-01-28\\\\\n AM~0644-741~& 3969 & 39.97 & ACIS S3 & 0.57 & 2003-11-17\\\\\n ESO~306-003~& 4994 & 22.75 & ACIS I3 & 6.60 & 2004-03-10\\\\\n \\hline\n \\end{tabular}\n \n\\medskip\nNotes: {\\it Chandra}~observation ID number, exposure time in kiloseconds, CCD on which the source was detected, the off-axis angle of the source in arcminutes and the observation date.\n\\end{minipage}\n\\end{table*}\n\nWe use \\textsc{Ciao} version 4.4 to process the {\\it Chandra}~observations, with the calibration files from \\textsc{caldb} version 4.5.0. We treat the {\\it Chandra}~observations as follows: first we update the event files with ACIS\\_{}PROCESS\\_{}EVENTS, then we use WAVDETECT to find the position of the ULX candidate. Sources within 3 arcmins of one of the ACIS aimpoints have a 90\\% confidence error circle around the absolute position with a radius of 0.6''; this is valid for the ULX candidates in NGC~533, NGC~741~and AM~0644-741. The candidate in ESO~306-003~has 25 counts and was observed at 6.6' off-axis, which means it has a 95\\% confidence error circle with a radius of $\\sim$1.5'' \\citep{hong05}. For the sources in NGC~533, NGC~741~and AM~0644-741~we extract the source counts in a circle with 6 pixel radius (90\\% encircled energy fraction) around the source positions using SPECEXTRACT. For the ULX candidate in ESO~306-003~we use a circle with a radius of 10 pixels to get the same encircled energy fraction, since it was observed at 6.6' off-axis. As background regions we use circles with 80 pixel radius on the same CCD but not containing any sources. We use \\textsc{XSpec} version 12.6.0 to fit an absorbed powerlaw (pegpwrlw) to the data in the 0.3-8 keV range. We then extrapolate to get the 0.2-12 keV flux to compare this with the values reported by \\citet{walton11b}. For consistency we adopt the same model parameters: a photon index of 1.7 and $N_H = 3 \\times 10^{20}$ cm$^{-2}$, and allow only the flux to vary. We find that all \\textit{Chandra} fluxes are consistent with those from XMM-{\\it Newton}~as reported by \\citet{walton11b}. The positions of the X-ray sources and their fluxes are summarized in Table \\ref{chansrc}.\n\n\\begin{table*}\n\\begin{minipage}{90mm}\n\\caption{The positions and unabsorbed 0.2-12 keV X-ray fluxes of the ULX candidates.}\\label{chansrc}\n \\begin{tabular}{ccccc}\n \\hline\n Host galaxy & Right Ascension & Declination & Source flux \\\\\n &&& (erg cm$^{-2}$ s$^{-1}$)\\\\\n \\hline\nNGC~533~& 01:25:33.63 & +01:46:42.6& $2.9 \\pm 0.2 \\times 10^{-14}$ \\\\\n NGC~741~& 01:56:16.14 & +05:38:13.2 & $2.5 \\pm 0.3 \\times 10^{-14}$ \\\\\n AM~0644-741~& 06:43:02.24 & -74:14:11.1 & $3.5 \\pm 0.2 \\times 10^{-14}$ \\\\\n ESO~306-003~& 05:29:07.21 & -39:24:58.4 & $2.4 \\pm 0.4 \\times 10^{-14}$ \\\\\n \\hline\n \\end{tabular}\n \n\\medskip\nNotes: the positions of the ULX candidates in NGC~533, NGC~741~and AM~0644-741~are accurate to within 0.6''(90\\% confidence level), for the source in ESO~306-003~this value is 1.3''. We fit the fluxes assuming an absorbed powerlaw with photon index 1.7 and $N_H = 3 \\times 10^{20}$ cm$^{-2}$ for consistency with the method used by \\citet{walton11b}. \n\\end{minipage}\n\\end{table*}\n\n\\subsection{Optical images and photometry}\nTo find the optical counterparts of the ULX candidates we use archival optical observations of their host galaxies. NGC~533~and NGC~741~were observed as part of the Sloan Digital Sky Survey (SDSS), and we use the SDSS $r'$-band images to identify the optical counterparts to the ULX candidates in these galaxies (Figure \\ref{fig:agn}). There is no photometric information for the source in NGC~533, so we use the aperture photometry tool in \\textsc{GAIA} to estimate the $r'$-band magnitude. SDSS does provide \\textit{u'}, \\textit{g'}, \\textit{r'}, \\textit{i'} and \\textit{z'} magnitudes for the object in NGC~741, but these are incorrect because the source is too close to the edge of the frame. Therefore we also use \\textsc{GAIA} to estimate the $r'$-band magnitude for this source. For both optical counterparts we find that $r' = 21 \\pm 1$.\n\nThe Hubble Space Telescope (HST) archive contains several observations of AM~0644-741~made with the Advanced Camera for Surveys (ACS). We use the V-band (F555W) image with exposure identifier j8my05o2q, observed on 2004-01-16 with an exposure time of 2200 seconds (see Figure \\ref{fig:agn}). We visually compare the position of point sources from the USNO CCD Astrograph Catalog (UCAC) 3 \\citep{zacharias09} with their counterparts in the HST image and find that the astrometric calibration of the image does not need to be improved. The ULX candidate has a counterpart that is in the DAOPHOT source list of this HST image in the Hubble Legacy Archive (HLA\\footnote{http:\/\/hla.stsci.edu}). It has a V-band magnitude of $21.79 \\pm 0.05$.\n\nWe identify the optical counterpart to the ULX candidate in ESO~306-003~in a 480 seconds R-band observation made on 2004-01-25 UT with VLT\/VIMOS that we retrieved from the ESO archive. Its R-band magnitude is approximately 21, with the caveat that this is an extended source in a region with a very high background level due to the galaxy, which means that this measurement is not very accurate. We also obtained a \\textit{g'}-band, 120 seconds exposure of this galaxy in our VLT\/FORS2 run (see Figure \\ref{fig:eso306}), of which we visually inspected the astrometric solution by comparing the positions of bright stars with those in the UCAC 3.\n\n\\subsection{Optical spectroscopy}\nWe obtained VLT\/FORS2 observations of NGC~741~($3 \\times 1800$ s), AM~0644-741~($3 \\times 1800$ s) and ESO~306-003~($2 \\times 2700$ s) on 2011-12-03 UT under programme 088.B-0076A using the GRIS\\_600V grism and a 1\\arcsec~slit width. This configuration covers the wavelength range 4430-7370 \\AA{}~with a dispersion of 0.74 \\AA{}\/pixel,  yielding a resolution of 4.25 \\AA{}~for the 1'' slit (measured at 6300 \\AA). This allows us to observe the H$\\alpha$, [N{\\sc II}] complex and the H$\\beta$ and [O{\\sc III}] lines if the sources are located at the same distance as their apparent host galaxies, with high enough resolution to separate them. The night was photometric so we also observed several spectrophotometric standard stars to perform a flux calibration. The seeing varied between 0.7 and 1.1\\arcsec.\nThe spectra of NGC~533~($3 \\times 1500$ s) were made in service mode on 2012-01-16 UT with the GRIS\\_300V+10 grism and a 0.5\\arcsec~slit width, giving a wavelength coverage from 4450-8700 \\AA{}~with a dispersion of 1.68 \\AA{}\/pixel and a spectral resolution of 6.4 \\AA{}~for the 0.5'' slit (measured at 6300 \\AA{}). The seeing varied during the night and we have no observations of spectrophotometric standards.\n\nTo reduce the spectra we use the \\textsc{starlink} software package \\textsc{Figaro} and the \\textsc{Pamela} package developed by Tom Marsh\\footnote{http:\/\/deneb.astro.warwick.ac.uk\/phsaap\/software}. We follow the steps outlined in the \\textsc{Pamela} manual to extract the spectra, using Keith Horne's optimal extraction algorithm \\citep{horne86}. We then use the software package \\textsc{Molly}, also by Tom Marsh\\footnotemark[\\value{footnote}], to perform the wavelength calibration and, for the data taken on 2011-12-03, the flux calibration. We do not correct for telluric absorption. Because we have multiple spectra of each source we average them to get a better signal-to-noise ratio. The two observations of ESO~306-003~were taken under varying seeing conditions. Because of this the continuum level is different in the two spectra, so we normalize these spectra before averaging them.\nWe use \\textsc{Molly}'s MGFIT task to fit Gaussian profiles to the emission lines in the spectra to determine the full width at half maximum (FWHM) of the lines and the redshift to the sources. \n\n\n\n\\section{Results}\n\n\\begin{table*}\n\\begin{minipage}{150mm}\n\\caption{Source properties of the background AGN}\\label{optinfo}\n \\begin{tabular}{lccccc}\n \\hline\n Source name & In galaxy & z & Line & FWHM & Log(F$_X$\/F$_{\\textrm{opt}}$) \\\\\n &&&& km\/s &\\\\\n \\hline\nCXOU J012533.3+014642 & NGC~533~& $1.8549 \\pm 0.0003$ & C\\textsc{IV} & $2300 \\pm 70 $ & $0.0 \\pm 0.5$\\\\\n& & & C\\textsc{III}] & $7800 \\pm 200$ & \\\\\n& & & Mg{\\sc II} & $5700 \\pm 200$ & \\\\\nCXOU J015616.1+053813 & NGC~741~& $0.8786 \\pm 0.0006$ or & Mg{\\sc II} or & $8400 \\pm 200$ & $0.0 \\pm 0.5$\\\\\n& & $1.7535 \\pm 0.0009$ & C{\\sc III}] & & \\\\\nCXO J064302.2-741411 & AM~0644-741~& $1.3993 \\pm 0.0001$ & C{\\sc III}] & $5100 \\pm 70$ &  $0.7 \\pm 0.1$\\\\\n & & & Mg{\\sc II} & $4220 \\pm 40$ &\\\\\n \\hline\n \\end{tabular}\n \n\\medskip\nNotes: Lines used for the redshift determination to the quasars, their FWHM in km\/s and the X-ray to optical flux ratio of these sources. The X-ray to optical flux ratios are calculated using the XMM-{\\it Newton}~0.2-12 keV fluxes from \\citet{walton11b} and the \\textit{r'}-band (for NGC~533~and NGC~741) or V-band (for AM~0644-741) optical fluxes.\n\\end{minipage}\n\\end{table*}\n\n\\begin{figure*}\n\\hbox{\n\\includegraphics[width=0.33\\textwidth]{ngc533_sdss}\n\\includegraphics[width=0.33\\textwidth]{ngc741_sdss}\n\\includegraphics[width=0.33\\textwidth]{a0644_hst}\n}\n\\hbox{\n\\includegraphics[width=0.33\\textwidth]{ngc533_avspec}\n\\includegraphics[width=0.33\\textwidth]{ngc741_avspec}\n\\includegraphics[width=0.33\\textwidth]{a0644_avspec}\n}\n\\caption{The finders and FORS2 spectra of the three ULX candidates that are background AGN. The 90\\% confidence error circles around the X-ray positions have a radius of 0.6'', for NGC~533~and NGC~741~we plot a larger circle for visual clarity. \\emph{Left:} The SDSS \\textit{r'}-band image of NGC~533~with a 1.2'' radius circle around the {\\it Chandra}~position of the ULX candidate and the spectrum in which the C\\textsc{IV}, C{\\sc III}] and Mg{\\sc II} emission lines, redshifted by $z=1.85$, are marked. The absorption features at 6200 \\AA{}~are caused by interstellar absorption, and those at 6900 \\AA{}~and 7600 \\AA{}~are telluric in origin. \\emph{Middle:} The SDSS \\textit{r'}-band image of NGC~741~with a 1.2'' radius circle around the {\\it Chandra}~position of the ULX candidate and the spectrum of the optical counterpart. The marked emission line can be either Mg{\\sc II} $\\lambda2798$ line redshifted by $z=0.88$ or C{\\sc III}] at $z = 1.75$. \\emph{Right:} An HST ACS V-band image of AM~0644-741~with the 0.6'' radius error circle around the {\\it Chandra}~position of the ULX candidate, and the spectrum with the Mg{\\sc II} and C{\\sc III}] lines, redshifted by $z=1.40$, marked.}\\label{fig:agn}\n\\end{figure*}\n\n\\subsection{NGC~533}\nNGC~533~is the dominant elliptical galaxy in a group with the same name at $z = 0.0185$ \\citep{smith00}. The ULX candidate is located at 78'' from the center of the galaxy that has a semi-minor axis of 90'' (based on the D25 isophote, \\citealt{nilson73}). The X-ray source has an unresolved optical counterpart that is visible in the image of the SDSS, with \\textit{r'}-band magnitude $\\approx 21$. Figure \\ref{fig:agn} shows the galaxy with the position of the ULX candidate and the FORS2 spectrum of the source. \n\nThree broad emission lines are visible. We identify these as C\\textsc{IV}, C{\\sc III}] and Mg{\\sc II} at $z = 1.8549 \\pm 0.0003$. This proves the ULX candidate to be a background AGN, not associated with NGC~533. The 0.2-12 keV X-ray luminosity calculated for this source by \\citet{walton11b} was $(2 \\pm 1) \\times 10^{40}$ erg s$^{-1}$, assuming a distance to the ULX of 73.8 Mpc. The true distance to this source is 4730 Mpc (using $H_0 = 75$ km\/s\/Mpc for consistency with \\citealt{walton11b}), which gives this AGN an X-ray luminosity of $(7 \\pm 4) \\times 10^{43}$ erg s$^{-1}$~using the flux as measured with XMM-{\\it Newton}.\n\n\n\\subsection{NGC~741}\nNGC~741~is an elliptical galaxy located at $z = 0.0185$ with a (D25) semi-major axis of 92.7'' \\citep{devaucouleurs91}. The ULX candidate is located 78'' West of the center of NGC~741~and has a counterpart that is unresolved in the SDSS image. Its \\textit{r'}-band magnitude is $\\sim 21$. Figure \\ref{fig:agn} shows the SDSS \\textit{r'}-band image of NGC~741~with the position of the ULX candidate and the FORS2 spectrum of the counterpart.\n\nThe spectrum shows one broad emission line, with a FWHM of 147 \\AA{}. We cannot say with certainty which line this is. The most likely options are that it is either the Mg{\\sc II} $\\lambda2798$ line or the C{\\sc III}] $\\lambda1909$ line. In the first case this ULX candidate would be a background AGN at a redshift of $z = 0.8786 \\pm 0.0006$ with an X-ray luminosity of $(1.6 \\pm 0.6) \\times 10^{43}$ erg s$^{-1}$. In the second case it would be at $z = 1.7535 \\pm 0.0009$, with $L_X = (4.2 \\pm 1.5) \\times 10^{43}$ erg s$^{-1}$. In both cases the source is not a ULX but a background AGN, unconnected to NGC~741.\n\n\\subsection{AM~0644-741}\nAM~0644-741~is a ring galaxy at $z = 0.022$ that shows signs of recent interaction with a smaller galaxy \\citep{few82, lauberts89}. The ULX candidate in this galaxy is located in between the core of the galaxy and the ring. A point-like object with a V-band magnitude of 21.8 is visible at the position of the X-ray source in archival HST images (see Figure \\ref{fig:agn}).\n\nThe FORS2 spectrum of the counterpart shows two emission lines that we identify as Mg{\\sc II} and C{\\sc III}] at $z=1.3993 \\pm 0.0001$. This ULX candidate is another background AGN with an 0.2-12 keV luminosity of $(8.1 \\pm 0.8) \\times 10^{43}$ erg s$^{-1}$.\n\n\n\\subsection{ESO~306-003}\nThe spiral galaxy ESO~306-003, at $z \\approx 0.016$ \\citep{dasilva06}, contains a ULX candidate that is located on the edge of the spiral structure (see Figure \\ref{fig:eso306}). An optical source is visible on the edge of the error circle. Visual inspection shows the profile of the counterpart to be  more extended than that of point sources in the same image, but because of the high background level and steep gradient it is not possible to perform an acceptable fit to the profile. The full width at half maximum (FWHM) of point sources in this image (provided by the seeing) is 0.8''. At the distance of ESO~306-003~this yields a lower limit to the size of the source of 240 pc. The two spectra that we obtained of this source show slightly different line ratios and continuum levels (for example, the H$\\beta$\/H$\\alpha$ ratio changes by 10\\%). This can be explained by seeing variations if the optical counterpart to this source is extended: then slit losses can cause the small changes in the line ratios if there are intrinsic spatial variations in the line ratios in the extended source.\n\nThe spectrum is similar to that of an H{\\sc II} region, with narrow Hydrogen emission lines and strong forbidden lines. The redshift of the lines equals that of the center of the galaxy, indicating that if the X-rays are associated with this optical source, this is a bona fide ULX with a luminosity of $1.4 \\pm 0.3 \\times 10^{40}$ erg s$^{-1}$~based on the XMM-{\\it Newton}~flux measured by \\citet{walton11b}. The X-ray flux is constant between the XMM-{\\it Newton}~and {\\it Chandra}~observations. The X-ray to optical flux ratio of the source is log(F$_X$\/F$_{\\textrm{opt}}$)$ = 0.3 \\pm 0.5$, based on the XMM-{\\it Newton}~0.2-12 keV flux from \\citet{walton11b} and the \\textit{r'}-band flux. The line ratios, especially the [O{\\sc I}] $\\lambda$6300\/H$\\alpha$ ratio, place the source among the transition objects in the diagnostic diagrams of \\citet{ho08} (see Figure \\ref{fig:dd}). The He{\\sc II} $\\lambda4686$ emission line has been detected in several ULX nebulae \\citep{pakull02, kaaret09}, but we do not detect it here, possibly because the sensitivity of the detector drops off steeply towards the blue end. The 2-$\\sigma$ upper limit for the equivalent width of this line is 1.0 \\AA{}. This corresponds to a flux of $\\sim 10^{-17}$ erg cm$^{-2}$ s$^{-1}$ or a luminosity of $\\sim 5 \\times 10^{36}$ erg s$^{-1}$.\n\n\\begin{figure*}\n\\hbox{\n\\includegraphics[width=0.4\\textwidth]{eso306_vlt_im}\n\\includegraphics[width=0.6\\textwidth]{eso306_avspec}\n}\n\\caption{\\emph{Left:} The FORS2 \\textit{g'}-band acquisition image of ESO~306-003~with the 1.3'' radius (90\\% confidence) error circle around the position of the ULX candidate. \\emph{Right:} The FORS2 spectrum of the candidate optical counterpart to the X-ray source. Several emission lines, redshifted by $z=0.016$, are marked.}\\label{fig:eso306}\n\\end{figure*}\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{dds_OI}\n\\caption{[O{\\sc I}] $\\lambda$6300\/H$\\alpha$ versus [O{\\sc III}] $\\lambda$5007\/H$\\beta$ line ratios for H{\\sc II} regions, AGN (LINERs\nand Seyferts) and transition objects (figure adapted from \\citealt{ho08}). The black dot represents the line ratios for the ULX in ESO~306-003.}\\label{fig:dd}\n\\end{figure}\n\n\\section{Discussion}\nWe obtained VLT\/FORS2 spectra of the optical counterparts of four bright ULX candidates with accurate positions obtained by us from archival {\\it Chandra}~observations. Two of these are located in elliptical galaxies NGC~533~and NGC~741. Another candidate is situated in AM~0644-741, a ring galaxy that recently interacted with a small elliptical galaxy, and in the spiral galaxy ESO~306-003. Three of our four targets turn out to be background AGN with X-ray luminosities ranging from 1 to 8 $\\times 10^{43}$ erg s$^{-1}$; one (in ESO~306-003) seems to be a bona fide ULX. \n\nThe fraction of background AGN in our sample is higher than the fraction estimated by \\citet{walton11b} for their catalog. Although this can be due to small number statistics since we only investigate four sources, it is in line with results from other spectroscopic studies of ULX candidates. Optical spectroscopy of a sample of 23 ULX candidates in total yielded 20 background AGN and three foreground stars (\\citealt{gutierrez05, gutierrez06, gutierrez07, gutierrez13}). Another study that targeted 17 ULX candidates from the catalog of \\citet{colbert02} found that 15 were background AGN and the other two objects were foreground stars \\citet{wong08}.\n\nAll these studies mainly target ULX candidates that are relatively isolated and have a bright optical counterpart, a selection effect induced by the relative ease with which spectroscopic observations can be carried out for these sources. Sources located in crowded areas, like the spiral arms of late type galaxies, are more difficult targets for ground based optical spectroscopic observations. This means that spectroscopic studies are aimed at ULX candidates that have a low X-ray to optical flux ratio and that are situated relatively far away from their suspected host galaxies. As the authors of these previous papers also note, these selection criteria introduce a bias towards background AGN.\n\nA possible method to select ULX candidates that are most likely to be real ULXs is to calculate the expected contribution of background sources based on the known density of AGN in X-ray and optical observations (\\citealt{lopez06, sutton12}). Alternatively it may be possible to use the X-ray to optical flux ratios of ULX candidates to select targets for future spectroscopic studies. All our sources have X-ray to optical flux ratios log(F$_X$\/F$_{\\textrm{opt}}$) in the range between -1 and 1, typical for AGN (e.g. \\citealt{barger03}). Most ULXs show values for log(F$_X$\/F$_{\\textrm{opt}}$) ranging from 2 - 3 (\\citealt{tao11, tao12, sutton12}). The low value that we find for the ULX in ESO~306-003~can be explained if we assume that we do not resolve the ULX counterpart but instead observe the optical flux of the entire HII region that it resides in, thus lowering log(F$_X$\/F$_{\\textrm{opt}}$).\n\nHowever, if we were to select candidates for spectroscopy on the basis of their X-ray to optical flux ratios only we run the risk of missing interesting sources. For instance, ULXs may display different values for log(F$_X$\/F$_{\\textrm{opt}}$) when observed in the high and low states, as was shown for M101 ULX-1 and M81 ULS1 (\\citealt{tao11}). For both sources log(F$_X$\/F$_{\\textrm{opt}}$) is between 2 and 3 during the high state, but around 0 during the low state, well inside the range of optical to X-ray flux ratios found for AGN. Therefore other source properties should be taken into account as well, such as galaxy morphology, the distance of the ULX to its apparent host galaxy and the absolute magnitude of its optical counterpart. \nThe source in AM~0644-741{} is a good example of a candidate with such favorable properties: situated in a ring galaxy, which is a strong sign of a recent interaction phase that triggered star formation, often linked to ULXs (e.g. \\citealt{swartz04}), and close to the center of its apparent host galaxy, decreasing the chance that it is a background AGN (\\citealt{wong08}). It has an optical counterpart of such magnitude that it is consistent with being a bright globular cluster if it is at the distance of AM~0644-741{}. Nevertheless our optical spectrum showed it to be a background object.\n\n\\subsection*{The ULX in ESO~306-003}\nThe X-ray source in ESO~306-003~is the only one of the four candidates in our sample that appears to be a bona fide ULX.  The extended nature of the source is confirmed by the fact that the emission line spectrum is consistent with that of an HII region. However, the [OI]\/H$\\alpha$ ratio indicates that some of the ionizing flux could come from an X-ray source. Potentially, we have found a ULX embedded in an HII region. \nAnother possibility is that this ULX candidate is a background AGN shining through an HII region in ESO~306-003. The X-ray to optical flux ratio is similar to that of the other AGN in our sample, so we would expect to see a contribution of redshifted emission lines from the AGN in the optical spectrum. The fact that we do not detect this makes this scenario implausible. \n\nWe find a 2-$\\sigma$ upper limit for the flux of a  HeII $\\lambda4686$ line of $10^{-17}$ erg cm$^{-2}$ s$^{-1}$. This corresponds to an upper limit to the luminosity in the line of $\\sim 5 \\times 10^{36}$ erg s$^{-1}$. The presence of this line would be a strong indication of ionization by an X-ray source. We can compare this upper limit with the strength of the HeII $\\lambda4686$ line in other ULX nebulae. For Holmberg II X-1, \\citet{pakull02} report a luminosity of $2.5 \\times 10^{36}$ erg s$^{-1}$. \\citet{kaaret09} report a flux for this line from the ULX in NGC 5408 of $3.3 \\times 10^{-16}$ erg cm$^{-2}$ s$^{-1}$, which translates to a luminosity of $9 \\times 10^{35}$ erg s$^{-1}$~at the distance to NGC 5408 (4.8 Mpc, \\citealt{karachentsev02}). Both these ULXs have an X-ray luminosity of $\\sim 10^{40}$ erg s$^{-1}$ -- similar to ESO~306-003~-- and a HeII $\\lambda4686$ to X-ray luminosity ratio of $\\sim 10^{-4}$. If the same is true for ESO~306-003~then we would expect a HeII $\\lambda4686$ flux of a few times $10^{-18}$ erg cm$^{-2}$ s$^{-1}$, which is just below our 2-$\\sigma$ upper limit. New observations of this source with greater sensitivity at the wavelength of the HeII $\\lambda4686$ line are needed to determine if the nebula is X-ray photo-ionized or not.\n\n\\section*{Acknowledgements}\nPGJ and MAPT acknowledge support from the Netherlands Organisation for Scientific Research. GM acknowledges support from the Spanish Plan Nacional de Astronom\\'{\\i}a y Astrof\\'{\\i}sica under grant AYA2010-21490-C02-02. This research is based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme ID 088B-0076A. This research has made use of software provided by the Chandra X-ray Center (CXC) in the application package CIAO and of the software packages Pamela and Molly provided by Tom Marsh.\n\n \\bibliographystyle{mn_new}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}