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+{"text":"\\section{\\label{}}\n\n\\vspace{-0.5in}\n\\section{INTRODUCTION}\nThe ATLAS combined reconstruction software uses several approaches to calculating the missing transverse energy (Missing\nET \\cite{csc-ref}). The simplest approach is to use as input all calorimeter cells, which are calibrated based on the cell energy density and position,\n and muons which leave little energy in the calorimeter. This approach is expected to be robust in early data and is constructed from three terms:\nThe \\emph{calo term} is calculated from all calorimeter cells, the \\emph{muon term} from muons measured in the muon system\nwith $|{\\eta}|$ $<$ 2.7 and the \\emph{cryostat term} corrects for energy deposited in the cryostat between the electromagnetic\nand hadronic calorimeters. A more sophisticated approach, called the Refined Missing ET, is to modify these calorimeter cell \nweights according to the object (electron, jet etc) that they are attached to. \n\n\\vspace{-0.2in}\n\\section{PERFORMANCE}\n\nThe linearity is defined to be:\n\\begin{equation}\n\\frac{|E_{TMiss}^{True}|-|E_{TMiss}|}{|E_{TMiss}^{True}|}\n\\end{equation}\nwhere $E_{TMiss}^{True}$ is the true Missing ET calculated from non-interacting truth particles. The performance is shown for a number of samples in Figure 1. In the left plot points with true Missing ET of 20 GeV are from $Z$ $\\rightarrow$ $\\tau^{+}\\tau^{-}$, at 35 GeV from $W$ $\\rightarrow$ e\/$\\mu\\nu$, at 68 GeV from semileptonic $t$$\\bar{t}$ events, at 124 GeV from $A^{0}$ $\\rightarrow$ $\\tau^{+}\\tau^{-}$ with $m_{A}$ = 800 GeV and at 280 GeV from events with supersymmetric particles with a mass scale of 1 TeV. The right plot shows a more detailed version for $A^{0}$ $\\rightarrow$ $\\tau^{+}\\tau^{-}$ with $m_{A}$ = 800 GeV - the non-linearity at low true Missing ET is due to the finite detector resolution and is not a bias in the algorithm.\n\\begin{figure}[h]\n\\begin{center}\n$\\begin{array}{c@{\\hspace{0.5in}}c}\n\\multicolumn{1}{l}{\\mbox{\\bf (a)}} &\n        \\multicolumn{1}{l}{\\mbox{\\bf (b)}} \\\\ [-0.53cm]\n\\hspace{-0.4in}\n\\includegraphics[width=7.cm,height=3.7cm]{.\/Fig3_1.eps} &\n        \\includegraphics[width=7.cm,height=3.7cm]{.\/fig3_2.eps} \\\\ [-0.4cm]\n\\end{array}$\n\\end{center}\n\\caption{Linearity as function of true Missing ET for different stages in the calibration. The global calibration refers to the hadronic calibration cell weights applied to the cells in the \\emph{calo term}. }\n\\label{ETMiss_Lin}\n\\end{figure}\n\nThe resolution is shown as a function of scalar missing transverse energy sum ($\\sum{E_{T}}$) for a number of processes in Figure 2. The resolution follows a stochastic behaviour, $a\\sqrt{\\sum{E_T}}$, with $a$ between 0.53 and 0.57.\n\\begin{figure}[h]\n\\begin{center}\n$\\begin{array}{c@{\\hspace{0.5in}}c}\n\\multicolumn{1}{l}{\\mbox{\\bf (a)}} &\n        \\multicolumn{1}{l}{\\mbox{\\bf (b)}} \\\\ [-0.53cm]\n\\hspace{-0.4in}\n\\includegraphics[width=7.cm,height=3.7cm]{.\/Fig4_1.eps} &\n        \\includegraphics[width=7.cm,height=3.7cm]{.\/Fig4_2.eps} \\\\ [-0.4cm]\n\\end{array}$\n\\end{center}\n\\caption{Resolution as function  scalar missing transverse energy ($\\sum{E_{T}}$) of for a number of physics processes. }\n\\label{ETMiss_Lin}\n\\end{figure}\n\\vspace{-0.3in}\n\\section{FAKE MISSING ET}\nThere are a number of sources of fake Missing ET: instrumental effects which can be understood using real data - beam gas, beam halo and noisy or dead cells, inefficiencies in reconstructing high $p_{T}$ muons, fake muons from e.g. jets punching through calorimeters and jet leakage from calorimeters. The effects of fake sources of Missing ET is shown for QCD di-jet events in the left plot of Figure 3 where there is a significant shape difference between the true Missing ET distribution and the fake Missing ET Distribution.  \n\n\\begin{figure}[h]\n\\begin{center}\n$\\begin{array}{c@{\\hspace{0.5in}}c}\n\\multicolumn{1}{l}{\\mbox{\\bf (a)}} &\n        \\multicolumn{1}{l}{\\mbox{\\bf (b)}} \\\\ [-0.53cm]\n\\hspace{-0.2in}\n\\includegraphics[width=7.cm,height=3.7cm]{.\/Fig10_1.eps} &\n        \\includegraphics[width=7.cm,height=3.7cm]{.\/Fig18.eps} \\\\ [-0.4cm]\n\\end{array}$\n\\end{center}\n\\caption{True and fake Missing ET distributions for di-jets sample with $p_{T}$ of the hard scatter between 560 and 1120 GeV (left). Missing ET resolution is shown as a function of $\\sum{E_{T}}$. $W$ $\\rightarrow$ $\\mu\\nu$ corresponds to $\\sum{E_T}$ of 150 GeV, $W$ $\\rightarrow$ $e\\nu$ to 165 GeV, $Z$ $\\rightarrow$ $\\tau^{+}\\tau^{-}$ to 210 GeV, $t$$\\bar{t}$ to 470 GeV, $A^{0}\/H^{0}$ $\\rightarrow$ $\\tau^{+}\\tau^{-}$ to 843 GeV, di-jets with $p_{T}$ of the hard scatter in the range 280 to 560 GeV corresponds to 800 GeV and SUSY to 906 GeV (right). }\n\\label{ETMiss_Lin}\n\\end{figure}\n\\vspace{-0.3in}\n\\section{VALIDATION WITH FIRST DATA}\nThe first data will be used to calibrate the calorimeter and understand instrumentation failures. Subsequently the $Z$ $\\rightarrow$ $\\tau^{+}\\tau^{-}$ process can be used to determine the Missing ET scale in situ with a systematic uncertainty of 8$\\%$. The $W$ $\\rightarrow$ $l\\nu$ decays can be used to test the Missing ET reconstruction over the energy range 20 to 150 GeV. Top pair production will allow the reconstruction of genuine Missing ET to be tested in an enviroment relevant to Supersummetry (SUSY). The Missing ET resolution is shown as a function of $\\sum{E_{T}}$ for various processes at four different stages of the Refined Missing ET calibration in the right plot of Figure 3.\n\nThe minimum bias sample will have large statistics in the first data and thus it will be very useful for commissioning the Missing ET reconstruction. Events are required to pass one of three minimum bias triggers, to have at least 20 semiconductor tracker space points to reject empty events and to have at least one good reconstructed track to reject beam gas and halo events. The mean true Missing ET is 0.06 GeV and mean expected Missing ET is 4.3 GeV because of the calorimeter energy resolution and acceptance. At low $\\sum{E_{T}}$ the stochastic term dominates the resolution, illustrated in the left plot of Figure 4. The Missing ET is observed to scale with the $\\sum{E_{T}}$ as expected when compared to QCD di-jet events (right plot in Figure 4).\n\\begin{figure}[h]\n\\begin{center}\n$\\begin{array}{c@{\\hspace{0.5in}}c}\n\\multicolumn{1}{l}{\\mbox{\\bf (a)}} &\n        \\multicolumn{1}{l}{\\mbox{\\bf (b)}} \\\\ [-0.53cm]\n\\hspace{-0.4in}\n\\includegraphics[width=7.cm,height=3.4cm]{.\/Fig19_1.eps} &\n        \\includegraphics[width=7.cm,height=3.4cm]{.\/Fig19_2.eps} \\\\ [-0.4cm]\n\\end{array}$\n\\end{center}\n\\caption{Resolution for different $\\sum{E_{T}}$ regimes in minimum bias events (left) and Missing ET resolution as function of $\\sum{E_{T}}$ for both minimum bias and di-jet events. }\n\\label{ETMiss_MinBias}\n\\end{figure}\n\nTo select the $Z$ $\\rightarrow$ $\\tau^{+}\\tau^{-}$ sample the Missing ET is required to be larger than 20 GeV to reject QCD events and the transverse mass, calculated from the Missing ET and lepton (defined to be the leading and isolated lepton, electron or muon, with $P_{T}^{l}$ $>$ 15 GeV and $|{\\eta}|$ $<$ 2.5), is required to be less than 50 GeV, to suppress semileptonic $W$ decays. A cut on $\\sum{E_{T}}$ $<$ 400 GeV further suppresses QCD events. No $b$-tagged jets are allowed in order to suppress bottom and top quark pair production. At least one tau-jet with $p_{T}$ $>$ 15 GeV, $|{\\eta}|$ $<$ 2.5 and a track multiplicity of one or three is required. The azimuthal distance between the isolated lepton and tau-jet is required to be in the range 1 - 2.8 radians to reject badly reconstructed events and further suppress backgrounds. The invariant mass of the $\\tau\\tau$ system shows good sensitivity to the Missing ET Scale (left plot in Figure 5) and allows the determination of this scale with 8$\\%$ accuracy.\nIn the semileptonic top pair sample 7000 events are selected in 200 $pb^{-1}$ when 3 jets with $p_{T}$ $>$ 40 GeV, one more jet with $p_{T}$ $>$ 20 GeV, Missing ET $>$ 20 GeV and an isolated lepton (electron or muon) with $p_{T}$ are required. The effect of scaling the true Missing ET by 0.8 and 1.2 is shown in the right plot of Figure 5 - Gaussian fits indicate the peak shifts by $\\pm$ 7 GeV with a statistical error of 0.5 $\\%$.\n\n\\begin{figure}[!h]\n\\begin{center}\n$\\begin{array}{c@{\\hspace{0.5in}}c}\n\\multicolumn{1}{l}{\\mbox{\\bf (a)}} &\n        \\multicolumn{1}{l}{\\mbox{\\bf (b)}} \\\\ [-0.53cm]\n\\hspace{-0.4in}\n\\includegraphics[width=7.cm,height=3.4cm]{.\/Fig20_2.eps} &\n        \\includegraphics[width=7.cm,height=3.4cm]{.\/Fig23_1.eps} \\\\ [-0.4cm]\n\\end{array}$\n\\end{center}\n\\caption{The invariant mass of the $\\tau\\tau$ system as function of the Missing ET scale (left). The transverse mass distribution for three semileptonic top quark pair samples, each with different Missing ET scales (right).}\n\\label{ETMiss_MinBias}\n\\end{figure}\n\n\\vspace{-0.4in}\n\\section{CONCLUSIONS}\nThe linearity of the Missing ET is expected to be within 5$\\%$ over a wide range of Missing ET at the beginning of data taking in ATLAS. The resolution follows a stochastic behaviour, $a\\sqrt{\\sum{E_T}}$, with $a$ between 0.53 and 0.57 except at very low and very high $\\sum{E_T}$. Minimum bias events will allow initial validation of the Missing ET reconstruction to be applied. The Missing ET scale can be measured with an 8$\\%$ accuracy using $Z$ $\\rightarrow$ $\\tau^{+}\\tau^{-}$ events. The semileptonic top quark pair sample will allow to check for shifts in the Missing ET scale in a kinematic regime relevant for SUSY.\n\n\\vspace{-0.2in}\n\n\\begin{acknowledgments}\nI would like to thank the organisers of the ICHEP08 for the invitation to present this poster and the ATLAS collaboration.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Background and Related Work}\nNetwork densification through the deployment of more access points(APs)\/base stations (BSs) per unit area  and connecting them in a multi-tier network architecture is a natural way to improve cellular network capacity~\\cite{HetNet_1}. However, network densification gives rise to critical issues such as increased  signal interference (both inter-cell interference [ICI] and inter-user interference [IUI]), as well as increased network complexity and capital and operating expenditures.\nMany interference-tolerant and low-cost technologies including massive multiple-input multiple-output antenna (mMIMO), transmit\/receive diversity (TRD), and distributed antenna system (DAS)~\\cite{massiveMIMO_1,R_Diversity_1,R_Diversity_2,Distributed_Ant_1} have therefore been developed. \nThe performance gains in these technologies are mainly achieved by increasing the received power of the desired signal without any coordination among  interference sources in other cells~\\cite{UltraDense_1}. The interference can however be minimized by coordinating wireless transmitters and\/or receivers (either APs or user equipment [UE]).  For instance, coordinated multi-point (CoMP) technique allows adjacent APs to coordinate their transmission and reception to serve cell-edge UEs. This will turn the strongest interference signals into a desired ones at the cost of decreased spectral efficiency \\cite{CoMP6146494}.\n\nRecently, the concept of cooperation among distributed APs has been generalized in which all APs serve all users  (i.e. both cell-edge and cell-center users) within the network coverage area. \nThis concept can be implemented through several techniques such as cloud radio access network (C-RAN)  \\cite{CRAN,CRAN_1}, cell-free networks (or cell-less networks) \\cite{Cell_Less_1,Cell_Less_2,Cell_Less_3,Cell_Less_4} and Generalized CoMP (GCoMP) networks~\\cite{DOMA,GCoMP}. \nIn a C-RAN, a group of distributed remote radio heads (RRHs) that are geographically distributed in the network coverage area are used to perform all radio functionalities such as high frequency amplification, frequency up\/down conversion and A\/D and D\/A conversion. \nThe users' baseband signals coming from all RRHs are then sent to a virtual baseband unit (BBU) pool that is located in the network cloud and is responsible to perform all baseband functionalities such as signal detection, etc. \nIn a cell-free network architecture, a group of distributed APs cooperate to serve all active users within the network coverage area simultaneously using the same frequency-time resources. This makes the entire network act as a massive distributed antenna system (DAS) that covers the entire network coverage area \\cite{DAS_1}. \nThis network architecture however will require a huge processing capability of the BBU pool (or the central processing unit [CPU]), especially with the massive increase on the number of users per unit area in beyond 5G\/6G networks \\cite{6G8412482,DBLP:journals\/corr\/abs-1901-07106}. \n\nSeveral studies on cell-free networks have investigated performance metrics such as per-user transmission rate \\cite{Cell_Less_3},  per-user packet delay \\cite{Cell_Less_Delay_1}, implementation issues such as pilot contamination \\cite{Cell_Less_1}, network backhauling\/fronthauling \\cite{Fronthaul_1,Fronthaul_2}, beamforming techniques \\cite{Cell_Less_Beamforming_1}. \nIn \\cite{Cell_Less_1}, it was found that channel estimation error resulting from non-orthogonal pilot sequences tends to significantly decrease the network performance.  \nThe authors in \\cite{Fronthaul_2} evaluated the per user performance under limited fronthaul link capacity and showed that the network performance degrades as the number of active users increases under limited fronthaul link capacity. Many uplink\/downlink beamforming techniques were developed for cell-free networks. In \\cite{Cell_Less_2}, the authors proposed the use of  conjugate beamforming on the downlink and matched filtering on the uplink to multiplex\/de-multiplex signals to\/from different users. \nIt was also shown that using these beamforming techniques, the inter-user interference (IUI) goes to zero as the number of APs goes to infinity. \nDifferent uplink\/downlink optimization techniques for cell-free decoding\/precoding vectors that are based on maximizing the weighted sum-rate (WSR) or users' minimum rate were proposed in the literature. \nFor instance, the authors in \\cite{Cell_Free_Sum_Rate_1} proposed a low-complexity algorithm that solve the problem of maximizing the WSR of downlink cell-free system (mixed non-convex and combinatorial optimization problem) while maximizing the minimum per-user rate was modeled as a quasi-convex problem and solved by different iterative algorithms (such as bisection search and gradient descent) \\cite{Cell_Less_1,Cell_Less_2,Cell_Less_3}. \n\n\nTo further enhance the performance of cell-free networks, non-orthogonal multiple access (NOMA) method can be used to suppress some of the in-band interfering signals\n\\cite{Cell_Free_NOMA_3,Cell_Free_NOMA_1,Cell_Free_NOMA_5}. \nIn \\cite{Cell_Free_NOMA_3}, downlink users were assumed to be divided into a number of NOMA clusters with successive interference cancellation (SIC) operation applied at every cluster members and the achievable per user transmission rate was calculated. The authors in \\cite{Cell_Free_NOMA_1} proposed an adaptive NOMA\/OMA selection scheme such that the downlink per-user transmission rate is maximized. It was also found that the performance of NOMA scheme outperforms that of cell-free OMA when the number of users is relatively high. Additionally, in \\cite{Cell_Free_NOMA_5}, a spectral efficiency maximization algorithm for uplink NOMA-enabled cell-free network was proposed where the authors showed that a better performance can be achieved by controlling the per-user transmission power. \n\n\\subsection{Motivation and Contributions}\n\nIn the existing literature, performance analysis of cell-free networks has focused only on the derivation of an approximate mathematical limits of the per-user transmission rate.\nThis is due to the fact that other important  performance metrics such as the probability of outage [or alternatively coverage probability] are significantly difficult to be derived in a closed-form. Furthermore, for the cell-free networks, beamforming techniques (uplink and downlink) were basically developed and tested for relatively small number of users and APs. The reason is that computing the beamforming vectors for massive numbers of users and APs using the conventional algorithms (such as bisection search and gradient descent) gives rise to real-time implementation issues such as convergence time and computational complexity. In  \\cite{GCoMP}, we proposed a downlink variable-order clustering scheme for APs that divides the APs into subgroups where every group is dedicated to serve a subset of the active users. This decreases the lengths of the beamforming vectors for every subgroup of APs by a factor that is linearly proportional to the clustering order. The performance degradation caused by clustering of the APs was compensated by adopting NOMA detection  in every cluster. Additionally,  clustering of APs was based on the channel gains of different users and APs.\n\nThis paper extends the work in \\cite{GCoMP} for massive {\\em uplink} multiple access in cell-free networks. For a static cell-free network, we first derive a closed-form expression of the per-user probability of outage. To reduce the complexity of joint processing of signals from all users in a static cell-free network, we propose a dynamic clustering scheme for APs. For real-time implementation of both dynamic AP clustering and uplink beamforming, we develop a deep reinforcement learning (DRL) scheme, namely, the hybrid Deep Deterministic Policy Gradient (DDPG)-deep double Q-network (DDQN)scheme. The major contributions on this paper can be summarized as follows:\n\n\\begin{itemize}\n    \\item For uplink static cell-free networks, we derive an accurate closed-form expression for the per-user probability of outage by exploiting the \\textit{Welch-Satterthwaite} approximation \\cite{Satterthwaite1946}.\n    \\item To  significantly decrease the signal processing complexity at the CPU for the static cell-free network, we propose a clustering scheme that dynamically partitions the APs into subsets with each subset acting as a virtual AP within a DAS system. We also propose an SIC-based signal detection scheme for non-orthogonal multiple access in a dynamic cell-free network and a modified DAS combining scheme that considers inter-user interference (IUI).\n    \\item We formulate a general problem to jointly optimize the clustering of APs and the beamforming vectors such that the uplink users' sum-rate is maximized or the minimum user rate is maximized.\n    \\item To solve the general optimization problem, we propose and design a novel hybrid DRL scheme based on DDPG-DDQN model.\n    \\item We study and compare different performance metrics of conventional static and those of dynamic cell-free networks under different number of users and APs.  \n\\end{itemize}\nThe rest of this paper is organized as follows.  The static cell-free network model and the analysis of outage performance for this network model are presented in Section II. In Section III, we present the dynamic cell-free network architecture and the corresponding outage performance analysis. For the dynamic cell-free network model, in Section IV, we present a successive interference cancellation (SIC)-aided signal detection scheme and a diversity combining scheme. Also, the joint optimization problem of AP clustering and  beamforming design for the dynamic cell-free model is formulated in this section. In Section V, we propose a novel hybrid DRL method that jointly performs AP clustering and optimization of the beamforming vectors. Numerical and simulation results are presented in Section VI\nbefore the paper is concluded in Section VII. \n\n\\textbf{Notations:} For a random variable (rv) $X$,  $F_X(x)$ and $f_X(x)$  represent cumulative distribution function (CDF) and probability density function (PDF), respectively.  $\\mathbb P ( \\cdot )$ and $\\mathbb E[\\cdot ]$ denote probability and  expectation. For a given matrix $\\bm{A}\\in \\mathbb{C}^{M\\times N}$, $A^H$ represents the Hermitian transpose of $\\bm{A}$. A PDF expression of Nakagami-$\\mathcal{M}$ rv is given by $f_X(x)=\\frac{2\\mathcal{M}^{\\mathcal{M}}}{\\Gamma(\\mathcal{M}) {\\Omega}^{\\mathcal{M}}}x^{2\\mathcal{M}-1}e^{\\frac{\\mathcal{M}}{\\Omega}x^2}$ while a Gamma  rv is  denoted by $X \\thicksim  \\mathcal G (\\alpha, \\beta)$, with PDF as $f_{X}(x) = \\frac{\\beta^\\alpha }{\\Gamma(\\alpha)} x^{\\alpha -1} e^{- \\beta x },  \\quad x >0$, where $ \\beta >0$, $ \\alpha \\geq  1 $, and $ \\Gamma(z)$ is the Euler's Gamma function. The base of $\\log(x) $ is  $2$. \n\\section{Static Cell-Free Network Architecture}\n\n\\subsection{Network Model}\nWe consider an uplink  network with $M$ single-antenna APs and $K$ single-antenna UEs that are located at fixed locations within a certain coverage area (Fig.~\\ref{System_Model}).\n\\begin{figure}[htb]\n\t\t\\centering\n\t\t\\includegraphics[height=6cm, width=6cm]{System_Model_Centr.jpg}\n\t\n\t\t\\caption{Static cell-free network model.}\\label{System_Model}\n\t\\end{figure}\nAll APs are connected to each other through backhaul links to form a static cell-free network architecture \\cite{Cell_Less_1}. \nThis network setup enables the distributed APs to collaborate simultaneously to serve all users within the network coverage area.\nThe network is equipped with a CPU pool that performs the last-stage processing tasks to detect\/decode signals from every user.\t\n\nThe channel state information (CSI), which will be required at the CPU for signal detection for individual UEs, is assumed to be achieved through training the pilot sequences that are not completely orthogonal.\nWe assume that the channel gain between the $k\\text{-th}$ user and the $m\\text{-th}$ AP follows the following probabilistic model:\n\\begin{equation}\\label{Channel_Model}\n    g_{mk}={L}_{mk}^{-\\kappa}\\,h_{mk},\n\\end{equation}\nwhere $L_{mk}=||d_{mk}||$ is the Euclidean distance between the $k\\text{-th}$ user and the $m\\text{-th}$ AP, $\\kappa$ is the path-loss exponent ($\\kappa\\geq 2$) and $h_{mk}$ is the small-scale channel fading gain between the $k\\text{-th}$ UE and the $m\\text{-th}$ AP. \nWe assume that $h_{mk}$ follows Nakagami-$\\mathcal{M}_{mk}$ distribution with spreading and shape parameters $\\mathcal{M}_{mk}$ and $\\Omega_{mk}$, respectively, i.e. $|h_{mk}|^2\\thicksim \\mathcal{G}(\\alpha_{mk},\\beta_{mk})$, where $\\alpha_{mk}=\\mathcal{M}_{mk}$ and $\\beta_{mk}=\\frac{\\mathcal{M}_{mk}}{\\Omega_{mk}}$. \nFor simplicity of analysis, we assume that $L_{mk}^{-\\kappa} (\\forall~m$ and $k$) is known. We have $|g_{mk}|^2\\thicksim \\mathcal{G}(\\alpha_{mk},\\beta_{mk}\/L_{mk}^{-2\\kappa})$.\nWe assume that $g_{mk}(\\forall m=1, \\dots, M$ and $k=1, \\dots, K$) belong to a set of independent but not identically distributed (i.n.d) rvs. \n\\subsection{Uplink Network Training (CSI Acquisition)}\nUnder the assumption that large-scale fading gain ($L_{ {m}k}^{-\\kappa}$) for all users is known, the aim of network training is to estimate the small-scale fading component of the overall channel gain ($h_{ {m}k}$). \nThis can be achieved by assigning different pilot sequences for every user denoted by ${\\bm{\\varphi}}_{ {m} k}=\\left[\\varphi_{mk, 1} \\dots \\varphi_{mk, \\tau_p} \\right]^H$ such that $||{\\bm{\\varphi}}_{ {m} k}||^2=1$, where $\\tau_p\\leq \\tau_c$ is the length of the length of pilot training sequence (in samples), which is less than or equal the channel coherence time ($\\tau_c$). Accordingly, the received pilot vector at $m$-th AP is given by \n\\begin{equation}\n\\label{Y_Pilot_Vector}\n\\bm{y}_{\\textit{p}, {m} }=\\sum_{k=1}^{K}\\sqrt{\\tau_p \\rho_k}g_{ {m} k}\\bm{\\varphi}_{mk}+\\bm{\\eta}_{ {m} },\n\\end{equation}\nwhere $\\rho_k$ is the normalized transmitted power for each symbol of the $k$-th user's pilot vector and $\\bm{\\eta}_m\\in \\mathbb{C}^{\\tau_p\\times 1}$ is the additive white Gaussian noise (AWGN) vector related to pilot symbols with independent and identically distributed (i.i.d) rvs, i.e. ${\\eta}_{i,m}\\thicksim \\mathcal{CG}\\left(\\mu_{i,m}, \\sigma_{i,m}^2\/2\\right), \\forall i=1, \\dots, \\tau_p$.\nThe goal is to find the best estimate of $g_{ {m} k}$ (denoted by $ \\hat{g}_{ {m} k}$) given the vector of observations ($\\bm{y}_{\\textit{p}, {m} k}$). \nThis can be optimally achieved by using the maximum {\\em a posteriori} decision rule (MAP).\nThe Bayesian estimator of $g_{ {m} k}$ is found to be identical to that of the minimum mean square method (MMSE) \\cite{Statistical_Signal_Processing, MMSE_1}.\nAccordingly, the best estimate of $g_{ {m} k}$ can be expressed as \\cite{Cell_Less_2}\n\\begin{dmath}\n    \\label{g_Estimate_1}\n     \\hat{g}_{ {m} k}=\\frac{\\sqrt{\\tau_p \\rho_k}L_{ {m} k}^{-\\kappa}}{\\rho_c\\sum_{l=1, l\\neq k}^{K}\\rho_kL^{-\\kappa}_{ {m} l}|\\bm{\\varphi}_{mk}^H\\bm{\\varphi}_{ml}|^2+1}\\bm{\\varphi}^H_{mk}\\bm{y}_{\\text{p}, {m} }\n     \\\\\n     = \\mathcal{E}_{mk}\\sqrt{\\tau_p\\rho_k}g_{mk}+\\sum_{l=1, l\\neq k}^K \\mathcal{E}_{mk}\\sqrt{\\tau_p\\rho_k}|\\bm{\\varphi}^H_{mk} \\bm{\\varphi}_{ml}|g_{ml}+\\mathcal{E}_{mk}|\\bm{\\varphi}^H_{mk} \\bm{\\eta}_m|,\n\\end{dmath}\nwhere $\\mathcal{E}_{mk}=\\frac{\\sqrt{\\tau_p \\rho_k}L_{ {m} k}^{-\\kappa}}{\\rho_c\\sum_{l=1, l\\neq k}^{K}\\rho_kL^{-\\kappa}_{ {m} l}|\\bm{\\varphi}_{mk}^H\\bm{\\varphi}_{ml}|^2+1}$. \nNote that, if all users assigned a set of mutually orthogonal pilot sequences (i.e. $|\\bm{\\varphi}^H_{mk} \\bm{\\varphi}_{ml}|=0, k\\neq l$), the estimated small-scale channel fading in (\\ref{g_Estimate_1}) reduces to a scaled version of the exact fading gain plus a relatively small AWGN noise portion.\nHowever, for certain applications (e.g. mMTC applications) and due to length limitations of $\\tau_p$, non-orthogonal pilot signals has to be used among some active users.\n\n\\subsection{Outage Performance}\n\\subsubsection{Distribution of Users' SINR}\nIn the static cell-free uplink network with $M$ APs and $K$ users, the distributed APs receive signals from all users within their coverage area and then forward the baseband signal components through the fronthaul links to the CPU pool. The CPU performs the detection of each user's signal separately. This process can be implemented by optimizing a user's transmission power and\/or the beamforming vectors at every user's detector.\nAt a certain CPU detector module, the received signal used in the detection of the $k\\text{-th}$ user's component is given by\n\\begin{dmath}\n y_k=\\sum_{m=1}^Mw_{mk}\\left[\\sum_{l=1}^K\\hat{g}_{ml}\\sqrt{p_l}x_l+\\tilde{\\eta}_m\\right]\n    =\n    \\underbrace{\n    \\sqrt{\\tau_p\\rho_kp_k}x_k\\sum_{m=1}^Mw_{mk}\\mathcal{E}_{mk}g_{mk}}\n    _{\\text{Desired Signal}}\n    +\n    \\sum_{m=1}^M\n    w_{mk}\n     \\left[\n     \\underbrace{    \\sum_{l=1, l\\neq k}^K\\sqrt{\\tau_p\\rho_lp_l}x_l\\mathcal{E}_{ml}g_{ml}\n    }\n    _{\\text{Inter-User Interference}}\n    \\\\\n    +\n    \\underbrace{\n    \\sum_{l=1, l\\neq k}^K\\sqrt{\\tau_p\\rho_kp_k}x_k\\mathcal{E}_{mk}|\\bm{\\varphi}_{mk}^H\\bm{\\varphi}_{ml}|g_{ml}\n    +\n    \\sum_{l=1, l\\neq k}^K \\sum_{\\tilde{l}=1, \\tilde{l}\\neq l}^K\\sqrt{\\tau_p\\rho_lp_l}x_l\\mathcal{E}_{ml}|\\bm{\\varphi}_{ml}^H\\bm{\\varphi}_{m\\tilde{l}}|g_{m\\tilde{l}}}\n    _{\\text{Non-Orthogonal Pilots' Related Estimation error}}\n        \\right]\n    \\\\\n    +\n    \\sum_{m=1}^M\n    w_{mk}\n    \\left[\n          \\underbrace{ \\sqrt{p_k}x_k|\\bm{\\varphi}_{mk}^H\\bm{\\eta}_m|\n    +\n    \\sum_{l=1, l\\neq k}^K\\sqrt{p_l}x_l|\\bm{\\varphi}_{ml}^H\\bm{\\eta}_m|\n    }\n    _{\\text{AWGN's Related Estimation Error}}\n    +\n              \\underbrace{  \\tilde{\\eta}_m}\n    _{\\text{AWGN Component}}\n    \\right]\n    ,\\label{Received_Signal_1} \n\\end{dmath}\nwhere $w_{mk}$ is the $m$-th element of the $k$-th user's beamforming vector such that $0\\leq w_{mk}\\leq 1$, $p_k$ is the uplink transmission power of the $k$-th user such that $0\\leq p_k\\leq P_k$, where $P_k$ is the maximum allowable transmission power for the $k\\text{-th}$ user, $x_k$ is the transmitted symbol from the $k\\text{-th}$ user such that $\\mathbb{E}[|x_k|^2]=1$ and ${\\tilde{\\eta}}_m$ is the AWGN at the input of the $m$-th AP such that $\\tilde{\\eta}_m\\thicksim \\mathcal{CG}\\left(\\tilde{\\mu}_m, \\tilde{\\sigma}_m^2\/2\\right)$. \nWe assume that $\\tilde{{\\eta}}_m, \\forall m=1, \\dots, M$ are from a set of i.i.d rvs.\n\nAccordingly, the SINR of the $k\\text{-th}$ user related to $y_k$ at (\\ref{Received_Signal_1}) can be expressed as\n\\begin{dmath}\n    \\gamma_k=\n    \\frac\n    {\n    \\sum_{m=1}^M|\\tilde{g}_{mk}|^2\n    }\n    {\n    \\sum_{m=1}^M\n    \\left[\n    \\sum_{l=1, l\\neq k}^K|\\tilde{g}_{ml}|^2\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    |\\tilde{g}_{m{\\ddot{l}}}|^2\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K|\\tilde{g}_{m{\\Breve{l}}}|^2\n    \\right]+1\n    }\n\n\n,\\label{SINR_1}\n\\end{dmath}\nwhere $|\\tilde{g}_{mk}|^2\\thicksim \\mathcal{G}\\left(\\tilde{\\alpha}_{mk},\\tilde{\\beta}_{mk} \\right)$, $|\\tilde{g}_{ml}|^2\\thicksim \\mathcal{G}\\left(\\tilde{\\alpha}_{ml},\\tilde{\\beta}_{ml} \\right)$, $|\\tilde{g}_{m{\\dot{l}}}|^2\\thicksim \\mathcal{G}\\left(\\tilde{\\alpha}_{m\\dot{l}},\\tilde{\\beta}_{m\\dot{l}} \\right)$,  $|\\tilde{g}_{m\\ddot{l}}|^2\\thicksim \\mathcal{G}\\left(\\tilde{\\alpha}_{m\\ddot{l}},\\tilde{\\beta}_{m\\ddot{l}} \\right)$ and \n$|\\tilde{g}_{m\\Breve{l}}|^2\\thicksim \\mathcal{G}\\left(\\tilde{\\alpha}_{m\\Breve{l}},\\tilde{\\beta}_{m\\Breve{l}} \\right)$\n with\n $\\tilde{\\alpha}_{mk}=\\mathcal{M}_{mk}$, \n $\\tilde{\\beta}_{mk}=\\frac{\\mathcal{M}_{mk}\\dot{\\sigma}_{mk}L_{mk}^{2\\kappa}}{\\Omega_{mk}    w_{mk}^2\\tau_p\\rho_kp_k\\mathcal{E}_{mk}^2}$, \n $\\tilde{\\alpha}_{ml}=\\mathcal{M}_{ml}$,  \n $\\tilde{\\beta}_{ml}=\\frac{\\mathcal{M}_{ml}\\dot{\\sigma}_{mk}L_{ml}^{2\\kappa} }{\\Omega_{ml}  w_{mk}^2\\tau_p\\rho_kp_k\\mathcal{E}_{mk}^2|\\bm{\\varphi}_{mk}^H\\bm{\\varphi}_{ml}|^2}$, \n $\\tilde{\\alpha}_{m\\ddot{l}}=\\mathcal{M}_{m\\ddot{l}}$,  \n $\\tilde{\\beta}_{m\\ddot{l}}=\\frac{\\mathcal{M}_{m\\ddot{l}}\\dot{\\sigma}_{mk}L_{m\\ddot{l}}^{2\\kappa}}{\\Omega_{m\\ddot{l}}   w_{mk}^2\\tau_p\\rho_{\\dot{l}}p_{\\dot{l}}\\mathcal{E}_{m{\\dot{l}}}^2|\\bm{\\varphi}_{m\\dot{l}}^H\\bm{\\varphi}_{m\\ddot{l}}|^2}$,\n $\\tilde{\\alpha}_{m\\Breve{l}}=\\mathcal{M}_{m\\Breve{l}}$,  \n $\\tilde{\\beta}_{m\\Breve{l}}=\\frac{\\mathcal{M}_{m\\Breve{l}}\\dot{\\sigma}_{mk}L_{m\\Breve{l}}^{2\\kappa}}{\\Omega_{m\\Breve{l}}   w_{mk}^2\\tau_p\\rho_{\\Breve{l}}p_{\\dot{l}}\\mathcal{E}_{m{\\dot{l}}}^2}$.\\\\ \n $\\dot{\\sigma}_{mk}=\\sum_{m=1}^M\\left[\\sum_{t=1}^{\\tau_p}\\frac{w_{mk}^2\\sigma_m^2}{2}\\left(p_k\\varphi_{mk,t}+\\sum_{l=1, l\\neq k}^Kp_l\\varphi_{ml,t}\\right)+ w_{mk}^2\\tilde{\\sigma}^2_m\/2\\right]$.\n \n \nNote that the cross-product terms between channel gains are averaged to zero due to the i.i.d assumption. \nFurthermore, the AWGN term in the denominator (denoted by $\\dot{\\sigma}_m$) is replaced by the PSD (variance) of the sum of $M$ i.i.d rvs. \nOur goal is to first find a closed-form expression for the PDF and CDF of $\\gamma_k$ and then utilize the derived expressions in deriving some fundamental performance limits for the static cell-free network. \\textbf{Lemma~\\ref{l_pdfs}} gives an accurate approximation for the PDF of $\\gamma_k$ in (\\ref{SINR_1}).\n\t\\begin{lemma} \\label{l_pdfs}\n    Let \n    $ |\\tilde{g}_{mk}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{mk}} {\\tilde{\\beta}_{mk}} $\n    , \n    $ |\\tilde{g}_{ml}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{ml}} {\\tilde{\\beta}_{ml}} $\n    , \n    $ |\\tilde{g}_{m\\ddot{l}}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{m\\ddot{l}}} {\\tilde{\\beta}_{m\\ddot{l}}} $ \n    and \n    $ |\\tilde{g}_{m\\Breve{l}}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{m\\Breve{l}}} {\\tilde{\\beta}_{m\\Breve{l}}} $\n    where \n    $|\\tilde{g}_{mk}|^2$\n    ,\n    $|\\tilde{g}_{ml}|^2$\n    ,\n    $|\\tilde{g}_{m\\ddot{l}}|^2$\n    and\n    $|\\tilde{g}_{m\\Breve{l}}|^2$ are independent rvs with $m=1, \\dots, M$ and $k=1, \\dots, K$.  The PDF of  $ \\gamma_k$ in (\\ref{SINR_1}) with relatively large $M$ and $K$ is \n    \\begin{dmath}\tf_{\\gamma_k}\\left(\\gamma\\right)=\n    \\frac{\\Dot{\\beta}_{mk}^{\\Dot{\\alpha}_{mk}}\\Dot{\\beta}_{mk'}^{\\Dot{\\alpha}_{mk'}}e^{-\\frac{\\Dot{\\beta}_{mk}\\gamma-\\Dot{\\beta}_{mk'}}{2}}}\n    {\\Gamma\\left(\\Dot{\\alpha}_{mk}\\right)\\left(\\Dot{\\beta}_{mk}\\gamma+\\Dot{\\beta}_{mk'} \\right)^{\\frac{\\Dot{\\alpha}_{mk}+\\Dot{\\alpha}_{ml}+1}{2}}}\n    W_{\\frac{\\Dot{\\alpha}_{mk}-\\Dot{\\alpha}_{mk'}+1}{2},\\frac{-\\Dot{\\alpha}_{mk}-\\Dot{\\alpha}_{mk'}}{2}}\\left(\\Dot{\\beta}_{mk}\\gamma+\\Dot{\\beta}_{mk'} \\right) \n    ,\\label{PDF_Centralized}\t\t\\end{dmath}\n\t\twhere  \n        $\\Dot{\\alpha}_{mk}$, $\\Dot{\\alpha}_{mk'}$, $\\Dot{\\beta}_{mk}$ and $\\Dot{\\beta}_{mk'}$ are defined in \\textbf{Appendix A} and  $W_{\\lambda,\\mu}(x)$ is the  Whittaker function with parameters $\\lambda$ and $\\mu$ \\cite[Eq. 9.222.1]{gradshteyn2000}.\n\t\\end{lemma}\n    \\begin{proof}\nSee \\textbf{Appendix A}.\n\\end{proof}\n\n\\subsubsection{SINR Outage}\nA receiver is in an outage if the SINR of the received signal falls bellow a certain predefined threshold value (denoted by $\\gamma_{\\text{th}}$). \n\\textbf{Theorem \\ref{Theorem_Outage}} below gives an accurate approximation for the average probability of outage of the $k\n$-th user.\n\t\\begin{theorem}\\label{Theorem_Outage} \\label{l_cdfs}\n    If \n    $ |\\tilde{g}_{mk}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{mk}} {\\tilde{\\beta}_{mk}} $\n    , \n    $ |\\tilde{g}_{ml}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{ml}} {\\tilde{\\beta}_{ml}} $\n    , \n    $ |\\tilde{g}_{m\\ddot{l}}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{m\\ddot{l}}} {\\tilde{\\beta}_{m\\ddot{l}}} $ \n    and \n    $ |\\tilde{g}_{m\\Breve{l}}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{m\\Breve{l}}} {\\tilde{\\beta}_{m\\Breve{l}}} $\n    where \n    $|\\tilde{g}_{mk}|^2$\n    ,\n    $|\\tilde{g}_{ml}|^2$\n    ,\n    $|\\tilde{g}_{m\\ddot{l}}|^2$\n    and\n    $|\\tilde{g}_{m\\Breve{l}}|^2$ are independent rvs with $m=1, \\dots, M$ and $k=1, \\dots, K$.  The probability of outage of the $k$-th user in a static cell-free network is \n    \\begin{dmath}\tP_{\\text{out}}^{(k)}=1-\\frac{\\left(\\frac{\\Dot{\\beta}_{mk'}}{\\gamma_{\\text{th}}\\Dot{\\beta}_{mk}} \\right)^{\\Dot{\\alpha}_{mk'}}\\Gamma\\left(\\Dot{\\alpha}_{mk}+\\Dot{\\alpha}_{mk'} \\right)}{\\Dot{\\alpha}_{mk'}\\Gamma\\left(\\Dot{\\alpha}_{mk} \\right)\\Gamma\\left(\\Dot{\\alpha}_{mk'} \\right)}\n    {}_{2}F_{1}\\left(\\Dot{\\alpha}_{mk'},\\Dot{\\alpha}_{mk}+\\Dot{\\alpha}_{mk'};1+\\Dot{\\alpha}_{mk'};-\\frac{\\Dot{\\beta}_{mk'}}{\\Dot{\\beta}_{mk}} \\frac{1}{\\gamma_{\\text{th}}}\\right)\n    ,\\label{e_outage_1}\t\t\\end{dmath}\n\t\twhere  \n        $\\gamma_{\\text{th}}$ is the SINR threshold value and  ${}_2F_{1}(.)$ is the  Gauss hypergeometric function function \\cite[Eq. 9.11.1]{gradshteyn2000}.\n\t\\end{theorem}\n    \\begin{proof}\nSee \\textbf{Appendix B}.   \n\\end{proof}\n\nOne major drawback of the static cell-free network architecture is the requirement of joint processing of signals related to all active users which will require the CSI for the links from all users to all APs. This becomes very challenging for massive numbers of active users and APs in the network. In the following section, we propose a dynamic cell-free network model for uplink access that reduces the complexity of joint processing, compared to that of static cell-free network model, in order to serve a massive number of users. \n\n\\section{Dynamic Cell-Free Architecture}\n\n\n\n\\subsection{Network Model and Assumptions}\\label{s_sys}\nIn the proposed dynamic cell-free network model (Fig.~\\ref{System_Model_2}), the  APs in the network are partitioned among a set of $\\tilde{M}$ subgroups (known as clusters) in which every cluster consists of $\\mathcal{N}_{\\tilde{m}}$ APs such that $1\\leq \\mathcal{N}_{\\tilde{m}}\\leq M-(\\tilde{M}-1)$ \\footnote{Note that when $\\tilde{M}=M$, every cluster will contain a single AP and this falls back to the conventional static cell-free network model.}.\nAdditionally, the $\\tilde{m}$-th cluster, where $\\tilde{m}=1, \\dots, \\tilde{M}$,  acts as an single virtual AP with a $\\mathcal{N}_{\\tilde{m}}$-antenna DAS that combine the overall received signals using a predefined combining technique (such as maximal ratio combining [MRC], equal gain combining [EGC], selection combining [SC], etc.)\\cite{simonmk05}.\nThe clustering of APs is assumed to be performed in each time slot (or transmission interval) based on the current CSI of the network.\nSimilar to the static cell-free network model, we assume that channel gain between the $k$-th user and the ${n}_{\\tilde{m}}$-th antenna at the $\\tilde{m}$-th cluster (the virtual AP) follows the following distribution:\n\\begin{equation}\n    g_{\\tilde{m}n_{\\tilde{m}}k}={L}_{\\tilde{m}n_{\\tilde{m}}k}^{-\\kappa}h_{\\tilde{m}n_{\\tilde{m}}k},\n\\end{equation}\nwhere $L_{\\tilde{m}n_{\\tilde{m}}k}=||d_{\\tilde{m}n_{\\tilde{m}}k}||$ is the Euclidean distance between the $k\\text{-th}$ user and the $n_{\\tilde{m}}$-th antenna at the $\\tilde{m}$-th cluster, $\\kappa$ is the path-loss exponent with a value depends on the propagation environment with $\\kappa\\geq 2$ and $h_{\\tilde{m}n_{\\tilde{m}}k}$ is the small-scale channel fading between the $k\\text{-th}$ user and the $n_{\\tilde{m}}$-th antenna at the $\\tilde{m}$-th cluster. \nSimilarly,  $h_{\\tilde{m}n_{\\tilde{m}}k}$ is assumed to follow a Nakagami-$\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}k}$ distribution, i.e. $|h_{\\tilde{m}n_{\\tilde{m}}k}|^2\\thicksim \\mathcal{G}(\\alpha_{\\tilde{m}n_{\\tilde{m}}k},\\beta_{\\tilde{m}n_{\\tilde{m}}k})$, where $\\alpha_{\\tilde{m}n_{\\tilde{m}}k}=\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}k}$ represents the shape parameter of Nakagami-$\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}k}$ fading and $\\beta_{\\tilde{m}n_{\\tilde{m}}k}=\\frac{\\mathcal{M}_{mnk}}{\\Omega_{\\tilde{m}n_{\\tilde{m}}k}}$ is defined as the shape parameter normalized by spreading parameter of Nakagami-$\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}k}$ fading. \nAgain, for simplicity of analysis, we assume that $L_{\\tilde{m}n_{\\tilde{m}}k}^{-\\kappa}, \\forall~\\tilde{m},n_{\\tilde{m}},k$ is known. Accordingly, we have $|g_{\\tilde{m}n_{\\tilde{m}}k}|^2\\thicksim \\mathcal{G}(\\alpha_{\\tilde{m}n_{\\tilde{m}}k},\\beta_{\\tilde{m}n_{\\tilde{m}}k}')$, where $\\beta_{\\tilde{m}n_{\\tilde{m}}k}'=\\beta_{\\tilde{m}n_{\\tilde{m}}k}\/L_{\\tilde{m}n_{\\tilde{m}}k}^{-2\\kappa}$. \nLet us denote $\\bm{\\mathcal{C}}=\\{\\mathcal{C}_1 \\dots \\mathcal{C}_N \\}$ as the set of all possible clustering configurations of APs such that every cluster contains at least one  AP. \nAs an example, with $M=8$ and $\\tilde{M}=3$, one possible set is\n\\[\n\\mathcal{C}_n=\\{ \\underbrace{ \\{ AP_3, AP_6, AP_8 \\}}_{\\mathcal{N}_{\\tilde{m}=1}=3}, \\dots, \\underbrace{\\{AP_2\\}}_{\\mathcal{N}_{\\tilde{m}=2}=1},\\underbrace{\\{ \\overbrace{AP_1}^{n_{\\tilde{m}=3}=1}, AP_4, AP_5, AP_7 \\}}_{\\mathcal{N}_{\\tilde{m}=\\tilde{M}=3}=4}  \\}.\n\\]\n\n\\begin{figure}[htb]\n\t\t\\centering\n\t\t\\includegraphics[height=6cm, width=6.5cm]{System_Model_Dist.jpg}\n\t\n\t\t\\caption{Dynamic cell-free network model.}\\label{System_Model_2}\n\t\\end{figure}\nNote that in this network setup, the CSI and the instantaneous clustering information will be only required at the CPU for decoding the signals from the UEs. \n\n\\subsection{Outage Performance}\nFor the proposed dynamic cell-free model, the received signal used in the detection of the $k$-th user's component is given by  \n\\begin{dmath}\n\\scriptsize\ny_k=\n\\sum_{\\tilde{m}=1}^{\\tilde{M}}w_{\\tilde{m}k}\n\\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{\\tilde{m}n_{\\tilde{m}}k}\n\\left[\\sum_{l=1}^K\\hat{g}_{\\tilde{m}n_{\\tilde{m}}l}\\sqrt{p_l}x_l+{\\tilde{\\eta}}_{\\tilde{m}n_{\\tilde{m}}}\\right]\n  =\n    \\underbrace{\n    \\sqrt{\\tau_p\\rho_kp_k}x_k\\sum_{\\tilde{m}=1}^{\\tilde{M}}w_{\\tilde{m}k}\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}k}\n    \\sum_{n_{n_{\\tilde{m}}}}^{\\mathcal{N}_{n_{\\tilde{m}}}}G_{{\\tilde{m}}n_{\\tilde{m}}k}\n    g_{\\tilde{m}n_{\\tilde{m}}k}}\n    _{\\text{Desired Signal}}+\n    \\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    w_{\\tilde{m}k}\n    \\sum_{n_{{\\tilde{m}}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{{\\tilde{m}}n_{\\tilde{m}}k}\n     \\left[\n     \\underbrace{    \\sum_{l=1, l\\neq k}^K\\sqrt{\\tau_p\\rho_lp_l}x_l\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}l}g_{\\tilde{m}n_{\\tilde{m}}l}\n    }\n    _{\\text{Inter-User Interference}}\n    \\\\\n    +\n    \\underbrace{\n    \\sum_{l=1, l\\neq k}^K\\sqrt{\\tau_p\\rho_kp_k}x_k\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}k}|\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}k}^H\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}l}|g_{\\tilde{m}n_{\\tilde{m}}l}\n    +\n    \\sum_{l=1, l\\neq k}^K \\sum_{\\tilde{l}=1, \\tilde{l}\\neq l}^K\\sqrt{\\tau_p\\rho_lp_l}x_l\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}l}|\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}l}^H\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}{\\tilde{l}}}|g_{\\tilde{m}n_{\\tilde{m}}\\tilde{l}}}\n    _{\\text{Non-Orthogonal Pilots' Related Estimation error}}\n        \\right]\n    \\\\\n    +\n    \\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    w_{\\tilde{m}k}\n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{{\\tilde{m}}n_{\\tilde{m}}k}\n    \\left[\n          \\underbrace{ \\sqrt{p_k}x_k|\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}k}^H\\bm{\\eta}_{\\tilde{m}n_{\\tilde{m}}}|\n    +\n    \\sum_{l=1, l\\neq k}^K\\sqrt{p_l}x_l|\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}l}^H\\bm{\\eta}_{\\tilde{m}n_{\\tilde{m}}}|\n    }\n    _{\\text{AWGN's Related Estimation Error}}\n    +\n              \\underbrace{  \\tilde{\\eta}_{\\tilde{m}n_{\\tilde{m}}}}\n    _{\\text{AWGN}}\n    \\right], \\label{Received_Signal_2} \n\\end{dmath}\nwhere $w_{\\tilde{m}k}$ is the $\\tilde{m}$-th element of the $k$-th user's beamforming vector such that $0\\leq w_{{\\tilde{m}}k}\\leq 1$, \n$G_{\\tilde{m}n_{\\tilde{m}}k}$ is a design gain parameter for the $\\mathcal{N}_{\\tilde{m}}$-antenna DAS at the $\\tilde{m}$-th virtual AP,\n$p_{k}$ is the uplink transmission power of the $k$-th user such that $0\\leq p_k\\leq P_k$, where $P_k$ is the power budget of the $k\\text{-th}$ user, $x_k$ is the transmitted symbol from the $k\\text{-th}$ user such that $\\mathbb{E}[|x_k|^2]=1$, and $\\tilde{\\eta}_{\\tilde{m}n_{\\tilde{m}}}$ is the AWGN at the $m$-th AP with $\\tilde{\\eta}_{\\tilde{m}n_{\\tilde{m}}}\\thicksim \\mathcal{CG}\\left(\\tilde{\\mu}_{\\tilde{m}n_{\\tilde{m}}}, \\tilde{\\sigma}_{\\tilde{m}n_{\\tilde{m}}}^2\/2\\right)$.\nWe assume that $\\tilde{\\eta}_{\\tilde{m}n_{\\tilde{m}}}, \\forall \\{\\tilde{m}~ \\&~ n_{\\tilde{m}}\\}$ are from a set of i.n.d rvs. \n\nFor uplink network training, we follow a procedure similar to that in Section II.B. \nTherefore, we have ${\\bm{\\varphi}}_{\\tilde{m}n_{\\tilde{m}}k}=\\left[\\varphi_{{\\tilde{m}n_{\\tilde{m}}k}, 1} \\dots \\varphi_{{\\tilde{m}n_{\\tilde{m}}k}, \\tau_p} \\right]^H$  as the set of $\\tau_p$-dimensional training symbols such that $||{\\bm{\\varphi}}_{\\tilde{m}n_{\\tilde{m}}k}||^2=1$ and $\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}k}=\\frac{\\sqrt{\\tau_p \\rho_k}L_{\\tilde{m}n_{\\tilde{m}}k}^{-\\kappa}}{\\rho_c\\sum_{l=1, l\\neq k}^{K}\\rho_kL^{-\\kappa}_{\\tilde{m}n_{\\tilde{m}}l}|\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}k}^H\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}l}|^2+1}$ is the MMSE channel estimation constant. \nNote that for every cluster set ($\\mathcal{C}_n, n=1, \\dots, N$), different SINR values ($\\gamma_k$) will result  for different users. \n\n\n\nThe SINR $\\gamma_k$ of the $k$-th user related to $y_k$ in (\\ref{Received_Signal_2}) and under the $n$-th cluster can be expressed as   \n\\begin{dmath}\n   \\text{\\hspace{-3mm}} \\gamma_k^{\\{\\mathcal{C}_n\\}}=\n    \\frac\n    {\n    \\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    \\sum_{n_{\\bar{m}}=1}^{\\mathcal{N}_{\\bar{m}}}\n    |\\tilde{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2\n    }\n    {\n    \\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n    \\left[\n    \\sum_{l=1, l\\neq k}^K|\\tilde{g}_{\\tilde{m}n_{\\tilde{m}}l}|^2\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    |\\tilde{g}_{{\\tilde{m}n_{\\tilde{m}}}{\\ddot{l}}}|^2\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K|\\tilde{g}_{{\\tilde{m}n_{\\tilde{m}}}{\\Breve{l}}}|^2\n    \\right]+1\n    } \n,\\label{SINR_2}\n\\end{dmath}\n where\n    $ |\\tilde{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k}} {\\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}k}} $\n    , \n    $ |\\tilde{g}_{\\tilde{m}n_{\\tilde{m}}l}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}l}} {\\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}l}} $\n    , \n    $ |\\tilde{g}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}} {\\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}} $ \n    and\\\\ \n    $ |\\tilde{g}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}} {\\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}} $ \n with\n $\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k}=\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}k}$, \n \n $\\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}k}=\\frac{\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}k}\\dot{\\sigma}_{\\tilde{m}n_{\\tilde{m}}k}L_{\\tilde{m}n_{\\tilde{m}}k}^{2\\kappa}}{\\Omega_{\\tilde{m}n_{\\tilde{m}}k}    w_{mk}^2G_{\\tilde{m}n_{\\tilde{m}}k}\\tau_p\\rho_kp_k\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}k}^2}$, \n $\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}l}=\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}l}$,  \n \n $\\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}l}=\\frac{\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}l}\\dot{\\sigma}_{\\tilde{m}n_{\\tilde{m}}l}L_{\\tilde{m}n_{\\tilde{m}}l}^{2\\kappa} }{\\Omega_{\\tilde{m}n_{\\tilde{m}}l}  w_{mk}^2G_{\\tilde{m}n_{\\tilde{m}}k}\\tau_p\\rho_kp_k\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}k}^2|\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}k}^H\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}l}|^2}$, \n $\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}=\\mathcal{M}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}$, \n \n $\\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}=\\frac{\\mathcal{M}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}\\dot{\\sigma}_{{\\tilde{m}n_{\\tilde{m}}}k}L_{{{\\tilde{m}n_{\\tilde{m}}}}\\ddot{l}}^{2\\kappa}}{\\Omega_{{{\\tilde{m}n_{\\tilde{m}}}}\\ddot{l}}   w_{mk}^2G_{\\tilde{m}n_{\\tilde{m}}k}\\tau_p\\rho_{\\dot{l}}p_{\\dot{l}}\\mathcal{E}_{{{\\tilde{m}n_{\\tilde{m}}}}{\\dot{l}}}^2|\\bm{\\varphi}_{{{\\tilde{m}n_{\\tilde{m}}}}\\dot{l}}^H\\bm{\\varphi}_{{{\\tilde{m}n_{\\tilde{m}}}}\\ddot{l}}|^2}$,\n \n $\\tilde{\\alpha}_{{{\\tilde{m}n_{\\tilde{m}}}}\\Breve{l}}=\\mathcal{M}_{{{\\tilde{m}n_{\\tilde{m}}}}\\Breve{l}}$,  \n \n $\\tilde{\\beta}_{{{\\tilde{m}n_{\\tilde{m}}}}\\Breve{l}}=\\frac{\\mathcal{M}_{{{\\tilde{m}n_{\\tilde{m}}}}\\Breve{l}}\\dot{\\sigma}_{{\\tilde{m}n_{\\tilde{m}}}k}L_{{{\\tilde{m}n_{\\tilde{m}}}}\\Breve{l}}^{2\\kappa}}{\\Omega_{{{\\tilde{m}n_{\\tilde{m}}}}\\Breve{l}}   w_{mk}^2\\tau_p\\rho_{\\Breve{l}}p_{\\dot{l}}\\mathcal{E}_{{{\\tilde{m}n_{\\tilde{m}}}}{\\dot{l}}}^2}$, and\n \\\\\n $\\dot{\\sigma}_{\\tilde{m}n_{\\tilde{m}}k}=\\sum_{\\tilde{m}=1}^{\\tilde{M}}\n w_{\\tilde{m}k}^2\n \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{\\tilde{m}n_{\\tilde{m}}k}^2\n \\left[\\sum_{t=1}^{\\tau_p}\\frac{\\sigma_{\\tilde{m}n_{\\tilde{m}}}^2}{2}\\left(p_k\\varphi_{{\\tilde{m}n_{\\tilde{m}}k},t}+\\sum_{l=1, l\\neq k}^Kp_l\\varphi_{{\\tilde{m}n_{\\tilde{m}}l},t}\\right)+ \\tilde{\\sigma}^2_{\\tilde{m}n_{\\tilde{m}}}\/2\\right]$.\\\\\nNote that $G_{\\tilde{m}nk}$ is a one-time design parameter while $w_{\\tilde{m}k}$ is an optimization parameter.\nGenerally, different performance metrics of the dynamic cell-free network will depend on the clustering algorithm that assigns different APs to different clusters.\n\\textbf{Corollary \\ref{Dynamic_Performance}} below states the outage performance of a dynamic cell-free network under a random clustering scheme.\n\\begin{corollary} \\label{Dynamic_Performance}\n     If \n    $ |\\tilde{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k}} {\\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}k}} $\n    , ~\n    $ |\\tilde{g}_{\\tilde{m}n_{\\tilde{m}}l}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}l}} {\\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}l}} $\n    , ~\n    $ |\\tilde{g}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}} {\\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}} $ \n    and \n    $ |\\tilde{g}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}|^2 \\thicksim \\gd {\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}} {\\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}} $\n    where \n    $|\\tilde{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2$\n    ,\n    $|\\tilde{g}_{\\tilde{m}n_{\\tilde{m}}l}|^2$\n    ,\n    $|\\tilde{g}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}|^2$\n    and\n    $|\\tilde{g}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}|^2$ are independent rvs with ${\\tilde{m}}=1, \\dots, \\tilde{M}$, $n_{\\tilde{m}}=1, \\dots, \\mathcal{N}_{\\tilde{m}}$ such that $1\\leq \\mathcal{N}_{\\tilde{m}}\\leq M-(\\tilde{M}-1)$ and $\\{k, l, \\ddot{l}, \\Breve{l}\\}=1, \\dots, K$ such that $k\\neq l\\neq \\ddot{l}\\neq \\Breve{l}$. The  probability of outage $P_{\\text{out}}^{(k)}$ of the $k$-th user in the proposed dynamic cell-fee network under random AP clustering is identical to that given in \\textbf{Theorem 1} with \n    \\[\n    \\Dot{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k}=\n    \\frac{\n    \\left(\n    \\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    \\sum_{n_m}^{\\mathcal{N}_{\\tilde{m}}}\n    \\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k}\/ \\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}k}\n    \\right)^2\n    }\n    {\\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n    \\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k}\/\\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}k}^2\n    },~~~~~~\n    \\Dot{\\beta}_{\\tilde{m}n_{\\tilde{m}}k}=\\frac{\\sum_{\\tilde{m}=1}^{\\tilde{M}}\n\\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k}\/\\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}k}^2}{\\sum_{\\tilde{m}=1}^{\\tilde{M}}\n\\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k}\/ \\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}k}},\n\\]\n\\[\n\\Dot{\\alpha}_{\\tilde{m}n_{\\tilde{m}}k'}=\\frac{\n\\left(\n\\sum_{\\tilde{m}=1}^{\\tilde{M}}\n\\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n    \\left[\n    \\sum_{l=1, l\\neq k}^K\n    \\frac{    \\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}l}}{ \\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}l}}\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    \\frac{\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}}{     \\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}}\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}}{        \\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}}\n    \\right]\n\\right)^2\n}\n{\n\\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n    \\left[\n    \\sum_{l=1, l\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}l}}{    \\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}l}^2}\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    \\frac\n    {\n    \\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}\n    }\n    {\n    \\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}^2\n    }\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K\n    \\frac{\n    \\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}\n    }\n    {\n    \\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}^2\n    }\n    \\right]\n},\n\\]\n\\[\n\\Dot{\\beta}_{\\tilde{m}n_{\\tilde{m}}k'}=\n\\frac{\n\\sum_{\\tilde{m}=1}^{\\tilde{M}}\n\\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n    \\left[\n    \\sum_{l=1, l\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}l}}{    \\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}l}^2}\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    \\frac{\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}}{\n    \\tilde{\\beta}_{m\\ddot{l}}^2}\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K\n    \\frac{    \\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}}{\\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}^2}\n    \\right]\n}\n{\n \\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n    \\left[\n    \\sum_{l=1, l\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{\\tilde{m}n_{\\tilde{m}}l}}{     \\tilde{\\beta}_{\\tilde{m}n_{\\tilde{m}}l}}\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    \\frac{\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}\n}}{ \\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}}\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}\n}}{\\tilde{\\beta}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}}\n    \\right]\n}\n.\n\\]\n\t\\end{corollary}\n    \\begin{proof}\nThis can be easily concluded due to the fact that equations (\\ref{SINR_1}) and (\\ref{SINR_2}) are different from each other only by the scaling parameters of the rvs.\n\\end{proof}\nNote that using the AP clustering scheme with DAS at every virtual AP, there will be no requirements for CSI from other virtual APs. Additionally, a significant reduction of the lengths of the beamforming vectors will be achieved with a direct positive effect on processing speed and complexity  for the beamforming optimization.\nIt is important to mention that the dynamic cell-free network model is a generalization of the static cell-free network model. \nThis can be inferred by setting $\\tilde{M}=M$ which will result in $\\mathcal{N}_{\\tilde{m}}=1,~\\forall \\tilde{m}=1, \\dots, \\tilde{M}(\\equiv M)$. Here,  $G_{\\tilde{m}n_{\\tilde{m}k}}$ will be merged with $w_{\\tilde{m}n_{\\tilde{m}}k}(\\equiv w_{mk})$.  \n\\section{Clustering Technique and Beamforming Design}\nThis section formulates the  general problem of jointly performing the clustering of APs and designing the beamforming vectors.\nHowever, before presenting the problem formulation, we describe the SIC technique for  detection of the signals received by the CPU and also the diversity technique to combine the signals from multiple antennas. SIC can be achieved by utilizing the fact that the signals from all users are superimposed in the power domain, and with the proper beamforming design, the signal related to every user can have a distinct power level. \n\\subsection{SIC-Enabled Signal Detection}\nNote that in a cell-free network, signals from different users use the same time-frequency resource. The main idea of power-domain SIC technique is that a proper power control is performed such that signals from different users have a distinct power levels\\footnote{This is similar to traditional power-domain non-orthogonal multiple access (NOMA), where signals from different users are multiplexed in the same time-frequency resource and SIC techniques are used at the receivers.}. \nAdditionally, to detect signal from a certain user, the SIC unit first detects signals of users with higher power levels, subtracts their contributions from the overall received signal, and then detects the intended signal.\nSpecifically, for the proposed dynamic cell-free network, when detecting the signal from the $k$-th user, the uplink beamforming vectors are designed such that $p_1\\sum_{\\tilde{m}=1}^{\\tilde{M}}~\n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{\\tilde{m}n_{\\tilde{m}}k}^2\n    |\\hat{g}_{\\tilde{m}n_{\\tilde{m}}1}|^2 \n    \\leq \\dots \\leq\n    p_k\\sum_{\\tilde{m}=1}^{\\tilde{M}} \n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{\\tilde{m}n_{\\tilde{m}}k}^2\n    |\\hat{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2~~\n    \\leq\n    \\dots\n    \\leq\np_K\\sum_{\\tilde{m}=1}^{\\tilde{M}} \n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{\\tilde{m}n_{\\tilde{m}}k}^2\n    |\\hat{g}_{\\tilde{m}n_{\\tilde{m}}K}|^2$.\n    \nAccordingly, the SINR $\\gamma_k$ of the $k$-th user can be expressed as\n\\begin{dmath}\n    \\gamma_k^{\\{\\mathcal{C}_n\\}}=\n    \\frac\n    {\n    \\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    w_{\\tilde{m}k}^2\n    \\sum_{n_{\\bar{m}}=1}^{\\mathcal{N}_{\\bar{m}}}\n    | \\check{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2\n    }\n    {\n    \\sum_{\\tilde{m}=1}^{\\tilde{M}}\n    w_{\\tilde{m}k}^2\n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}\n    \\left[\n    \\sum_{l=1, l\\neq k}^K|\\check{g}_{\\tilde{m}n_{\\tilde{m}}l}|^2\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    |\\check{g}_{{\\tilde{m}n_{\\tilde{m}}}{\\ddot{l}}}|^2\n    +\n    \\sum_{\\Breve{l}=1}^{k-1}|\\check{g}_{{\\tilde{m}n_{\\tilde{m}}}{\\Breve{l}}}|^2\n    \\right]+1\n    } \n,\\label{SINR_3}\n\\end{dmath}\nwhere $|\\Breve{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2=\\frac{\n    G_{\\tilde{m}n_{\\tilde{m}}k}\\tau_p\\rho_kp_k\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}k}^2}\n    {   \n \\dot{\\sigma}_{\\tilde{m}n_{\\tilde{m}}k} }|{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2$, \n$|\\Breve{g}_{\\tilde{m}n_{\\tilde{m}}l}|^2\n=\\frac{\n   G_{\\tilde{m}n_{\\tilde{m}}k}\\tau_p\\rho_kp_k\\mathcal{E}_{\\tilde{m}n_{\\tilde{m}}k}^2|\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}k}^H\\bm{\\varphi}_{\\tilde{m}n_{\\tilde{m}}l}|^2}\n    {\n     \\dot{\\sigma}_{\\tilde{m}n_{\\tilde{m}}l} \n    }|{g}_{\\tilde{m}n_{\\tilde{m}}l}|^2\n$,\\\\\n$\n|\\Breve{g}_{{\\tilde{m}n_{\\tilde{m}}}{\\ddot{l}}}|^2\n=\n\\frac{\nG_{\\tilde{m}n_{\\tilde{m}}k}\\tau_p\\rho_{\\dot{l}}p_{\\dot{l}}\\mathcal{E}_{{{\\tilde{m}n_{\\tilde{m}}}}{\\dot{l}}}^2|\\bm{\\varphi}_{{{\\tilde{m}n_{\\tilde{m}}}}\\dot{l}}^H\\bm{\\varphi}_{{{\\tilde{m}n_{\\tilde{m}}}}\\ddot{l}}|^2\n}\n{\n \\dot{\\sigma}_{{\\tilde{m}n_{\\tilde{m}}}k}\n}|{g}_{{\\tilde{m}n_{\\tilde{m}}}{\\ddot{l}}}|^2\n$, \nand \n$\n|\\Breve{g}_{{\\tilde{m}n_{\\tilde{m}}}{\\Breve{l}}}|^2\n=\n\\frac{\n\\tau_p\\rho_{\\Breve{l}}p_{\\dot{l}}\\mathcal{E}_{{{\\tilde{m}n_{\\tilde{m}}}}{\\dot{l}}}^2\n}\n{\n \\dot{\\sigma}_{{\\tilde{m}n_{\\tilde{m}}}k}\n}|{g}_{{\\tilde{m}n_{\\tilde{m}}}{\\Breve{l}}}|^2\n$.\nNote that     \n    $ |{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2$\n    , \n    $ |{g}_{\\tilde{m}n_{\\tilde{m}}l}|^2 $\n    , \n    $ |{g}_{{\\tilde{m}n_{\\tilde{m}}}\\ddot{l}}|^2$, \n    and \n    $ |{g}_{{\\tilde{m}n_{\\tilde{m}}}\\Breve{l}}|^2$\nrepresent different order statistical rvs of gamma rvs which prevents the direct utilization of \\textit{Welch-Satterthwaite} approximation as in \\textbf{Appendix A}. \nThis is due to the introduction of random variable ordering required for SIC-based detection which requires the pre-ordering of different users' signals based on their different power levels. However, a closed-form expression for the outage probability can be derived using order statistics theory as in \\cite{GCoMP}.\nWe can also notice from (\\ref{SINR_3}) that the CSI estimation error contains portions from all users including those with higher power levels than that of the $k$-th user. This is because the SIC operation is also affected by channel estimation errors \\cite{SIC_IMPERFECT_CSI}.\n\\subsection{Diversity Combining Scheme}\nSeveral diversity techniques can be used at multi-antenna transmitters\/receivers to combat multipath fading. However, maximal ratio combining (MRC) and selection combining (SC) are the most common diversity combining techniques \\cite{simonmk05}.\nIn this work, we propose the use of a modified version of \\textit{Wiener-Hopf} multiple antennas combining scheme under the existence of co-channel interference \\cite{iet:\/content\/books\/ew\/sbew046e}.\nAccordingly, the gain factor $\\bm{G}_{\\tilde{m}k}=\\left[G_{\\tilde{m}1k} \\dots G_{\\tilde{m}{\\mathcal{N}}_{\\tilde{m}}k}\\right]^H$ that is used in combining signals at the $\\tilde{m}$-th AP for the detection of $k$-th user signal is defined as \\cite{iet:\/content\/books\/ew\/sbew046e}\n\\begin{equation}\n    \\bm{G}_{\\tilde{m}k}=\\bm{R}_{\\tilde{m}}^{-1}\\hat{\\bm g}_{\\tilde{m}k},\n\\end{equation}\nwhere $\\bm{R}_{\\tilde{m}}=\\text{Cov}\\left(\\sum_{l=1, l\\neq k}^K\\sqrt{{p}_l}\\bm{\\hat g}_{\\tilde{m}l}+\\tilde{\\bm \\eta}_{\\tilde{m}}\\right)$,   $\\bm{\\hat g}_{\\tilde{m}l}=\\left[{\\hat g}_{\\tilde{m}1l} \\dots {\\hat g}_{\\tilde{m}\\mathcal{N}_{\\tilde{m}} l} \\right]^H$, \nand\n$\\tilde{\\bm \\eta}_{\\tilde{m}}=\\left[\\tilde{ \\eta}_{\\tilde{m}1} \\dots \\tilde{ \\eta}_{\\tilde{m}\\mathcal{N}_{\\tilde{m}}} \\right]^H$.  \nFurthermore, the covariance matrix $\\bm{R}_{\\tilde{m}}$ conditioned on instantaneous CSI can be written as\n\\begin{equation}\n\\bm{R}_{\\tilde{m}}=\\sum_{l=1, l\\neq k}^K p_l\\hat{\\bm g}_{\\tilde{m}l}\\hat{\\bm g}_{\\tilde{m}l}^H+ \\bm{\\tilde{\\Sigma}}_{\\tilde{m}},\n\\end{equation}\nwhere $\\bm{\\tilde{\\Sigma}}_{\\tilde{m}}\\in \\mathbb{C}^{\\mathcal{N}_{\\tilde{m}}\\times \\mathcal{N}_{\\tilde{m}}}$ is a diagonal matrix with $n_{\\tilde{m}}$-th diagonal element given by ${\\tilde{\\Sigma}}_{\\tilde{m}n_{\\Tilde{m}}}=\\tilde{\\sigma}_{\\tilde{m}n_{\\tilde{m}}}^2\/2$.\nNote that, when $K=1$, we have $G_{\\tilde{m}k}=2\\hat{g}_{\\tilde{m}k}\/\\tilde{\\sigma}_{\\tilde{m}n_{\\tilde{m}}}$ which is identical to the MRC scheme that represents the optimal combining scheme under no IUI. \nAdditionally, note that we use the estimated values of CSI ($\\hat{g}_{\\tilde{m}n_{\\tilde{m}}k}$) without considering the CSI estimation error. However, several works in the literature discussed the optimal diversity combining scheme under imperfect CSI estimation \\cite{DAS_IMPERFECT_CSI_1,DAS_IMPERFECT_CSI_2}, which is however not investigated in this paper for brevity.  \n\\subsection{Formulation of Optimization Problem }\nFor the proposed dynamic cell-free network, the process of AP clustering needs to be performed jointly with beamforming design.\nThis can be achieved by forming a general optimization problem that jointly finds the optimal cluster set-beamforming vector pairs such that a certain objective is satisfied.\nTo achieve the best gain from SIC detection, we propose to order users based on their overall weighted power gain of the communication channel such that\n$p_1\\sum_{\\tilde{m}=1}^{\\tilde{M}}~\n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{\\tilde{m}n_{\\tilde{m}}k}^2\n    |\\hat{g}_{\\tilde{m}n_{\\tilde{m}}1}|^2 \n    \\leq \\dots \\leq\n    p_k\\sum_{\\tilde{m}=1}^{\\tilde{M}} \n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{\\tilde{m}n_{\\tilde{m}}k}^2\n    |\\hat{g}_{\\tilde{m}n_{\\tilde{m}}k}|^2~~\n    \\leq\n    \\dots\n    \\leq\np_K\\sum_{\\tilde{m}=1}^{\\tilde{M}} \n    \\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}}G_{\\tilde{m}n_{\\tilde{m}}k}^2\n    |\\hat{g}_{\\tilde{m}n_{\\tilde{m}}K}|^2$.\nAccordingly, the general problem of AP clustering and beamforming optimization can be formulated as\n\\begin{equation}\n\\begin{aligned}\n& ~\\textbf{P}_1: \\underset{n, \\bm{W}}{\\text{max}}~\n& \\text{\\hspace{-40mm}} \\sum_{k=1}^K\\log_2\\left(1+\\gamma_{k}^{\\{\\mathcal{C}_n\\}} \\right)\\\\\n& ~\\text{Subject to:} \\\\\n& ~\\textbf{C}_1: \\sum_{m=1}^{M} \\left(w_{m \\delta_l}^2-\\sum_{i=\\delta_l+1}^{l}w_{mi}^2 \\right)\\bar{\\gamma}_{\\tilde{m}n_{\\tilde{m}}l}\\geq P_{\\text{s}}, \\\\\n& ~  \\textbf{C}_2: 0\\leq w_{\\tilde{m}k}\\leq 1 ,\n\\\\\n&  ~  \\textbf{C}_3:||\\bm{w}_{k}||^2\\leq 1,\n\\\\\n&~~\\forall k,  \\forall~\\delta_l= 1, \\dots, l-1\\;\\mbox{and}\\; l=2, \\dots, K,\\\\\n\\end{aligned}\\label{OptimizationProb}\n\\end{equation}\nwhere $\\bar{\\gamma}_{\\tilde{m}n_{\\tilde{m}}l}=\\frac{2p_lG_{\\tilde{m}n_{\\tilde{m}l}}}{\\tilde{\\sigma}_{\\tilde{m}n_{\\tilde{m}}}^2}|\\hat{g}_{\\tilde{m}n_{\\tilde{m}l}}|^2$. \nThe constraints $\\textbf{C}_1$ refer to the set of $\\sum_{l=2}^K(l-1)=\\frac{K(K-1)}{2}$ conditions required for successful SIC operation with receiver sensitivity of $P_{\\text{s}}$. For maximizing the minimum rate, the objective function of the optimization problem will be replaced by $\\underset{k}{\\text{min}} \\log_2\\left(1+\\gamma_{k}^{\\{\\mathcal{C}_n\\}} \\right)$.\nNote that the receiver deals only with measured channel values (estimated) which include the estimation error as well as the AWGN part. However, the achieved SINR value after the SIC procedure will be decreased by pilot contamination components (as can be noticed from the second part of Eq. (\\ref{SINR_3})).  \n\nTo optimally solve the problem $\\textbf{P}_1$ in (\\ref{OptimizationProb}), \nwe need to simultaneously solve for $\\mathcal{C}_n$ and $\\bm{W}$.\nSpecifically, for every possible clustering configuration, there is a related optimal beamforming vector that maximizes the objective function.\n{\\em The globally optimal solution is then the one that gives the best performance among all clustering configurations and their corresponding optimized beamforming vectors ($\\bm{W}$).\nSuch a problem belongs to the set of complete non-deterministic polynomial-time hard (NP-hard) problems.\nFurthermore, given that a certain clustering configuration is selected, solving $\\textbf{P}_1$ w.r.t $\\bm{W}$ grows exponentially with $\\tilde{M}$ and\/or $K$ which makes the cell-free network model  impractical for a network with a massive number of devices communicate through a massive number of APs within a  geographic area.\nTherefore, for  practical implementation of a dynamic cell-free network, we design a novel hybrid DDPG-DDQN-based DRL system that produces the best clustering configuration and the beamforming vectors such that the objective function in $\\textbf{P}_1$ is maximized.}\n\n\n\\section{DRL Implementation of AP Clustering and Beamforming Design}\n\nIn this section, we introduce a novel DRL model that solves the general optimization problem (i.e. problem $\\textbf{P}_1$) of jointly optimizes the clustering of APs and the beamforming vectors. \n\n\\subsection{Theoretical Preliminaries}\nThe concept of reinforcement learning (RL) refers to the learning process of an agent interacting with its environment after receiving certain observations. The environment provides a reward to the agent for every interaction and\nthe RL agent aims to select the right action for the next interaction in order to maximize the discounted reward over a time horizon. \nThis problem can be formulated as a \\textit{Markov decision\nprocess} (MDP). An MDP is a tuple ($\\mathcal{\\bm S}$, $\\mathcal{A}$, $\\mathcal{P}$, $\\mathcal{R}$, $\\zeta$), where $\\mathcal{S}$ represents the states space which contains the set of $K$-dimensional states (with each state at time $t$ denoted by $\\bm{s}_t$), $\\mathcal{A}$ is the action space that contains a finite set of actions from which the agent can choose, $\\mathcal{P}$ : $\\mathcal{S}$ $\\times$ $\\mathcal{A}$ $\\times$ $\\mathcal{S}$ $\\rightarrow$ [0, 1] is a transition probability in which $\\mathcal{P}(\\bm{s}, a, \\bm{s}')$ defines the probability of observing state $\\bm{s}'$ after executing action $a$ in the state $\\bm{s}$, $\\mathcal{R}: \\mathcal{S} \\times \\mathcal{A} \\rightarrow \\mathbb{R}$ is the expected reward after being in state $\\bm{s}$ and taking action $a$, and $\\zeta$ $\\in [0, 1)$ is the discount factor.\nTo solve the MDP,  RL algorithms have been developed to learn and find a discrete value function or a ``policy\". Such a discretization can lead to lack of generalization and significantly increase the problem's dimensionality. \nTherefore, deep RL (DRL) algorithms based on function approximation by deep neural networks (DNNs) have been proposed.\n\nDRL algorithms can be classified into three types: (i) \\textit{value-based} methods such as  deep Q-learning (DQL) and SARSA which only learn the so-called  value function to find a policy, (ii) \\textit{policy-based} methods which learn the policy directly by following the gradient with respect to the policy, and (iii) \\textit{actor-critic} methods which are a hybrid of the value-based for the critic and policy-based methods for the actor.\n\\iffalse\n\\cite{DRL_1}:\n\\begin{itemize}\n    \\item \\textit{Value-based} methods such as  deep Q-learning (DQL) and SARSA which only learn the so-called  value function to find a policy.\n    \\item \\textit{Policy-based} methods which learn the policy directly by following the gradient with respect to the policy itself in  order to keep improving the policy constantly. These methods suffer from noisy gradients and high variance.\n    \\item \\textit{Actor-critic} methods which are a hybrid of the value-based methods for the critic and the policy-based methods for the actor. The critic reduces the variance of policy gradient methods by estimating the action-value function $Q(\\bm{s},a)$.\n\\end{itemize}\n\\fi\n\nThe standard DQL method represents the most popular update algorithm in the literature due to the availability of a mature theory. Basically, the DQL update equation at time $t$ for a network agent with parameters $\\theta^Q$ after taking action $a_t$ in state $\\bm{s}_t$ and observing the\nimmediate reward $r_{t+1}$ and resulting state $\\bm{s}_{t+1}$ is:\n\\begin{equation}\n\\label{eq:dqn-update equation}\n\\begin{aligned}\nQ(\\bm{s}, a \\,|\\, \\theta^Q_{t+1}) &= Q(\\bm{s}, a\\,|\\, \\theta^Q_t) + \\nu \\bigg[r_{t+1} + \\zeta  \\max_{a'} Q(\\bm{s}_{t+1}, a'\\,|\\, \\theta^Q_t) - Q(\\bm{s}_t, a_t\\,|\\, \\theta^Q_t)\\bigg]\\\\\n&\\text{\\hspace{-30mm}}= Q(\\bm{s}, a\\,|\\, \\theta^Q_t) + \\nu \\bigg[r_{t+1} + \\zeta \\max_{a'} Q(\\bm{s}_{t+1}, \\argmax_{a'} Q(\\bm{s}_{t+1},a'\\,|\\, \\theta^Q_t)\\,|\\, \\theta^Q_t) - Q(\\bm{s}_t, a_t\\,|\\, \\theta^Q_t)\\bigg],\n\\end{aligned}\n\\end{equation}\nwhere $\\nu$ is the learning rate.\nComputing the term $ \\underset{a'}{\\text{max}}~ Q(\\bm{s}_{t+1}, \\underset{a'}{\\text{argmax}}  Q(\\bm{s}_{t+1},a'\\,|\\, {\\theta}^Q_t)\\,|\\, \\theta^Q_t)$ introduces a systematic overestimation of the Q-values during the learning that is accentuated by the use of bootstrapping, i.e. learning estimates from estimates. The Q-learning update in (\\ref{eq:dqn-update equation}) uses the same Q-network $Q(\\bm{s}, a\\,|\\, {\\theta}^Q_t)$ both to select and to evaluate an action. After highlighting the overestimation bias in experiments across different Atari game environments,  Hasselt et al. \\cite{van2016deep} decoupled the action selection and evaluation by introducing two deep Q-networks,  a  $Q$ network and a target network $Q'$ with different parameters $\\theta^Q$ and ${\\theta}^{Q'}$, respectively, to avoid the maximization bias.\nThe  $Q'$ network is used for action selection while the $Q$ network is used for action evaluation.\nThis is known as {\\em deep double Q-learning algorithm (DDQL)}.\nThe DDQL update equation of the network can be expressed as:\n\\begin{dmath}\n\\text{\\hspace{-2mm}}Q(\\bm{s}, a \\,|\\, {\\theta}^Q_{t+1}) = Q(\\bm{s}, a\\,|\\, {\\theta}^Q_t) \n\\\\\n+ \\nu \\bigg[r_{t+1} + \\zeta \\max_{a'} Q(\\bm{s}_{t+1}, \\argmax_{a'} Q'(\\bm{s}_{t+1},a'\\,|\\, {\\theta}^{Q'}_t)\\,|\\, {\\theta}^Q_t) - Q(\\bm{s}_t, a_t\\,|\\, {\\theta}^Q_t)\\bigg].\n\\label{eq:ddqn-update-q1-equation}\n\\end{dmath}\nThe parameters ${\\theta}^{Q'}$ of the  $Q'$ network periodically hard-copy the parameters $\\theta^Q$ of $Q$ network after $t_0$ time steps using the Polyak averaging method with parameter $\\tau \\in [0,1]$:\n\\begin{equation}\n\\label{eq:ddqn-update-q2-equation}\n{\\theta}^{Q'}_{t+t_0} = (1-\\tau) \\,{\\theta}^{Q'}_t + \\tau \\,\\theta^Q_t.\n\\end{equation}\n\nThe DDQL networks show a better performance that standard DQL \\cite{van2016deep}; however, due to the discretization requirements of the DNN outputs (the action space $\\mathcal{A}$), it results in a huge expansion of the action space dimensionality when used in the optimization of an objective function of continuous dependent variables.\n{\\em This dimensionality issue makes it an unattractive solution for solving the beamforming problem under massive number of users and APs.}\nHowever, it is a relevant candidate for the clustering problem of the APs since it avoids the need for an extremely inefficient exhaustive search method. \nThis  motivates us to utilize the ``DDPG\" policy for the beamforming design problem.\n\nDDPG belongs to the class of actor-critic algorithms. It concurrently learns a Q-function network approximation $Q(\\bm{s}, a|\\theta^{Q})$ called the critic, and a policy network approximation $\\mu(\\bm{s}|\\theta^{\\mu})$ called the actor. \nThe Q-function network is trained using the Bellman equation, while the policy network is learnt using the Q-function. \nUnlike the DQL policies which output the probability distribution $\\pi(a|\\bm{s})$ across a discrete action space $\\mathcal{A}$,\nthe policy network of DDPG directly maps states to actions. \nSpecifically, at every time step $t$, it maximizes its loss function defined as:\n\\begin{equation}\n\\label{eq:loss-function-actor}\nJ(\\theta) = \\mathbb{E}\\bigg[Q(\\bm{s}, a) \\;|\\; \\mathcal{S}=\\bm{s}_t, a=\\pi(a|\\bm{s}_t)\\bigg]\n\\end{equation}\nand updates its weights $\\theta$ by following the gradient of (\\ref{eq:loss-function-actor}):\n\\begin{equation}\n\\label{eq:gradient-loss-function-actor}\n\\nabla J_{\\theta^{\\mu}}(\\theta) \\approx \\nabla_{a} Q(\\bm{s},a) \\,\\nabla \\mu(s|\\theta^{\\mu}).\n\\end{equation}\nThis update rule represents the deterministic policy gradient (DPG) theorem, rigorously proved by Silver et al. in the supplementary material of \\cite{silver2014deterministic}.\nThe term $\\nabla_{a} Q(\\bm{s},a)$ is obtained from a Q-network  $Q(\\bm{s}, a|\\theta^{Q})$ called the critic by backpropagating its output w.r.t. the action input $\\mu(\\bm{s}|\\theta^{\\mu})$. When the number of actions is very large, this actor-critic training procedure solves the intractability problem of DQN \\cite{mnih2015human}  by using the following approximation:\n\\begin{equation}\n\\label{eq:max-approx-ddpg}\n\\max_{a} Q(\\bm{s}, a) \\approx Q(\\bm{s}, a |\\theta^{Q})|_{a=\\mu(\\bm{s}|\\theta^{\\mu})}.\n\\end{equation}\nSimilar to DQN, two tricks are employed to stabilize the training of the DDPG actor-critic architecture, namely, 1) the experience replay buffer $R$ to train the critic, and 2) target networks for both the actor and the critic which are updated using the polyak averaging in the same way it was done in (\\ref{eq:ddqn-update-q2-equation}).\nNow that we have provided a brief DRL background on the two methods used by our proposed algorithm (DDQL and DDPG), \na detailed description of the proposed DRL-based AP clustering and beamforming design for a dynamic cell-free network can be presented in the following subsection.\n\\subsection{DRL Agent Design for AP Clustering and Beamforming Optimization}\nOur goal is to design a DRL system that jointly optimizes the clustering of APs and the beamforming vectors given a certain CSI matrix\n$\\bm{H}=\n\\left[\n{\\tilde{\\bm{g}}}_{1}\n\\dots \n{\\tilde{\\bm{g}}}_{K}\n\\right]$.\nIn this context, we develop a hybrid DDPG-DDQL DRL scheme that simultaneously learns the best clustering subsets-beamforming vector pairs given a certain CSI matrix $\\bm{H}$ (or any other metric related to $\\bm{H}$).\nFig. \\ref{AI_Model} shows a schematic block diagram of the developed hybrid DDPG-DDQL DRL scheme.\n   \t\\begin{figure}[htb]\n\t\t\\centering\n\t\t\\includegraphics[scale=0.35]{DQL_MODEL.jpg}\n\t\n\t\t\\caption{Hybrid DDPG-DDQL model for AP clustering and uplink beamforming design. }\\label{AI_Model}\n\t\\end{figure} \nTo find the beamforming decoding matrix of the $k$-th user $\\bm{W}_k=\\text{Reshap}(\\bm{w}_{k})$, where $\\bm{W}_k\\in \\mathbb{R}^{\\nint*{\\sqrt{\\sum_{\\tilde{m}=1}^{\\tilde{m}}\\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}} \\bm{w}_{k}}}\n\\times\n\\nint*{\\sqrt{\\sum_{\\tilde{m}=1}^{\\tilde{m}}\\sum_{n_{\\tilde{m}}=1}^{\\mathcal{N}_{\\tilde{m}}} \\bm{w}_{k}}}}$\\footnote{Note that the empty elements on $\\bm{W}_k$ are zero padded.} and  $\\bm{w}_{k}=\\left[w_{1k} \\dots w_{\\tilde{M}k}\\right]^H$, we opt for the actor-critic DDPG algorithm \\cite{lillicrap2015continuous} since all the  elements $w_{i,j}$ of $\\bm{W}_k$ are in the continuous range $[0,1]$.\nFor the AP clustering problem, we use the well-known DDQL algorithm \\cite{van2016deep} to find the best access point clustering configuration since the possible number of clustering configurations is finite and the index of each possible configuration is an integer number. \n\nAs shown in Fig.~\\ref{AI_Model}, the two algorithms interact with a simulated cell-free network environment to solve the optimization problem $\\textbf{P}_1$ in (\\ref{OptimizationProb}).\nThe design of the cell-free network environment involves the specification of the environment state $\\bm{s}$ and the definition of immediate reward function $r$ required by the DRL algorithms to approximate the policies and the Q-values. \nThe state of the environment is a vector of the SINR values of users, where the state $\\bm{s}_k$ of the $k^{th}$-user is chosen to be the SINR value $\\gamma_{k}^{\\mathcal{C}_n}$ which is a varying function of the instantaneous CSI matrix.\nUnder this setup, the action space $\\mathcal{A}$ is the pair of actions $a=(a^{b}, a^{\\mathcal{C}}) = (\\bm{W}_k,\\mathcal{C}_n)$ that the DDPG and the DDQN output separately. \nThe superscripts $b$ and $\\mathcal{C}$ refer to  ``\\textit{beamforming}\" and ``\\textit{clustering}\", respectively. \nAfter receiving the environment's state $\\bm{s}$, the DDPG algorithm outputs the action $a^{b} = \\bm{W}_k$ and the DDQL outputs  the action $a^{\\mathcal{C}}$ representing the cluster partition $\\mathcal{C}_n$.\n\nTable \\ref{Table3} summarizes the environment design by specifying the additional problem parameters.\n\\begin{table}[h!]\n    \\centering\n      \\caption{DRL agent design}\n\\begin{tabular}{|c|c|}\n\\hline\nEnvironment Variables & System Equivalence  \\\\\n\\hline\n\\hline\nState ${\\bm{s}}=\\{{s}_1, \\dots, {s}_K \\}$ & $\\{\\gamma_{1}^{\\mathcal{C}_n}, \\dots, \\gamma_{K}^{\\mathcal{C}_n}\\}$\\\\\n\\hline\nReward $r$ & $ \\sum_{k=1}^K  \\;\\log\\left(1+\\gamma_{k}^{\\mathcal{C}_n}\\right)$\\\\\n\\hline\nAction $\\mathcal{A}$ & $(a^{b}, a^{\\mathcal{C}}) = (\\bm{W}_k, \\mathcal{C}_n)$ \\\\\n\\hline\n$n=1, \\dots, N$ & Index of possible Clustering \\\\\n\\hline\n$\\mathcal{C}_n$ & $n$-th clustering configuration \\\\\n\\hline\n$K$ & Number of active users \\\\\n\\hline\n\\end{tabular}\n    \\label{Table3}\n\\end{table}\n{\\em Note that when the number of clusters equals to the total number of APs ($\\tilde{M}=M$), the network will reduce to the static cell-free network with $\\mathcal{C}_n=\\mathcal{C}, ~  n=1$}. \nAccordingly, only DDPG model will be required to optimize the beamforming vectors ($\\bm{W}_k$).\\\\\nOur TensorFlow\/Keras implementation of the actor and critic networks (including their corresponding target networks) have two hidden layers of 256 and 128 neurons, respectively. The DDQN networks have two \nfully-connected layers of 64 neurons followed with an activation function \\textit{relu} each, and a final linear fully-connected layer. We use a discount factor $\\zeta = 0.99$, a learning rate $\\nu = 5 \\,\\cdot\\,10^{-5}$, a Polyak averaging parameter $\\tau=10^{-3}$, and an experience replay buffer of size $R=20000$. The critic optimizer is Adam with its default hyperparameters $\\beta_1=0.9$ and $\\beta_2=0.999$. We train all the networks on 2500 episodes, with 500 time step each.\n\\subsection{Description of the Hybrid DDPG-DDQL Algoritm}\n\nFor the static cell-free network, only the DDPG algorithm is trained (one clustering configuration). However, for the dynamic cell-free network, both the DDPG and DDQL networks are trained.\n\\begin{itemize}\n    \\item \\underline{DDPG network}:\n    \\begin{itemize}\n        \\item \\textit{The actor network $\\mu (\\bm{s} | \\theta^{\\mu})$} maps the SINR values of users to the beamforming matrix $\\bm{W}_k$. The output of the network is $a^{b}$, i.e. a flatten list of all the elements $w_{ij} \\in [0,1]$.\n        \\item \\textit{The target actor network $\\mu' (\\bm{s} | \\theta^{\\mu'})$}: time-delayed copy of the actor network $\\mu (\\bm{s} | \\theta^{\\mu})$.\n        \\item \\textit{The critic network $Q(\\bm{s}, a^b | \\theta^{Q})$}: maps the SINR values and the output action of $\\mu (\\bm{s} | \\theta^{\\mu})$ to their corresponding Q-value.\n        \\item \\textit{The target critic network $Q'(\\bm{s}, a^b | \\theta^{Q'})$}: time-delayed copy of the critic network $Q(\\bm{s}, a^b | \\theta^{Q})$.\n    \\end{itemize}\n    \\item \\underline{DDQL network}:\n    \\begin{itemize}\n        \\item \\textit{The $Q_c$-network $Q_c(\\bm{s}, a^{\\mathcal{C}} | \\theta^{Q_c})$} maps the SINR values of users to the the Q-values of the state and all the clustering partitions.\n        \\item \\textit{The target $Q_c$-network $Q_c'(\\bm{s}, a^{\\mathcal{C}} | \\theta^{Q_c'})$}: time-delayed copy of the $Q_c$-network $Q(\\bm{s}, a^{\\mathcal{C}} | \\theta^{Q_c})$.\n    \\end{itemize}\n\\end{itemize}\n\nWe describe all of the training steps of our algorithm in \\textbf{Algorithm \\ref{algo:DRL-for-free-cell-network}}. We start by initializing all neural networks and their targets for the beamforming and clustering problems as well as a replay buffer $R$ (lines \\ref{algo:init1}--\\ref{algo:init5}). For every episode, we initialize the environment with $N$ access points and $K$ users by setting the initial state $\\bm{s}_0$ to a random vector of SINR values of size $K$ (line \\ref{algo:init-env}). At every time step $t$ of the episode, the DDQL and DDPG agents pick an action $a^{\\mathcal{C}}_t$ and $a^b_t$ respectively (lines \\ref{algo:pick-ab}--\\ref{algo:pick-ac}). The combined action $a_t = (a^b_t, a^{\\mathcal{C}}_t)$ is sent to the free-cell network environment which will transit to a new state $\\bm{s}_{t+1}$, and this new state will be returned together with the immediate reward $r_t$ (lines \\ref{algo:combine-ab-ac}--\\ref{algo:get-state-reward}). After storing the transition tuple $(\\bm{s}_{t}, a_{t}, r_{t}, s_{t+1})$ in the replay buffer $R$ (line \\ref{algo:store-transition}), we randomly sample from the experience replay buffer $N$ transitions to train the DDPG and DDQL networks (line \\ref{algo:sample-replay-buffer}). We start the DDPG training in line (\\ref{algo:ddpg-td-target}) by computing the TD target for the Q-network $Q(\\bm{s}, a | \\theta^{Q})$ using the target Q-network $Q'(\\bm{s}, a | \\theta^{Q'})$. We update the critic $Q(\\bm{s}, a | \\theta^{Q})$'s parameters $\\theta^{Q}$ in line \\ref{algo:update-critic} using the gradient of the MSE of the loss function of the TD target and the output of the critic. The update of the actor's parameters $\\theta^{\\mu}$ uses the Monte Carlo approximation of gradient in line \\ref{eq:gradient-loss-function-actor}. \nThe target critic and target policy networks get updated slowly every $P$ iterations (lines \\ref{algo:update-critic-target}--\\ref{algo:update-actor-target}). Finally, we update the parameters $\\theta^{Q_c}$ of the DDQL Q-network using the Bellman equation in line (\\ref{algo:update-ddql-q-network}) after selecting the action using the target Q-network $Q'_c(\\bm{s}, a | \\theta^{Q'_c})$ in line \\ref{algo:ddql-action-selection}. Similar to the DDPG target network, we update in line \\ref{algo:update-ddql-target} the DDQL target Q-network every $P$ iterations.\n\\begin{algorithm}\n\\small\n\\caption{Hybrid DDPG-DDQL algorithm for uplink beamforming and clustering}\\label{algo:DRL-for-free-cell-network}\n\\begin{algorithmic}[1]\n\\State Randomly initialize the critic $Q(\\bm{s}, a | \\theta^{Q})$ and the actor $\\mu(\\bm{s} | \\theta^{\\mu})$ with weights $\\theta^{Q}$ and $\\theta^{\\mu}$ \\label{algo:init1}\n\\State Initialize target network $Q'$ and $\\mu'$ with weights $\\theta^{Q^{\\prime}} \\leftarrow \\theta^{Q}$, $\\theta^{\\mu^{\\prime}} \\leftarrow \\theta^{\\mu}$ \\label{algo:init2}\n\\State Randomly initialize the $Q_c$-network $Q_c(\\bm{s}, a | \\theta^{Q_c})$ \\label{algo:init3}\n\\State Initialize the target network $Q'_c(\\bm{s}, a | \\theta^{Q'_c})$ with weights $\\theta^{Q_c^{\\prime}} \\leftarrow \\theta^{Q_c}$ \\label{algo:init4}\n\\State Initialize replay buffer R \\label{algo:init5}\n\\For {$episode=1,\\dots, E$}\n\\State Receive initial observation state $\\bm{s}_1$ after initializing the environment \\label{algo:init-env}\n\\For {$t=1,\\dots, T$}\n\\State  Select the beamforming action $a^{b}_{t}=\\mu\\left(\\bm{s}_{t} | \\theta^{\\mu}\\right)$ \\label{algo:pick-ab}\n\\State  Select the clustering action $a^{\\mathcal{C}}_t=\\arg \\max _{a^{\\mathcal{C}}} Q_c\\left(\\bm{s}_t, a^{\\mathcal{C}}\\right)$ \\label{algo:pick-ac}\n\\State Define $a_t = (a^b_t, a^{\\mathcal{C}}_t)$ \\label{algo:combine-ab-ac}\n\\State Execute action $a_{t}$ and observe reward $r_{t}$ and observe new state $\\bm{s}_{t+1}$ \\label{algo:get-state-reward}\n\\State Store transition $(\\bm{s}_{t}, a_{t}, r_{t}, \\bm{s}_{t+1})$ in $R$ \\label{algo:store-transition}\n\\State Sample a random minibatch of $\\mathcal{L}$ transitions $(\\bm{s}_{i}, a_{i}, r_{i}, \\bm{s}_{i+1})$ from $R$ \\label{algo:sample-replay-buffer}\n\\State Get $a^b_i$ and $a^{\\mathcal{C}}_i$ from $a_i$\n\\Statex\\LeftComment{2}{\\underline{Training the DDPG networks}}\n\\State Compute the TD target $y_{i}=r_{i}+\\zeta\\, Q^{\\prime}\\left(\\bm{s}_{i+1}, \\mu^{\\prime}\\left(\\bm{s}_{i+1} | \\theta^{\\mu^{\\prime}}\\right) | \\theta^{Q^{\\prime}}\\right)$  \\label{algo:ddpg-td-target}\n\\State Update the critic $Q(\\bm{s}, a | \\theta^{Q})$ by minimizing the loss: $L=\\frac{1}{\\mathcal{L}} \\sum_{i}\\left(y_{i}-Q\\left(\\bm{s}_{i}, a^b_{i} | \\theta^{Q}\\right)\\right)^{2}$ \\label{algo:update-critic}\n\\Statex \\qquad \\quad  Update the actor policy $\\mu(\\bm{s} | \\theta^{\\mu})$ using a monte-carlo approximation of (\\ref{eq:gradient-loss-function-actor}):\n\\State \\qquad \\qquad$\\left.\\left.\\nabla_{\\theta^{\\mu}} J \\approx \\frac{1}{\\mathcal{L}} \\sum_{i} \\nabla_{a} Q\\left(\\bm{s}, a | \\theta^{Q}\\right)\\right|_{\\mathcal{S}=\\bm{s}_{i}, a=\\mu\\left(\\bm{s}_{i}\\right)} \\nabla_{\\theta^{\\mu}} \\mu\\left(\\bm{s} | \\theta^{\\mu}\\right)\\right|_{\\mathcal{S}=\\bm{s}_{i}}$\n\\Statex \\qquad \\quad Update the DDPG target networks $Q'$ and $\\mu'$ \\textbf{if} $mod(t, P)=0$:\n\\State \\qquad \\qquad ${\\theta^{Q^{\\prime}} \\leftarrow \\tau \\theta^{Q}+(1-\\tau) \\theta^{Q^{\\prime}}}$ \\label{algo:update-critic-target}\n\\State \\qquad \\qquad ${\\theta^{\\mu^{\\prime}} \\leftarrow \\tau \\theta^{\\mu}+(1-\\tau) \\theta^{\\mu^{\\prime}}}$ \\label{algo:update-actor-target}\n\\Statex\\LeftComment{2}{\\underline{Training the DDQL networks}}\n\\State select $a^{*}=\\arg \\max _{a} Q'_c\\left(\\bm{s}_{i+1}, a| \\theta^{Q'_c}\\right)$   \\label{algo:ddql-action-selection}\n\\Statex \\qquad \\quad Update the $Q_c$ usin{}g:\n\\State \\qquad \\qquad $Q_c(\\bm{s}_i, a^c_i | \\theta^{Q_c}) \\leftarrow Q_c(\\bm{s}_i, a^c_i| \\theta^{Q_c})+\\nu\\,\\left(r_i+\\zeta\\, Q_c\\left(s_{i+1}, a^{*}| \\theta^{Q_c}\\right)-Q_c(\\bm{s}_i, a^{\\mathcal{C}}_i| \\theta^{Q_c})\\right)$ \\label{algo:update-ddql-q-network}\n\\Statex \\qquad \\quad Update the DDQL target networks $Q_c'$ \\textbf{if} $mod(t, P)=0$:\n\\State \\qquad \\qquad ${\\theta^{Q_1^{\\prime}} \\leftarrow \\tau \\theta^{Q_1}+(1-\\tau) \\theta^{Q_1^{\\prime}}}$ \\label{algo:update-ddql-target}\n\\EndFor\n\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\nNote that the DRL model can be easily extended to  optimize also the  uplink transmission power for users $P_k, \\forall k=1, \\dots, K$. However, the power control information will then need to be transmitted to the users, which will increase the signaling traffic significantly.\n\\section{Numerical Results and Discussions}\n\n\\subsection{Parameters and Assumptions}\n\n\n\n\n\n\\begin{table}[h!]\n    \\centering\n      \\caption{Simulation parameters}\n\\begin{tabular}{|c|c|}\n\\hline \nParameter & Value  \\\\\n\\hline\n\\hline\nAWGN PSD per UE &\n$-169$ dBm\/Hz  \\\\\n\\hline\nPath-loss exponent, $\\kappa$ & $2$\\\\\n\\hline\nNakagami parameters, $(\\mathcal{M},\\Omega)$ & $(1,1)$\\\\\n\\hline\nTraining sequence length, $\\tau_p$ & $K$ Samples\\\\\n\\hline\nPilot transmission power, $\\rho_k$ & $100$ mW, ${\\forall k}$\\\\\n\\hline\nSIC sensitivity, $P_s$  & $1$ dBm\n\\\\\n\\hline\n\\end{tabular}\n    \\label{Table2}\n\\end{table}\n\nTable \\ref{Table2} presents  the  main  system  parameters  used\nto obtain simulation and analytical results. To simplify simulation and analysis of this work and to concentrate on the most insightful conclusions, we establish some operating assumptions related to channel training and CSI fading models.\nWe assume that users' pilot signals are mutually orthonormal such that $\\tau_p=K\\leq \\tau_c$ with zero AWGN component in CSI estimation, i.e. $|\\bm{\\varphi}_{lmn}^H\\bm{\\varphi}_{xyz}|=1$ if $(l,m,n)=(x,y,z)$ or zero otherwise.\n$\\forall~(l,x)=1, \\dots ,\\tilde{M}$, $(m,y)=1, \\dots, \\mathcal{N}_{(l,x)}$ and $(n,z)=1, \\dots, K$.\nThis will result in removal of the CSI estimation error parts (due to both pilot contamination and AWGN component, $\\bm{\\eta}_{\\tilde{m}n_{\\tilde{m}}}$) and considering $\\hat{g}_{xyz}={g}_{xyz}$.\nAdditionally, we assume that all channel fading gains $h_{\\tilde{m}n_{\\tilde{m}}k}$ are independent and identically distributed (i.i.d) with $\\mathcal{M}_{\\tilde{m}n_{\\tilde{m}}k}=\\mathcal{M}=1$ and $\\Omega_{\\tilde{m}n_{\\tilde{m}}k}=\\Omega=1$.\nSimilarly, we assume that all AWGN values belong to a set of i.i.d rvs with PSD $\\sigma_{\\tilde{m}n_{\\tilde{m}}}\/2=-169$ dBm\/Hz. \nAs for large-scale fading, we assume that all APs and users are uniformly distributed over a disc of radius 18m (corresponding to a network total coverage area of $1 \\text{km}^2$.\n\nFirst, we  investigate the accuracy of \\textit{Welch-Satterthwaite} sum of gamma rvs approximation method \n(Fig. \\ref{Welch_Figure_1}).\n\\begin{figure}[htb]\n\t\t\\centering\n\t\n\t\t\\includegraphics[scale=0.6]{Sum_Gamma.pdf}\n\t\t\\caption{An illustrative example of \\textit{Welch-Satterthwaite} approximation with different $\\#$ of rvs ($K$).}\\label{Welch_Figure_1}\n\t\\end{figure}\n\tIt can be noticed from Fig. \\ref{Welch_Figure_1} that a satisfactory accuracy for the sum of non-identically distributed (different $\\beta$) gamma rvs can be achieved by \\textit{Welch-Satterthwaite} approximation for small, medium, and high numbers of random variables.\n\tThis justifies the use of such an approximation instead of the use of the central limit theorem which requires a relatively large number of random variables.\n\\subsection{Outage Performance}\nThis section discusses the outage performance of cell-free network in a massive communication regime. \nEach simulation value is obtained via  $2\\times10^6$ Monte-Carlo simulation runs. In Fig. \\ref{Figure_Outage_DRL_1} (a),\nwe evaluate the  outage probability for a user in a static cell-free  network under different values of $K$.\n\\begin{figure}[htb]\n\t\t\\centering\n\t\t\\includegraphics[height=9cm, width=14cm]{Figure_Outage_DRL_1.pdf}\n\t\t\\caption{User performance: (a) Probability of outage,  (b) Transmission rate per user. }\\label{Figure_Outage_DRL_1}\n\t\\end{figure}\nFig.~\\ref{Figure_Outage_DRL_1} (a) shows that an excellent outage performance can be achieved when the number of APs ($M$) is much grater than the number of users $K$, i.e. $M>>K$. \nHowever, when $K$ becomes comparable to $M$, the outage performance deteriorates significantly with almost total blockage at $K\\approx 0.5 M$.\nNote that this figure is produced assuming no precoding scheme, i.e. $w_{mk}=c, \\forall m, k$. A better performance can be achieved by first solving problem $\\textbf{P}_1$ to obtain $\\bm{W}_k$ and then substituting $\\bm{W}_k$ into (\\ref{e_outage_1}).\n\n\n\\subsection{DRL Model}\nIn this section, we study the performance of using the proposed DRL model in solving problem $\\textbf{P}_1$.\nFig. \\ref{Figure_Outage_DRL_1} (b) shows the per-user transmission rate in the cell-free network with the proposed DRL method.\nWe can notice that the performance reaches a certain level after around $500$ training episodes and then stays around that level.\nCompared with the max-min objective, we notice a significant performance gain with the max-sum objective.\nThis is due to the fact that, with the max-sum objective, we optimize $\\bm{W}_k, \\forall k=1, \\dots, K$ while with max-min objective, we maximize only $\\bm{W}_{k'}$ and use $W_k, k\\neq k'~\\&~k=1, \\dots, K$, where $k'$, denoting the index of the user with minimum achievable rate, is found from previous iterations.\n\nTo evaluate the performance of the proposed dynamic cell-free clustering scheme with SIC detection, Fig. \\ref{Figure_Two_DRL_1} a shows the per-user transmission rate for both static and dynamic cell-free networks.  \nIt can be noticed that utilizing SIC at the receiver side significantly compensates for the performance loss caused by clustering of the APs. Note that the per-user transmission rate achieved without SIC in presence of AP clustering is significantly lower than that in a static cell-free network.\n \nAdditionally, in Fig. \\ref{Figure_Two_DRL_1}(b), we compare the performance of the proposed DRL-based solution approach compared to traditional methods.\n\\begin{figure}[htb]\n\t\t\\centering\n\t\t\\includegraphics[height=9cm, width=14cm]{Figure_Two_DRL_1.pdf}\n\t\t\\caption{Per-user transmission rate: (a) with SIC, (b) optimal vs. DRL.}\\label{Figure_Two_DRL_1}\n\\end{figure}\nWe can notice that DRL can achieve per user transmission rate of around $78\\%$ of that with conventional bisection method (subgradient method). \nThis performance degradation comes with a remarkable decrease in the computational complexity. With the proposed DRL-based design, the beamforming vectors can be obtained in an online manner for practical implementation of non-orthogonal multiple access in cell-free networks.\n\nFigs. \\ref{RMSE} (a) and (b) evaluate the convergence rate of the proposed DRL algorithm for two different network setups.\nSpecifically, we plot the absolute value of the difference between the normalized rate values generated by the DRL model and that by the bisection method (using the Matlab optimization toolkit).\n\\begin{figure}[htb]\n\t\t\\centering\n\t\t\\includegraphics[height=9cm, width=14cm]{DRL_Conv_Figure.pdf}\n\t\t\\caption{DRL convergence rate: (a) Medium-scale scenario, (b) large-scale scenario.}\\label{RMSE}\n\\end{figure}\nIt can be noticed that the proposed DRL model shows a better convergence rate than the bisection method under medium-scale scenario with $M = 150$ and $K = 50$ (Fig. \\ref{RMSE} [a]) compared to that of large-scale scenario with $M = 200$ and $K = 60$ (Fig. \\ref{RMSE} [b]).\nHowever, the proposed DRL model under large-scale scenario is found to achieve a better performance by achieving an approximate of $92\\%$ of the normalized rate achieved by bisection method compared to only $78\\%$ achieved under medium-scale scenario.   \n\nFinally, to evaluate the performance of the cell-free network under different values of per-user transmission power at the large-scale regime, Fig. \\ref{DRL_Vs_Power} illustrates the variation in per-user transmission rate in a large-scale static cell-free network for different transmit power values.\n\\begin{figure}[htb]\n\t\t\\centering\n\t\t\\includegraphics[height=10cm, width=11cm]{DRL_Vs_Power.pdf}\n\t\t\\caption{Per-user transmission rate vs. $P$.}\\label{DRL_Vs_Power}\n\\end{figure}\nIt can be noticed that rate enhancement becomes smaller when the transmission power is increased in a high $P$ regime (e.g. from $P = 37$ dBm to $P = 50$ dBm). \nHowever, a good rate enhancement still can be achieved when $P$ is changed from a relatively small value to a significantly larger value (e.g. from $P = 20$ dBm to $P = 37$ or $P=50$ dBm). Furthermore, the DRL model is found to show a faster convergence with larger values of $P$.  \n\\section{Conclusion}\n We have first derived closed-form expressions for the  probability of outage for an uplink user in a static  cell-free network.\nNext, we have proposed a novel dynamic cell-free network that partitions the distributed APs among a set of subgroups with each subgroup acting as a virtual AP equipped with a distributed antenna system (DAS).\nSuch a clustering is performed based on the current channel state information (CSI) among the APs and all users within the network coverage area. The dynamic cell-free network model can reduce the complexity for joint signal processing. For the dynamic cell-free model, we have  formulated the general optimization problem of  clustering the APs and designing the beamforming vectors. To solve this optimization problem, we have proposed a hybrid deep deterministic policy gradients (DDPG)-double deep Q-network(DDQN) scheme that jointly selects the optimal network clustering scheme with its optimal beamforming vector values. Possible extensions of this work would include the design and evaluation of more comprehensive DRL models that jointly estimate CSI, select the best clustering configuration for APs, and optimize different beamforming vectors. Also, benchmarking different DRL-based algorithms for optimizing resource allocation in different cell-free network architectures will be valuable.\n\n\n\t\\appendices\n\t\\section{}\n\n\\setcounter{equation}{0}\nFirst, let us define $X=\\sum_{m=1}^M|\\tilde{g}_{mk}|^2$ and\n\\[\nY =    \\sum_{m=1}^M\n    \\left[\n    \\sum_{l=1, l\\neq k}^K|\\tilde{g}_{ml}|^2\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    |\\tilde{g}_{m{\\ddot{l}}}|^2\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K|\\tilde{g}_{m{\\Breve{l}}}|^2\n    \\right].\n    \\]\n    Let $W=Y+1\\approx Y$. (This assumption is valid for large values of $K$ and $M$.) \nThe derivation of closed-form expression for $X$ and $Y$ is very complicated due to the non-equal rate parameters ($\\tilde{\\beta}_{m,k}$ and $\\tilde{\\beta}_{m,l}$) which makes the MGF method unusable to derive $f_X(x)$ and $f_Y(y)$. Here, we use an accurate approximation of the sum of independent Gamma rvs with different shape and rate parameters, which is referred to as the \\textit{Welch-Satterthwaite} approximation \\cite{Satterthwaite1946}. Accordingly, we have $X\\thicksim \\mathcal{G}\\left(\\Dot{\\alpha}_{mk}, \\Dot{\\beta}_{mk} \\right)$ and \n$Y\\thicksim \\mathcal{G}\\left(\\Dot{\\alpha}_{mk'}, \\Dot{\\beta}_{mk'} \\right)$, \nwhere \n\\[\\Dot{\\alpha}_{mk}=\\frac{\\left(\\sum_{m=1}^M\\tilde{\\alpha}_{mk}\/ \\tilde{\\beta}_{mk}\\right)^2}{\\sum_{m=1}^M\\tilde{\\alpha}_{mk}\/\\tilde{\\beta}_{mk}^2},~~~~\n\\Dot{\\beta}_{mk}=\\frac{\\sum_{m=1}^M\\tilde{\\alpha}_{mk}\/\\tilde{\\beta}_{mk}^2}{\\sum_{m=1}^M\\tilde{\\alpha}_{mk}\/ \\tilde{\\beta}_{mk}},\n\\]\n\\[\\Dot{\\alpha}_{mk'}=\\frac{\n\\left(\n\\sum_{m=1}^M\n    \\left[\n    \\sum_{l=1, l\\neq k}^K\n    \\frac{    \\tilde{\\alpha}_{ml}}{ \\tilde{\\beta}_{ml}}\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    \\frac{    \\tilde{\\alpha}_{m\\ddot{l}}}{ \\tilde{\\beta}_{m\\ddot{l}}}\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{m\\Breve{l}}}{        \\tilde{\\beta}_{m\\Breve{l}}}\n    \\right]\n\\right)^2\n}\n{\n\\sum_{m=1}^M\n    \\left[\n    \\sum_{l=1, l\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{ml}}{    \\tilde{\\beta}_{ml}^2}\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    \\frac{    \\tilde{\\alpha}_{m\\ddot{l}}}{\\tilde{\\beta}_{m\\ddot{l}}^2}\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{m\\Breve{l}}}{    \\tilde{\\beta}_{m\\Breve{l}}^2}\n    \\right]\n},\n\\]\n\\[\\Dot{\\beta}_{mk'}=\n\\frac{\n\\sum_{m=1}^M\n    \\left[\n    \\sum_{l=1, l\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{ml}}{\\tilde{\\beta}_{ml}^2}\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    \\frac{    \\tilde{\\alpha}_{m\\ddot{l}}}{\\tilde{\\beta}_{m\\ddot{l}}^2}\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K\n    \\frac{\\tilde{\\alpha}_{m\\Breve{l}}}{    \\tilde{\\beta}_{m\\Breve{l}}^2}\n    \\right]\n}\n{\n \\sum_{m=1}^M\n    \\left[\n    \\sum_{l=1, l\\neq k}^K\n    \\frac{    \\tilde{\\alpha}_{ml\n}}{\\tilde{\\beta}_{ml}}\n    +\n    \\sum_{\\dot{l}=1, \\dot{l}\\neq k}^K\n    \\sum_{\\ddot{l}=1, \\ddot{l}\\neq \\dot{l}}^K\n    \\frac{\\tilde{\\alpha}_{m\\ddot{l}\n}}{    \\tilde{\\beta}_{m\\ddot{l}}}\n    +\n    \\sum_{\\Breve{l}=1, \\Breve{l}\\neq k}^K\n    \\frac{    \\tilde{\\alpha}_{m\\Breve{l}\n}}{\\tilde{\\beta}_{m\\Breve{l}}}\n    \\right]\n}.\n\\]\nNote that this approximation becomes identical to the exact expression even for small to moderate number of APs and\/or UEs. This makes it useful for modeling large-scale cell-free architectures. \nThe PDF of the ratio $Z=\\frac{x}{w}$ can be defined as\n\\begin{equation}\n    f_Z(z)=\\frac{d}{dz}\\int_{0}^{\\infty}\\text{P}\\left(x\\leq z  w\\right)f_Y(w)dw=\\int_{0}^{\\infty}wf_X(zw)f_Y(w)dw.\\label{PDF_Integral_1}\n\\end{equation}\n Substituting $f_X(zw)$ and $f_Y(w-1)$ into \\textbf{A}.1, utilizing \\cite[Eq. 3.383.4]{gradshteyn2000}, and simplifying, we obtain  Eq. (\\ref{PDF_Centralized}).\n\\section{}\n\\setcounter{equation}{0}\nThe CDF of the ratio distribution ($Z=\\frac{X}{W}$) can be expressed as\n\\begin{equation}\n    F_Z(z)=\\text{P}\\left(x\\leq z W\\right) =\\int_{0}^{\\infty}\\text{P}\\left(x\\leq z w\\right)f_Y(w)dw.\\label{CDF_Integral_1}\n\\end{equation}\nBy substituting $f_Y(w)$ and $P\\left(x\\leq zw \\right)=1-\\gamma\\left(\\Dot{\\alpha}_{mk},\\Dot{\\beta}_{mk}zw\\right)$, where $\\gamma(.)$ is the lower incomplete gamma function \\cite[Eq. 6.5.2]{1965}, and utilizing \\cite[Eq. 6.455.2]{gradshteyn2000} and \\cite[Eq. 15.3.7]{1965}, and simplifying, we obtain Eq. (\\ref{e_outage_1}).\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAn intersection representation $\\calR$ of a graph $G$ is a collection\nof sets $\\{R(v) : v \\in V(G)\\}$ such that\n$R(u) \\cap R(v) \\neq \\emptyset$ if and only if $uv \\in\nE(G)$. Important classes of graphs are obtained by restricting the\nsets $R(v)$ to some specific geometric objects.  In an \\emph{interval\n  representation} of a graph, each set $R(v)$ is a closed interval of\nthe real line; \nand in a\n\\emph{circular-arc representation}, the sets $R(v)$ are closed arcs of\na circle;\nsee Fig.~\\ref{fig:cK222}.\n A graph is an \\emph{interval graph} if it admits an interval representation\n and it is a \\emph{circular-arc graph} if it admits a circular-arc\n representation.  We also denote the corresponding classes of graphs by \\hbox{\\rm \\sffamily INT}\\xspace\n and \\hbox{\\rm \\sffamily CA}\\xspace, respectively.\n \n\nIn many cases, the availability of a geometric representation makes\ncomputational problems tractable that are otherwise \\hbox{\\rm \\sffamily NP}-complete.  For\nexample, maximum clique can be solved in polynomial time for both\ninterval graphs and circular-arc graphs.  Another example is the\ncoloring problem, which can be solved in polynomial time for interval\ngraphs but remains \\hbox{\\rm \\sffamily NP}-complete for circular-arc graphs~\\cite{GareyJMP80}.\n\nA key problem in the study of geometric intersection graphs is the\n\\emph{recognition problem}, which asks whether a given graph has a\nspecific type of intersection representation.  It is a classic result\nthat interval graphs can be recognized in linear time.  For\ncircular-arc graphs the first polynomial-time recognition algorithm\nwas given by Tucker~\\cite{tucker1980efficient}.  McConnell gave a linear-time\nrecognition algorithm~\\cite{mcconnell2003linear}.\n\nIn this paper, we are interested in a generalization of the\nrecognition problem.  For a class $\\calX$ of intersection representations, the\n\\emph{partial representation extension\n  problem for $\\calX$} ({\\textsc{RepExt}}{$(\\calX)$} for short) is defined as follows.\n  In addition to a graph $G$, the input consists of a\n\\emph{partial representation}~$\\calR'$ that is a representation of an\ninduced subgraph $G'$ of $G$.  The question is whether there exists a\nrepresentation $\\calR \\in \\calX$ of $G$ that \\emph{extends} $\\calR'$ in the\nsense that $R(u) = R'(u)$ for all $u \\in V(G')$.  The recognition\nproblem is the special case where the partial representation is empty.\nThe partial representation extension problem has been recently studied\nfor many different classes of intersection graphs, e.g., interval\ngraphs~\\cite{KlavikKOSV17}, proper\/unit interval\ngraphs~\\cite{klavik2017extending}, function and permutation\ngraphs~\\cite{klavik2012extending}, circle graphs~\\cite{CFK13}, chordal\ngraphs~\\cite{KKOS15}, and trapezoid graphs~\\cite{KrawczykW17}.\nRelated extension problems have also been considered, e.g., for planar\ntopological~\\cite{adfjk-tppeg-15,jkr-kttpp-13} and\nstraight-line~\\cite{p-epsld-06} drawings, for contact\nrepresentations~\\cite{chaplick2014contact}, and rectangular\nduals~\\cite{abs-2102-02013}.\n\nIn many cases, the key to solving the partial representation extension\nproblem is to understand the structure of all possible\nrepresentations.  For interval representations, the basis for this is\nthe characterization of Fulkerson and Gross~\\cite{maximal_cliques},\nwhich establishes a bijection between the combinatorially distinct\ninterval representations of a graph $G$ on the one hand and the linear\norderings~$\\preceq$ of the maximal cliques of $G$ where for each\nvertex $v$ the cliques containing $v$ appear consecutively\nin~$\\preceq$ on the other hand.  This not only forms the basis for the\nlinear-time algorithm using PQ-trees by Booth and\nLueker~\\cite{PQ_trees}, but also shows that a PQ-tree can compactly\nstore the set of all possible interval representations of a graph.\nThe partial representation problem for interval graphs can be solved\nefficiently by searching this set for one that is compatible with the\ngiven partial representation.\n\nDespite the fact that circular-arc graphs straightforwardly generalize\ninterval graphs, the structure of their representations is much less\nunderstood.  It is not clear whether there exists a way to compactly\nrepresent the structure of all representations of a circular-arc\ngraph.  There are two structural obstructions to this aim.  First, in\ncontrast to interval graphs, it may happen that two arcs have\ndisconnected intersection, namely in the case when their union covers\nthe entire circle.  Secondly, intervals of the real line satisfy the\n\\emph{Helly property}: if any pair of sets in a set system intersects,\nthen the intersection of the entire set system is non-empty.\nConsequently, the maximal cliques of interval graphs can be associated\nto distinct points of the line and also the number of maximal cliques\nin an interval graph is linear in the number of its vertices.  In\ncontrary, arcs of a circle do not necessarily satisfy the Helly property\nand indeed the number of maximal cliques can be exponential.  The\ncomplement of a perfect matching $nK_{2}$ is an example of this\nphenomenon, see Fig.~\\ref{fig:cK222}b.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=1.0,page=1]{cK222}\n\\caption{(a) The graph $\\overline{3K_{2}}$ and (b) its circular-arc representation.\n(c) A non-Helly representation of $K_4$.}\n\\label{fig:cK222}\n\\end{figure}\n\nTo capture the above properties, that may have substantial impact on\nexplorations of  circular-arc graphs, the following specific subclasses of\ncircular-arc graphs have been defined and intensively\nstudied~\\cite{Tucker1971MatrixCO,gavril1974algorithms,LinS2009survey,lin2013normal}:\n\n\\begin{compactitem}\n\\item \\emph{Normal circular-arc graphs} ($\\hbox{\\rm \\sffamily NCA}\\xspace$) are circular-arc graphs that\n  have an intersection representation in which the intersection of any two arcs\n  is either empty or connected. \n\\item \\emph{Helly circular-arc graphs} ($\\hbox{\\rm \\sffamily HCA}\\xspace$) have an intersection\n  representation that satisfies the Helly property, i.e there are no $k\\ge 3$\n  pairwise intersecting arcs without a point in common.\n\\item \\emph{Proper circular-arc graphs} ($\\hbox{\\rm \\sffamily PCA}\\xspace$) are circular-arc graphs that\n  have an intersection representation in which no arc properly contains\n  another.\n\\item \\emph{Unit circular-arc graphs} ($\\hbox{\\rm \\sffamily UCA}\\xspace$) are circular-arc graphs with an\n  intersection representation in which every arc has a unit length. \n\\end{compactitem}\n\nThe above properties can be combined together in the sense that a\nsingle representation shall satisfy more properties simultaneously,\ne.g. \\emph{Proper Helly circular-arc graphs} ($\\hbox{\\rm \\sffamily PHCA}\\xspace$) are\ncircular-arc graphs with an intersection representation that is both\nproper and Helly~\\cite{lin2007proper}. This is stronger than requiring\nthat a graph is a proper circular-arc graph as well as a Helly\ncircular-arc graph (with each property guaranteed by a different\nrepresentation), i.e., $\\hbox{\\rm \\sffamily PHCA}\\xspace \\subsetneq \\hbox{\\rm \\sffamily PCA}\\xspace \\cap \\hbox{\\rm \\sffamily HCA}\\xspace$.\n\nAnalogously, since $C_4$ has a unique representation, the wheel $W_4$\nis a graph with a Helly representation (the universal vertex covers\nall four clique points) or a normal representation (it covers three\ncliquepoints) but not normal Helly representation. Thus also\n$\\hbox{\\rm \\sffamily NHCA}\\xspace \\subsetneq \\hbox{\\rm \\sffamily NCA}\\xspace \\cap \\hbox{\\rm \\sffamily HCA}\\xspace$.\n\nMoreover, Tucker~\\cite{Tucker1974} proved that every representation of a proper\n(Helly) circular-arc graph that is not normal can be transformed into a normal\nrepresentation.\nHence, the following graph classes coincide $\\hbox{\\rm \\sffamily PCA}\\xspace=\\hbox{\\rm \\sffamily NPCA}\\xspace$ and $\\hbox{\\rm \\sffamily PHCA}\\xspace=\\hbox{\\rm \\sffamily NPHCA}\\xspace$.\nFig.~\\ref{fig:ca_inclusions}a shows inclusions between the defined graph classes.\n\nWe use an analogous notation for the classes of possible representations, \ni.e., \nfor $X\\subseteq\\{\\hbox{\\rm \\sffamily N,P,H}\\}$ the symbol $X$\\hbox{\\rm \\sffamily CAR}{} for the\nclass of all $X$\\hbox{\\rm \\sffamily CA}\\xspace{} representations, see Fig.~\\ref{fig:ca_inclusions}b.  We\nnote that whether a graph $G$ with a partial representation $\\cal R'$ admits an\nextension depends crucially on the class of allowed representations, as\nillustrated by the example of $W_4$ above.\n \n\\myparagraph{Our results.}\nWhile for many classes efficient algorithms for the representation extension\nproblem have been found, the problem has been open for circular arc graphs for\nnine years~\\cite{DBLP:journals\/corr\/abs-1207-6960}.\nWe prove that ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily CAR})$ is \\hbox{\\rm \\sffamily NP}-hard. To the best of our knowledge,\nit is the first known representation class for which the extension problem is\n\\hbox{\\rm \\sffamily NP}-hard while the recognition problem is in \\hbox{\\rm \\sffamily P}.\nOur reduction also works for ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily HCAR})$.\n\n\\begin{restatable}{theorem}{hcanpc}\\label{thm:hca_npc}\n  The problems ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily HCAR})$ and ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily CAR})$ are \\hbox{\\rm \\sffamily NP}-hard.\n  ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily CAR})$ is also \\hbox{\\rm \\sffamily NP}-hard if the predrawn arcs have pairwise distinct endpoints.\n\\end{restatable}\n\nWe complement this result by showing tractability for several subclasses,\nincluding all Helly variants; see Figure~\\ref{fig:ca_inclusions}b. \nLinear-time algorithms for recognizing\nHelly circular-arc graphs~\\cite{lin2006characterizations,joeris2011linear} use\nMcConnell's~\\cite{mcconnell2003linear} algorithm to construct a circular-arc\nrepresentation and transform it to a Helly circular-arc representation. This\ncannot be exploited in the case of partial representation extension.\n\n \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=1,page=2]{ca_inclusions}\n\\caption{ (a) Relationships between classes of circular-arc graphs.  (b)\n  Relationships between classes of circular-arc representations. Classes\n  studied in this paper are underlined. {\\textsc{RepExt}}{} is polynomial for blue, while\n  \\hbox{\\rm \\sffamily NP}-complete for red.  \n}\n\\label{fig:ca_inclusions}\n\\end{figure}\n\nDeng et al.~\\cite{deng1996linear} and Lin et\nal.~\\cite{lin2006characterizations} characterize proper and proper Helly\ncircular-arc representations in terms of vertex orderings of the graph.  They\nshow that these orderings are unique under certain conditions.  Building on\nthese results, we prove the following two theorems.\n\n\\begin{restatable}{theorem}{nphcapoly}\n  \\label{thm:nphca_poly}\n  The problem ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NPHCAR})$ can be solved in linear time.\n\\end{restatable}\n\n\\begin{restatable}{theorem}{phcared}\n  \\label{thm:phca_red}\n  The problem ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily PHCAR})$ can be solved in linear time.\n\\end{restatable}\n\nRecall that in the case of interval graphs, PQ-trees can be used to capture all\nplausible linear orderings of the maximal cliques.   Klav\\'{\\i}k et\nal.~\\cite{klavik2017extending} use this to solve ${\\textsc{RepExt}}$ for interval\nrepresentation by determining an order that is represented by the PQ-tree and\nthat extends a partial order that is derived from the partial representation.\n\nThe fact that Gavril~\\cite{gavril1974intersection} shows that a graph $G$ is a Helly\ncircular-arc graph if and only if there exists a \\emph{cyclic\n  ordering} $\\lhd$ of its maximal cliques such that for every vertex\n$v$, the maximal cliques containing $v$ appear consecutively in\n$\\lhd$ and that Hsu and McConnell~\\cite{hsu2003pc} use PC-trees to capture\nall plausible cyclic orderings of the maximal cliques of a Helly\ncircular-arc graph makes it tempting to simply apply the same techniques to\ngeneralize the algorithm of Klav\\'ik et al.  However, this cannot be\nstraightforwardly applied for two reasons.  First, the clique ordering carries\nlittle information about whether a representation is normal or not, and, even\nmore severely, extending a partial cyclic ordering is \\hbox{\\rm \\sffamily NP}-complete, even\nwithout requiring that the order be additionally represented by some given\nPC-tree~\\cite{gm-con-77}. We overcome this by working with suitably linearized\npartial orders to show the following results.\n\n\\begin{restatable}{theorem}{nhcapoly}\\label{thm:nhca_poly}\nThe problem ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NHCAR})$ can be solved in $\\calO(n^3)$ time.\n\\end{restatable}\n\n\\begin{restatable}{theorem}{hcapoly}\\label{thm:hca_poly}\nThe problem ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily HCAR})$ can be solved in $\\calO(n^3)$ time if the partial\nrepresentation consists of arcs with pairwise distinct endpoints.\n\\end{restatable}\n\nIt follows that Helly representations used in our reduction essentially\ninvolve arcs that share endpoints. \nThis is surprising since non-degeneracy assumptions like this are often made\nwithout much consideration of the impact on the problem when working with graph\nrepresentations.\n\n\nThe bottleneck of our NHCA-algorithms is the testing of the consecutivity\nconstraints for all universal pairs of vertices.  A closer exploration of the\nstructure of the set of universal pairs may yield improvements of the running\ntime upper bound.\n\nFinally, we show that involving the most tight constraints on arc lengths, the\nproblem becomes again computationally difficult.\n\n\n\\begin{restatable}{theorem}{ucanpc}\\label{thm:uca_npc}\nThe problem ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily UCAR})$ is \\hbox{\\rm \\sffamily NP}-complete.\n\\end{restatable}\n\nThe \\hbox{\\rm \\sffamily NP}-hardness of Theorems~\\ref{thm:hca_npc} and\n\\ref{thm:uca_npc} follows by a reduction from the $3$-${\\textsc{Partition}}$\nproblem~\\cite{garey1975complexity}.  For the unit case, the membership\nin \\hbox{\\rm \\sffamily NP} can be seen by a linear programming argument.\n\n\\section{Preliminaries}\n\n\\myparagraph{Cyclic order.}\nLet $<\\ = v_0,\\dots,v_{n-1}$ and $<'\\ = u_0,\\dots,u_{n-1}$ be two linear orders\non a finite set $S$.\nWe say that $<$ and $<'$ are \\emph{cyclically equivalent} if there is\n$k\\in\\{0,\\dots,n-1\\}$ such that $v_i = u_{i+k}$, where the addition is modulo\n$n$.\nClearly, this is an equivalence relation on the set of all linear orders on $S$.\nA \\emph{cyclic order} $\\lhd$ on $S$ is an equivalence class of this relation.\nFor a linear order $<$, we denote the corresponding cyclic ordering by $[<]$.\n\n\nEvery linear order $<$ on $S$ induces a linear order $<'$ on a subset\n$S'\\subseteq S$ by omitting all ordered pairs in which the elements of\n$S\\setminus S'$ occur.\nIn this case we say that $<$ \\emph{extends} $<'$ and similarly that the cyclic\norder $[<]$ \\emph{extends} $[<']$.\n\n\\iffalse\nFor a cyclic order $\\lhd$ a subset $S$ of elements is \\emph{consecutive} if it\nis consecutive in some linear order $<$ with $[<]=\\lhd$.  We say $b$ is\n\\emph{between} $a$ and $c$ if $a<b<c$ for a linear order $<$ with $[<]=\\lhd$.\nIf no element is between $a$ and $c$, then $a$ and $c$ are \\emph{consecutive}\nand we also say that $a$ is \\emph{immediately before} $c$ and that $c$ is\n\\emph{immediately after} $a$. \n\\fi\n\n\\myparagraph{Circular-arc representations.}\nFor any circular-arc representation $\\calR$ and each connected component $C$ \nof a graph $G$ the set $\\bigcup_{v \\in V(C)}R(v)$ is a connected subset of the circle.\nTherefore, if $G$ is a disconnected circular-arc graph, then each connected component of\n$G$ has to be an interval graph. These cases can be treated with the\ncorrepsonding algorithms for interval graphs\nof~\\cite{klavik2017extending,KlavikKOSV17}. Hence without loss of generality we\nrestrict ourselves to connected graphs in this paper.\n\nLet $\\calR$ be a representation of a circular-arc graph $G$.\nFor a vertex $v$ of $G$, we call the \\emph{tail} $R(v)_t$ and the \\emph{head}\n$R(v)_h$ the two endpoints of $R(v)$. We use the convention of traversing the\narc from the tail to the head in the clockwise direction along the circle. \nWe denote such an arc as $R(v)=[R(v)_t,R(v)_h]$, and its complement\n$(R(v)_h,R(v)_t)$ as $R(v)^c.$\n\n\nLet $\\calR$ be a Helly representation of a circular-arc graph $G$.\nDenote by $\\calC$ the set of maximal cliques of $G$.\nWe assign every maximal clique $C\\in \\calC$ a unique point\n$\\cp(C)\\in\\bigcap_{v\\in C} R(v)$ and call it the \\emph{clique-point} of $C$. \n\n\\begin{lemma}[Gavril~\\cite{gavril1974intersection}]\n\\label{lem:gavril_hca}\nA graph $G$ is a Helly circular-arc graph if and only if there exists a cyclic\nordering $\\lhd$ of its maximal cliques such that for every vertex $v$,\nthe maximal cliques containing $v$ appear consecutively in $\\lhd$.\n\\end{lemma}\n\nNote that if we distribute clique points on the circle according to a cyclic\nordering $\\lhd$ of Lemma~\\ref{lem:gavril_hca}, then a representation $\\calR$ of\n$G$ can be obtained by choosing for each vertex $v$ an arc $R(v)$ that covers\nexactly the clique-points $v$ belongs to.\n\n\\myparagraph{PC-Trees and the Reordering Problem.}\nA \\emph{PC-tree} $T$ on a set $L$ of leaves is a tree whose inner\nnodes have one of two types: \\emph{P-nodes} and \\emph{C-nodes}.  The\nneighbors around a P-node can be permuted arbitrarily, whereas the order of the\nneighbors of a C-node is fixed up to reversal.  In this way a PC-tree\nrepresents a set of cyclic orderings of its leaf set $L$.  The\nusefulness of PC-trees derives from the fact that they can represent\ncyclic orderings subject to consecutivity constraints.  Namely, given\na set $L$ and sets $X_1,\\dots,X_r \\subseteq L$, a PC-tree that\nrepresents precisely those cyclic orderings of $L$ where each of the subsets\n$X_1,\\dots,X_r$ is consecutive can be computed in\n$\\calO(|L| + \\sum_{i=1}^r |X_i|)$ time~\\cite{hsu2003pc}.\nIn our setting, in the spirit of Lemma~\\ref{lem:gavril_hca}, the leaf set $L$\nwill always be the set of all maximal cliques of a Helly circular-arc graph.\n\nWe make use of the following reordering problem, which can be solved\nanalogously to the topological sorting of PQ-trees~\\cite{klavik2017extending}.\n The input consists of a PC-tree $T$, a leaf $u$ of $T$, and a partial ordering\n (not cyclic) $<$ of the remaining leaves $L' = L\\setminus\\{u\\}$ of $T$. The\n question is whether $T$ represents a cyclic order $<'$ which induces a linear\n extension of $<$ on $L'$. \n If this is the case, $T$ is called \\emph{compatible with $<$ with respect to $u$}.  \nWe denote an instance of this problem by ${\\textsc{Reorder}}(T,u,<)$.\n\n\\begin{restatable}{lemma}{reorderingLinear} \n  \\label{lem:reordering_linear}\nAn instance ${\\textsc{Reorder}}(T, u, <)$ can be solved in time $\\calO(\\ell + c)$, where\n$\\ell$ is the number of leaves of $T$ and $c$ is the number of comparable pairs\nin the partial ordering $<$.\n\\end{restatable}\nFor a detailed proof see Section~\\ref{sec:PQtree}.\n\\newcommand{\\reorderingLinearProof}{\n\\begin{proof}\nWe root $T$ by $u$. We represent the ordering $<$ by a digraph $D$ having $c$\nedges. The algorithm reorders the nodes from the bottom to the root and modifies\n$D$ by contractions. Once we finish reordering a subtree, it is never modified\nlater.  After reordering a subtree, the corresponding vertices in $D$ are\ncontracted. We process a node of $T$ when all its subtrees are finished and the\ncorresponding digraphs are contracted to single vertices. Note that since the\ntree $T$ is now rooted, the equivalent transformations (i) and (ii) correspond\nto permuting the children of a P-node and reversing the children of\na C-node.\n\nFor a P-node, we then permute its children according to any topological sort of\nthe subdigraph $D'$ induced by the vertices corresponding to the children of the\nP-node. Note that $D'$ is acyclic since $<$ is a partial ordering. For a C-node,\nthere are two possible orderings of its children and we check whether one of\nthem is feasible. The resulting PC-tree $T'$ is compatible with $<$.\n\\relax\n\\end{proof}}\n\n\n\n\\section{Complexity}\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[page=2]{HCA_Reduction}\n  \\caption{Illustration of the proof of Theorem~\\ref{thm:hca_npc}. (a) predrawn\n    universal vertices (b) the instance of ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily CAR})$ obtained from\n    $\\{1,1,2,2,2,3,3,4\\}$ of  {\\sc 3-Partition}. For clarity we have omitted 24\n    predrawn universal vertices and instead indicated their enpoints.   \n    (c) the corresponding solution (d) variant without shared endpoints.\n  }\n  \\label{fig:hca_npc}\n\\end{figure}\n\n\\begin{proof}[Sketch of proof for Theorem~\\ref{thm:hca_npc}]\nWe first reduce the strongly \\hbox{\\rm \\sffamily NP}-complete\nproblem {\\sc 3-Partition}~\\cite{garey1975complexity} to ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily HCAR})$.\nFigure~\\ref{fig:hca_npc} illustrates the proof.  Figure~\\ref{fig:hca_npc}a\nshows a representation with four universal vertices (blue).  The green dots\nindicate the endpoints that are shared by these arcs.  The red \\emph{blocker\nvertex} covers all other points of the circle shared by all universal vertices.\nThe key insight is that each vertex that is not adjacent to the blocker vertex\nmust not be contained in the complement of a universal vertex and thus contain\nat least one of the green points.\nFigure~\\ref{fig:hca_npc}b shows an instance resulting from our reduction, and\nFigure~\\ref{fig:hca_npc}c shows a corresponding solution from the instance\n$\\{1,1,2,2,2,3,3,4\\}$.\n\nFor ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily CAR})$, the same construction works. However, we can avoid \nshared endpoints in the partial representation with a simple modification. \nNamely, we slightly shorten the arc for each universal vertex; see\nFigure~\\ref{fig:hca_npc}d. \nThen we have for each former green dot a green area between the corresponding\narc ends. Each vertex that is not adjacent to the blocker vertex must now contain \na green area.\nNote that in every solution  each leaf of a star in $G$ must contain exactly\none green area. It thus violates the Helly property with the two universal arcs\nending there.\n\\end{proof}\n\nFor details see Section~\\ref{ssec:CARnp}.\n\n\n\\section{Normal Proper Helly Circular-Arc Graphs}\n\nWe show how to extend partial representations of normal proper Helly\ncircular-arc graphs in linear time.  To do this, we give a characterization of\nall partial representations that are extendable.  This generalizes the\ncharacterization of the extendable partial representations of proper interval\ngraphs~\\cite[Lemma 2.4]{klavik2017extending}.\nWe first simplify the possible instances by reducing the number of vertices\nwith the same neighborhood as follows.\n\n\\myparagraph{Twin vertices.}\nRecall that vertices $u,v\\in V(G)$ are called twins if $N[u] = N[v]$.  It is\npossible to find the equivalence classes of twin vertices in linear\ntime~\\cite{rose1976algorithmic}.  If vertices $u$ and $v$ are twins and if\neither $u$ is not predrawn or if both $u$ and $v$ are predrawn with the same\narc, then we may remove $u$ and in the final representation we can set $R(u) =\nR(v)$.  This allows us to assume that each twin class either consists of a\nsingle vertex, which is not predrawn, or it consists only of predrawn\nvertices that are represented by distinct arcs.\n\n\\myparagraph{Consecutive orderings of vertices.}\nLet $\\lhd = [v_0, \\dots, v_{n-1}]$ be a cyclic ordering of the vertices of some\ngraph $G$. \nWe say that $\\lhd$ is \\emph{consecutive} if the closed neighborhood $N[v]$ of\neach $v\\in V(G)$ is consecutive in $\\lhd$.\nNote that $N[v_i]$ is consecutive in $\\lhd$ if there exist positive integers\n$a$ and $b$ such that $a + b \\leq n$ and \\[N[v_i] = \\{v_{i-a}, \\dots, v_{i-1},\nv_i, v_{i+1}, \\dots, v_{i+b}\\},\\] where the addition is performed modulo $n$.\n\nRoberts characterized connected proper interval graphs in terms of consecutive linear\norderings~\\cite{roberts1968representations} that are unique up to permuting\ntwin vertices and a complete reversal.  The following lemma, proved in slightly\ndifferent terminology by Deng et al. as Corollaries 2.7 and 2.9.\nin~\\cite{deng1996linear} give a characterization of proper circular-arc\ngraphs by cyclic ordering:\n\n\\begin{lemma}\\label{lem:pca_charac}\n  A graph $G$ belongs to $\\hbox{\\rm \\sffamily PCA}\\xspace$ if and only if $V(G)$ has a consecutive cyclic\n  ordering.\n\\end{lemma}\n\nThey also showed that this ordering is unique up to permuting twin vertices and\na complete reversal only in the case when the complement is connected or\nnonbipartite. For example, the complement of a perfect matching\n$K_{2,2,\\dots,2}$ on at least 6 vertices allows several distinct consecutive\norderings, like $[1,2,3,4,5,6]$ and $[1,3,2,4,6,5]$ for the graph depicted in\nFig.~\\ref{fig:cK222}a. \nFor normal proper Helly circular-arc graphs, the following strengthening of\nLemma~\\ref{lem:pca_charac} follows from the results summarized in Lin et\nal.~\\cite{lin2013normal}:\n\n\\begin{lemma}\\label{lem:nphca_unique}\nEvery $\\hbox{\\rm \\sffamily NPHCA}\\xspace$ graph $G$ has a \\emph{unique} consecutive cyclic ordering of\n$V(G)$, up to permuting twins and a reversal. Such an ordering can be obtained\nin  $O(m+n)$ time, where $n$ and $m$ stand for the number of vertices and\nedges, respectively.\n\\end{lemma}\n\nNote that each such consecutive cyclic ordering of vertices corresponds to the\ncyclic order on tails of the arcs in some normal proper Helly representation.\n\n\\myparagraph{Characterization of extendable instances.}\nLet $G \\in \\hbox{\\rm \\sffamily NPHCA}\\xspace$ and let $\\calR'$ be a partial representation of $G$.\nBy the discussion above, we may assume that if $u$ and $v$ are twins, then both\nare predrawn with $R(u)\\neq R(v)$. \n\nA constraint specific to the representation extension of proper circular-arc\ngraphs is imposed by any pair of touching arcs, for which  the tail of one\ncoincides with the head of the other, i.e., $R'(v_i)_h = R'(v_j)_t$. In this\nsituation, it is impossible to place another tail or head between $R'(v_i)_h$\nand $R'(v_j)_t$ -- in fact on this point of the circle -- as the resulting\nrepresentation would not be proper.\n\nWe use this fact as well as the uniqueness of the ordering of\nLemma~\\ref{lem:nphca_unique} to characterize all instances that allow a\nrepresentation extension; see Section~\\ref{sec:DetailsNPHCA}.\n\n\\begin{restatable}{lemma}{nphcaChar}\\label{lem:nphca_char}\nA partial normal proper Helly circular-arc representation $\\calR'$ of a\nconnected graph $G$, where all twins are predrawn by distinct arcs, is\nextendable if and only if $V(G)$ has a consecutive cyclic ordering $\\lhd =\n[v_0,\\dots,v_{n-1}]$ satisfying:\n\\begin{compactenum}[(1)]\n  \\item \\label{itm:nphca_prop1}\nThe ordering $\\lhd$ extends the cyclic ordering $\\lhd'$ of the predrawn\nvertices corresponding to the clockwise cyclic ordering of the tails of their\npredrawn arcs.\n  \\item \\label{itm:nphca_prop2}\nIf $R'(v_i)$ and $R'(v_j)$, are distinct touching predrawn arcs such that\n$R'(v_i)_h = R'(v_j)_t$, then\n$N[v_i] = \\{v_a, \\dots,v_i,\\dots,v_j\\} \\quad \\text{and} \\quad N[v_j] =\n\\{v_i,\\dots,v_j,\\dots,v_{b}\\}$, for some non-negative integers $a, b$.\n\\end{compactenum}\n\\end{restatable}\n\n\n\\newcommand{\\nphcaCharProof}{\n\\begin{proof}\nSuppose that there is a normal proper Helly representation $\\calR$ extending\n$\\calR'$.  Let $\\lhd$ be the cyclic ordering $[v_0,\\dots,v_{n-1}]$ of $V(G)$\ninduced by the clockwise cyclic ordering $[t_0,\\dots,t_{n-1}]$ of tails\n$t_i=R(v_i)_t$.  For convenience, we also write $h_i$ for $R(v_i)_h$.\nThe ordering $\\lhd$ extends $\\lhd'$ since $\\calR'$ is contained in $\\calR$, so\nthe condition~(\\ref{itm:nphca_prop1}) is satisfied.\n\nTo verify condition~(\\ref{itm:nphca_prop2}), suppose that there are two\npredrawn arcs $R(v_i)$ and $R(v_j)$ such that $h_i = t_j$.\nIf $v_i$ is a universal vertex (adjacent to all other vertices), then the\ncondition on $N[v_i]$ holds trivially.\nOtherwise, since $N[v_i]\\neq V(G)$ and $N[v_i]$ is consecutive, $\\lhd$ induces\na linear order on $N[v_i]$, where the minimum and maximum elements $v_a$ and\n$v_b$ correspond to the first and the last neighbor of $v_i$ in clockwise\ndirection.\nSince $h_i = t_j$, it must be that $b = j$. An analogous argument holds for $v_j$.\n\n\nFor the opposite implication, let $\\lhd = [v_0,\\dots,v_{n-1}]$ be a consecutive\ncyclic ordering of $V(G)$ satisfying all three conditions. \nWithout loss of generality we may assume that $G$ is not a complete graph, as\notherwise the problem is trivial -- we pick any predrawn arc and replicate it\nfor every non-predrawn vertex.\nTo construct the representation $\\calR = \\{R(v_0),\\dots,R(v_{n-1})\\}$ of $G$,\nwe need to determine the position of the endpoints of $R(v_i)$, for every $v_i$\nthat is not predrawn. \n\nFirst, we construct a cyclic order $\\prec$ on the auxiliary set\n\\[A = \\{t_0,\\dots,t_{n-1}, h_0,\\dots,h_{n-1}\\}.\\]\nWe construct $\\prec$ in several steps.\nInitially, we set $\\prec\\ = [t_0,\\dots,t_{n-1}]$.\nNext, we insert $h_i$ into $\\prec$, for $i = 0,\\dots,n-1$, in the following way.\nFor each $i=0,\\dots,n-1$ we distinguish two cases, depending on whether $v_i$\nis universal or not, see Fig.~\\ref{fig:nphca_order}:\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=1,page=1]{nphca_order}\n\\caption{(a) Position of the head $h_i$ in $\\prec$ for a non-universal $v_i$,\n and (b) for a universal $v_i$.}\n\\label{fig:nphca_order}\n\\end{figure}\n\n\\begin{compactenum}[(a)]\n\\item If $v_i$ is not universal, then we determine the last neighbor $v_b$ of\n  $v_i$, i.e., the vertex satisfying $N[v_i] = \\{v_a, \\dots,v_i,\\dots,v_b\\}$\n  and insert $h_i$ into $\\prec$ immediately before $t_{b+1}$.\n\\item Otherwise, i.e., when $v_i$ is universal, we determine the first\n  non-universal vertex $v_a$ for which $N[v_a] = \\{\\dots, v_a,\n    \\dots,v_i,\\dots\\}$, i.e., the arc $R(v_a)$ shall contain the tail\n    $R(v_i)_t$, and insert $h_i$ into $\\prec$ immediately before $t_a$. The\n    phrase \\emph{the first} means that all other non-universal vertices $a'$\n    which have $v_{a'}$ before $v_i$ in their neighborhood $N(v_{a'})$ are\n    between $v_a$ and $v_i$ in $\\lhd$.\n\\end{compactenum} \n\nWe compute the order $\\prec$ on the set $A$ from the cyclic vertex order $\\lhd$\nand from adjacencies between vertices of the graph $G$. As desribed, the order\n$\\prec$ depends uniquely on $\\lhd$ and $G$.\n\nThe representation $\\calR'$ imposes a cyclic ordering on the points\nrepresenting the tails and heads of the predrawn arcs. We extend this cyclic\nordering to the set of tails and heads of the predrawn arcs, i.e. on\n$\\{t_i,h_i: R'(v_i)\\in \\calR'\\}$. In the case when a single point of the circle\nis the tail and the head of touching arcs, i.e., when $R'(v_i)_h=R'(v_j)_t$, we\ninsert $h_i$ immediately after $t_j$. Note that as $G$ is a proper circular-ar\ngraph, only a single pair of arcs may touch on a single point.  Denote the\nresulting cyclic ordering by $\\prec'$.\n\nSince $\\calR'$ represents an induced subgraph of $G$ and as the cyclic ordering\n$\\lhd'$ on tails is a sub-ordering of $\\lhd$ due to (1), it follows that\n$\\prec'$ is a sub-ordering of $\\prec$. In particular, it could be obtained from\n$\\lhd'$ and $G'$ by the same process as $\\prec$ was constructed from $\\lhd$ and\n$G$.\n\nNow, we use the cyclic order $\\prec$ to construct the representation $\\calR$.\nWe identify every $t_i \\in A$ with $R(v_i)_t$ and $h_i \\in A$ with $R(v_i)_h$\nand with a slight abuse of notation we view the elements of $A$ as the points\nof the circle.\n\nConsider now two predrawn elements $e$ and $e'$ that are consecutive in\n$\\prec'$ and that corresponds to distinct points of the circle --- in other\nwords, the interior of the arc $[e,e']$ contains no other predrawn head or\ntail. \nEspecially we exclude here the case when $e$ and $e'$ represent the head and\nthe tail of touching arcs.  For any such pair $e$ and $e'$, let\n$e_0,\\dots,e_{\\ell-1}$ be the linear ordering of the elements in $\\prec$ that\nare between $e$ and $e'$ (if any). We place $e_0,\\dots,e_{\\ell-1}$\nequidistantly between $e$ and $e'$ on the circle. This way we obtain the\nrepresentation $\\calR$. By the same argument we have used for $\\calR'$, the\ncyclic ordering on the points representing the tails and heads can be extended\nto $\\lhd$ by putting heads of touching arcs immediately after the matching\ntails.\n\nSince we do not modify the endpoints of predrawn arcs, $\\calR$ extends\n$\\calR'$.\n\nConsider now any pair of vertices $v_i, v_j\\in V(G)$.\nWhen $v_iv_j \\in E(G)$, we distinguish the following cases: \n\\begin{compactitem}\n  \\item If both $v_i$ and $v_j$ are not universal, then we may without loss of\n    generality assume that $v_j$ appears after $v_i$ in $N[v_i]= \\{v_a,\n    \\dots,v_i,\\dots, v_j, \\dots, v_b\\}$.  In this case the head $h_i$ has been\n    inserted in $\\prec$ immediately before the tail $t_{b+1}$. As the tail $t_b$\n    is between $t_i$ and $t_{b+1}$ and $t_j=t_b$ or $t_j$ is between $t_i$ and\n    $t_b$, it follows that the tail $t_j$ is inside the arc $R(v_i)$, see\n    Fig.~\\ref{fig:nphca_order}a.\n  \\item If $v_i$ in universal but $v_j$ not, then either $v_i$ appears before\n    $v_j$ or after it in $N[v_j]$.  When $v_i$ is after $v_j$ in $N[v_j]$ we\n    deduce that $t_i$ is inside the arc $R(v_j)$ as in the previous case.\n    Otherwise the head $h_i$ has been inserted in $\\prec$ immediately before\n    $t_a$, the first non-universal vertex which has in $N[v_a]$ the vertex\n    $v_j$ positioned after $v_a$. Therefore $h_j$ is between $h_i$ and $t_i$\n    and the tail $t_j$ is inside the arc $R(v_i)$, see\n    Fig.~\\ref{fig:nphca_order}b.\n  \\item Finally, when both $v_i$ and $v_j$ are universal, then they are twins\n    and hence are predrawn and intersecting already in $\\calR'$. \n\\end{compactitem}\nIn all three cases, the arcs $R(v_i)$ and $R(v_j)$ intersect.\n\nOn the other hand, when $v_iv_j \\notin E(G)$, then none of these vertices is universal. \nIn the moment of insertion of $t_i$ in $\\prec$, i.e., immediately before\n$t_{b+1}$ for $N[v_i]= \\{v_a, \\dots,v_i,\\dots,v_b\\}$, the tail $t_i$ appeared\nbetween $h_i$ and $h_j$ as otherwise $N[v_i]$ would not be consecutive.\nAnalogously, the tail $t_j$ was inserted between $h_j$ and $h_i$.  Therefore,\nthe arcs $R(v_i)$ and $R(v_j)$ are disjoint.\n\\relax\n\\end{proof}}\n\nWe are ready to prove that ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NPHCAR})$ can be solved in time $\\mathcal{O}{}(n+m)$,\nwhere~$n$ is the number of vertices and~$m$ is the number of edges of the given\ngraph $G$.\n\n\\nphcapoly*\n\\begin{proof}\nThe procedure is given as Algorithm~\\ref{alg:nphca}.  Its correctness follows\ndirectly from ~Lemma~\\ref{lem:nphca_char}. \n\n\\begin{algorithm}[tb]\n\\KwData{A graph $G$ and a partial representation $\\calR'$.}\n\\KwResult{A \\hbox{\\rm \\sffamily NPHCA}\\xspace{} representation $\\calR$ of $G$ extending $\\calR'$ or a message that it does not exist.}\n\\Begin{\nDetermine the twin classes and prune not predrawn twins\\;\nFind the ordering $<$  by Lemma~\\ref{lem:nphca_unique}\\;\nTransform $<$ to $\\lhd$ by reversal and permutation of predrawn twins\\;\n\\ForAll{arcs $R'(v_i)$ and $R'(v_j)$ such that \n$R'(v_i)_h = R'(v_j)_t$}{\n\\lIf{$N[v_i] \\cap N[v_j] \\ne \\{v_i,\\dots,v_j\\}$}{\\Return{$\\calR'$ has no extension.}}\n}\nConstruct the cyclic order $(A, \\prec)$ from the proof of Lemma~\\ref{lem:nphca_char}\\;\nConvert $(A, \\prec)$ into the representation $\\calR$ as descibed in  Lemma~\\ref{lem:nphca_char}\\;\nReplicate the pruned twins into $\\calR$ from the non-pruned ones\\;\n\\Return{$\\calR$}\n}\n\n\\caption{The algorithm for the ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NPHCAR})$ problem.}\\label{alg:nphca}\n\\end{algorithm}\n\nFor the computational complexity note that:\n\\begin{compactitem}\n\\item Step 3 could be achieved in $O(n+m)$ time by Lemma~\\ref{lem:nphca_unique}. \n\\item Step 4. To check whether reversal is necessary it suffices to check two consecutive vetices belonging to discinct twin classes.\nThe correct ordering of the predrawn twins can be achieved by a single traversal of $<$.\n\\item\nThe for-loop at lines 5--7 has at most $n$ iterations since we check the head of every predrawn arc exactly once.\n\\item\nThe test at line $6$ can be done in constant time since it suffices to check whether $v_{i-1}v_j \\in E(G)$ or $v_{i}v_{j+1} \\in E(G)$.\n\\item\nStep 8.\nFollowing the proof of Lemma~\\ref{lem:nphca_char}, we perform a single\ntraversal of the initial order $[t_0,\\dots, t_{n-1}]$ and insert the elements\n$h_0,\\dots,h_{n-1}$ appropriately according to $\\lhd$.\n\\item\nFor step 9, the positions of heads and tails of non-predrawn arcs can be determined in $\\calO(n)$ by a single traversal of $(A, \\prec)$.\n\\item\nAt line 10, each pruned twin requires only constant time to be replicated.\n\\end{compactitem}\n\\relax\n\\end{proof}\n\n\nFor Theorem~\\ref{thm:phca_red}, which deals with ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily PHCA}\\xspace)$, note that\neven though all proper Helly circular-arc graphs allow also a normal proper\nrepresentation, we cannot reduce ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily PHCAR})$ to ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NPHCAR})$\ndirectly, as the given partial representation need not to be normal.\n\nHowever, the existence of a proper Helly representation extending a pair of arcs in a\nnot-normal position imposes strong conditions on the structure of $G$: after\npruning all universal vertices only two disjoint cliques remain.\nSuch instances can be solved in linear time. For details see\nSection~\\ref{sec:PHCA}.\n\n\n\n\\section{Normal Helly Circular-Arc Graphs}\n\\label{sec:NHCA}\n\nWith the following lemma, we can avoid universal vertices since they allow \nconsidering instances of ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NHCAR})$ as instances for the interval case. \n\\begin{lemma}\n  Let $G$ be a graph with a universal vertex $u$. Then for every normal Helly\n  circular-arc representation of $G$ there exists a point on the circle that is\n  contained in no arc.\n  \\label{lem:nhcaGuniversal}\n\\end{lemma}\n\\begin{proof}\n  Let $\\calR$ be a normal Helly circular-arc representation of $G$.\n  Assume that every point of the circle is contained in some arc. \n  Without loss of generality, we may assume that $R(u)$ is not strictly contained in\n  any other arc of $\\calR$. We consider the complement $R^c(u)$ of $R(u)$.\n  Let $v_t$, $v_h$ be vertices whose arcs contain $R^c(u)_t$, $R^c(u)_h$\n  respectively, and whose arcs maximize the intersection with $R^c(u)$.\n\n  Note that, since $\\calR$ is normal, neither $R(v_t)$ nor $R(v_h)$ contain $R^c(u)$.\n  Assume that one of $R(v_t)$, $R(v_h)$, say $R(v_t)$ contains both endpoints of $R(u)$.\n  We then have $R^c(u)\\subseteq R(v_t)$ which by the maximality of $R(u)$\n  implies $R(v_t)=R(u)$. \n  By the choice of $v_t$, we find in this case a point close to $R(u)_t$ that\n  is not contained in any arc.\n\n  In the other case, each of $R(v_t)$ and $R(v_h)$ contains exactly one endpoint of $R(u)$.\n  In particular, we have $v_t\\ne v_h$.\n  Since $u$ is universal and since every point of $R^c(u)$ is contained in some arc,\n  it follows that $R^c(u)\\subseteq R(v_t)\\cup R(v_h)$. We obtain $R(v_t)\\cap\n  R(v_h) \\cap R^c(u)\\ne \\emptyset$ (recall that our arcs are closed sets).\n  Due to the normal property, it follows that $R(v_t)\\cap R(v_h) \\cap R(u)=\n  \\emptyset$, which contradicts the Helly property.\n  We conclude that there exists a point $p$ that is not contained in any arc.\n\\end{proof}\n\nWe assume for the rest of this section that our graph contains no universal vertices.\n\nLet $G=(V,E)$ be a graph. Then two adjacent vertices $u, v\\in V$ form a\n\\emph{universal pair} if each vertex $w\\in V$ is adjacent either to $u$ or to $v$.\n\n\\begin{lemma}\n  A graph $G$ without a universal vertex is a normal Helly circular-arc graph\n  if and only if there exists a cyclic ordering $\\lhd$ of its maximal cliques\n  such that \n  \\begin{compactenum}[(i)]\n    \\item for every vertex $v$, the maximal cliques containing $v$ appear\n  consecutively in $\\lhd$.\n    \\item for every universal pair $u$, $w$, the maximal cliques\n      containing $u$ and $w$ appear consecutively in $\\lhd$.\n  \\end{compactenum}\n\n  \\label{lem:nhcaG}\n\\end{lemma}\n\\begin{proof}\n If $G$ has a normal Helly circular-arc representation, then, due to the Helly\n property, we have for each maximal clique $C$ a clique point $\\cp(C)$ where\n the arcs of all vertices in $C$ intersect.\n For each vertex $v$ and for each universal pair $u$, $w$ the corresponding\n cliques are consecutive in $\\lhd$ since $R(v)$ and $R(u)\\cap R(w)$ each are\n connected (the latter due to the normal property).\n\n Next assume that we have a cyclic ordering $\\lhd$ of the maximal cliques\n of $G$ with properties (i) and (ii).\n We obtain a normal Helly circular-arc representation as follows;\n see Fig.~\\ref{fig:lem16}.\n We first arrange the maximal clique points on the circle according to $\\lhd$.\n Then, for each vertex $v$, we define the arc $R(v)$ as the smallest arc that\n contains exactly the clique points of the maximal cliques that contain $v$. Note\n that $R(v)$ is well-defined since $v$ is not universal.\n This defines a circular-arc representation $\\calR$ of $G$ since any two\n intersecting arcs share a clique point. Moreover, the existence of the clique\n points shows the Helly property. It remains to show that $\\calR$ is\n normal.\n\n Assume that there are two nodes $u$, $w$ such that $R(u)\\cap R(w)$ is not connected.\n Then $u$, $w$ is a universal pair. Thus the cliques containing $u$ and\n $w$ are consecutive in $\\lhd$ which contradicts $R(u)\\cap R(w)$ being not connected.\n Hence, the representation is normal.\n\\end{proof}\n\n\n\\myparagraph{Extendable Partial Representations.}\nWe characterize all partial representations of a normal Helly circular-arc\ngraph $G$ that are extendable. Let $\\calR'$ be a partial representation of $G$\nand let $C$ be a maximal clique of $G$. We define $\\Pre(C) = \\{R'(v) : v \\in\nC \\cap V(G')\\}$ to be the \\emph{predrawn arcs} corresponding to the vertices in $C$.\n\n\\begin{definition}\nFor a maximal clique $C\\in\\calC$, we define sets\n$$\\Reg^+(C) = \\bigcap\\limits_{w\\in\\Pre(C)} R'(w) \\qquad\\text{ and }\\qquad \\Reg^-(C) =\n\\bigcup\\limits_{w\\in \\calR'\\setminus \\Pre(C)} R'(w).$$\nThe \\emph{region} of $C$ is the set $\\Reg(C) =\n\\Reg^+(C)\\setminus \\Reg^-(C)$.\n\\end{definition}\n\nThe set $\\Reg^+(C)$ means the set of possible locations of $\\cp(C)$ with respect to predrawn arcs, \nwhile $\\Reg^-(C)$ means the forbidden locations.\nThus if the region $\\Reg(C)$ is empty for some clique $C\\in\\calC$, then the\ngiven partial representation is not extendable. We thus assume in the following\nthat no region is empty. The following lemmas give some useful properties that\nhold for the regions of maximal cliques.\n\n\\begin{lemma}\\label{lem:region_equivalence}\nFor maximal cliques $C$ and $D$, we have either $\\Reg(C) \\cap \\Reg(D) =\n\\emptyset$, or $\\Reg(C) = \\Reg(D)$.\n\\end{lemma}\n\n\\begin{proof}\nIf $\\Pre(C) = \\Pre(D)$, then clearly $\\Reg(C) = \\Reg(D)$. So, we assume that $\\Pre(C) \\neq\n\\Pre(D)$. We can further assume that there exists an arc $R'\\in\n\\Pre(C)\\setminus \\Pre(D)$.\nSince the clique point $\\cp(C)$ must be placed on the point of $R'$ and the clique point $\\cp(D)$ cannot be placed on any point\nof $R'$, we have that $\\Reg(C)$ and $\\Reg(D)$ are disjoint.\n\\relax\n\\end{proof}\n\nFor every maximal clique $C$, we call the connected components of $\\Reg(C)$\n\\emph{islands} and the connected components of its complement $\\Reg^c(C)$ we call \\emph{gaps of $C$}.\nWe say an island and a gap are \\emph{neighboring}, if they share an endpoint\n(where one end is open and the other is closed.) Note that every island has two\nneighboring gaps and every gap has two neighboring islands.\n\nObserve that if two maximal cliques $C,D\\in \\calC$ satisfy $\\Pre(C)=\\Pre(D)$\nthen $\\Reg(C)=\\Reg(D)$ by definition. In the other\ncase we obtain the following relationship:\n\n\n\\begin{lemma}\n\\label{lem:singleGap}\nLet $C$ and $D$ be two maximal cliques with $\\Pre(C)\\ne\\Pre(D)$.\nThen $D$ has a gap $J$ with $\\Reg(C)\\subseteq J$.\n\\end{lemma}\n\n\\begin{proof}\nFrom $\\Pre(C)\\ne\\Pre(D)$ we obtain $\\Reg(C)\\subseteq \\Reg^c(D)$.\nAssume that there are two gaps $J_1$, $J_2$ of $D$ that contain points $j_1,\nj_2\\in\\Reg(C)$. Let $I_1,I_2$ be the islands of $D$ neighboring $J_1$.\nLet $i_1\\in I_1$, $i_2\\in I_2$. \nFor $v\\in\\Pre(C)$ it holds that $j_1,j_2\\in R'(v)$ and thus $i_1\\in R'(v)$ or\n$i_2\\in R'(v)$ since $i_1$, $i_2$ separate $j_1$, $j_2$ on the circle; see\nFig.~\\ref{fig:lem15}.\nThis implies \n$\\Pre(C)\\subseteq\n\\{v\\in V(G'): i_1 \\in R'(v) \\lor i_2 \\in R'(v)\\} =\n\\{v\\in V(G'): i_1 \\in R'(v)\\} \\cup \\{v\\in V(G'): i_2 \\in R'(v)\\} \n= \\Pre(D)$ as both sets in the union are equal to $\\Pre(D)$.\nLikewise, we obtain $\\Pre(D)\\subseteq\\Pre(C)$. This contradicts $\\Pre(C)\\ne\\Pre(D)$.\nHence, $\\Reg(C)$ is contained in a single gap of $D$.\n\\end{proof}\n\nThis means that for any two maximal cliques $C$ and $D$, the clique point $\\cp(C)$\nmust be placed in a given gap of $D$. We obtain additional consecutivity constraints.\nFor every gap $J$, we define the set $S_J = \\{C\\in\\calC \\mid \\Reg(C)\\cap J\\ne \\emptyset\\} \n= \\{C\\in\\calC \\mid \\Reg(C)\\subseteq J\\}$. Recall that we assumed that no region is empty.\n\n\n\n\n\\begin{lemma}\n  \\label{lem:gapConsecutivity}\n  Let $\\calR$ be a NHCA extension of $\\calR'$. Let $J$ be a gap of some maximal clique $D$.\n  Let $\\le$ be the clique ordering derived from $\\calR$. Then in $\\le$ the set $S_J$\n  is consecutive.\n\\end{lemma}\n\\begin{proof}\n  Direct consequence of Lemma~\\ref{lem:singleGap} since all clique points of\n  that set must be placed on $J$ and all other clique points must be placed on\n  its complement $J^c$.\n\\end{proof}\n\n\n\n\\begin{lemma}\n  \\label{lem:singleIsland}\n  There exists a maximal clique $D$ with a single island.\n\\end{lemma}\n\\begin{proof}\nLet $D$ be a maximal clique with $|\\Pre(D)|$ maximal.\nThen $\\Reg^+(D)$ and $\\Reg^-(D)$ are disjoint as otherwise there would\nexist a point $p$ in $\\Reg^+(D)\\cap\\Reg^-(D)$ with \n$\\Pre(D)\\subsetneq \\{v\\in V(G'): p \\in R'(v)\\}$.\nThis implies the existence of a (maximal) clique $C$ in $G$ with\n$\\Pre(D)\\subsetneq \\Pre(C)$ in contradiction to the choice of $D$. \n\nSince $\\calR'$ is NHCA, this\nyields that $\\Reg(D) = \\Reg^+(D)$ is connected. In other words, $D$ has a single island.\n\\end{proof}\n\nFor the rest of the section, let $D$ be a maximal clique with a single island. \nLet $p_D$ be a point in the middle of $\\Reg(D)$\nand let $\\le$ be the linear order of points on the circle obtained by starting\nat $p_D$. We consider a point $p$ to be \\emph{on the left} of a point $q$ if we\nhave $p\\le q$.\n\nWe define a partial order $\\prec$ on the maximal cliques $\\calC$ such that \nfor each $C \\in \\calC\\setminus\\{D\\}$ we have $D\\prec C$ and \nfor each $C, C' \\in \\calC\\setminus \\{D\\}$ satisfying $\\forall p\\in\n\\Reg(C),q\\in \\Reg(C')\\colon p < q$ we set $C\\prec C'$.\nNote that every linearization of a clique ordering of an extension of $\\calR'$\nthat starts with $D$ extends $\\prec$.\nFor any vertex $v$ let $M_v$ denote the set of maximal cliques containing $v$.  \n\n\n\\begin{theorem}\n  \\label{the:NHCArepextChar}\n  Let $G$ be a graph without universal vertices and let $\\calR'$ be a partial\n  normal Helly circular-arc representation of $G$.  \n  %\n  There exists a normal Helly circular-arc\n  representation of $G$ that extends $\\calR'$ if and only if there exists a\n  linear extension $<$ of $\\prec$\n such that\n  \\begin{compactenum}\n    \\item  \\label{itm:linCh0} For any pair of distinct maximal cliques $C\\ne C'$ with $\\Reg(C)=\\Reg(C')$, the region $\\Reg(C)$ is not a single point.\n     \n    \\item \\label{itm:linCh1} For every vertex $v$, the set $M_v$ is consecutive in $[<]$. \n    \\item \\label{itm:linCh2} For every universal pair $u$, $w$, the set $M_{u}\\cap M_{w}$ is consecutive in $[<]$. \n    \\item \\label{itm:linCh4} For every gap $J$ of some $C \\in \\calC$, the set $S_J$ is consecutive in $[<]$. \n  \\end{compactenum}\n\\end{theorem}\n\n\\begin{proof}\nWe first show that, if there is an NHCA-extension of $\\calR'$, then these properties are satisfied.\nWe obtain $<$ as the linearization of a clique ordering of an extension of $\\calR'$\nstarting with $D$ where the clique point of $D$ is $p_D$. \nBy the construction of $\\prec$, the order $<$ is a linear extension of $\\prec$.\nBy Lemmas~\\ref{lem:nhcaG} and \\ref{lem:gapConsecutivity}, we obtain\nproperties~\\ref{itm:linCh1},~\\ref{itm:linCh2} and~\\ref{itm:linCh4}.\nProperty~\\ref{itm:linCh0} is necessary, since no two clique points can be\nplaced at the same point.\n    \nFor the opposite implication, let $< = C_1,C_2,\\dots,C_k$ be a linear extension of $\\prec$ such that\nproperties~\\ref{itm:linCh0},~\\ref{itm:linCh1},~\\ref{itm:linCh2} and~\\ref{itm:linCh4} are\nsatisfied. We show that each $C_i\\in \\calC$ can be assigned its clique point $\\cp(C_i)\\in \\Reg(C_i)$ such that\n$\\cp(C_j)<\\cp(C_i)$ whenever $j<i$.\n\nLet $\\varepsilon>0$ be the $\\frac{1}{2n+1}$-fraction of the length of the\nshortest nontrivial island. This allows to draw all new endpoints at\ndistance at least $\\varepsilon$ but still within any chosen island or side of\nisland $\\Reg(D)$.\nFor $C_1=D$ we place $\\cp(C_1)$ on $p_D$. In a greedy way, when the location of the clique points $\\cp(C_1),\\dots,\\cp(C_{i-1})$ is settled, we determine the set $P$ of feasible points for $\\cp(C_i)$ that is \n$P=\\Reg(C_i) \\cap \\{p: p> \\cp(C_{i-1})+\\varepsilon\\}$. If $P$ has minimum, we place $\\cp(C_i)$ there, otherwise\nwe put $\\cp(C_i)$ at $\\inf(P)+\\varepsilon$.\nWe argue that such choice always exists.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=1,page=1]{nhca_char.pdf}\n\\caption{Argument for the existence of a placement of $\\cp(C_i)$.}\n\\label{fig:nhca_char}\n\\end{figure}\n\nAssume for a contradiction that $C_i$ is the first maximal clique in the order $<$ whose \nclique point $\\cp(C_i)$ cannot be properly placed. \nNote that a clique $C\\ne C_i$ can only have an island $I_C$ consisting of a\nsingle point if we have $I_C=\\Reg(C)$, since $\\Reg^+$ is a closed arc and all islands\nseparated by gaps within $\\Reg^+$ have an open end.  Hence, by the choice of\n$\\varepsilon$, we only place a clique $C$ at the very last point of $\\Reg(C)$, if\n$\\Reg(C)$ consists of a single point. With property~\\ref{itm:linCh0}, this can\nnot happen if $\\Reg(C) = \\Reg(C_i)$.\nTherefore, one clique point must be placed to the right of $\\Reg(C_i)$ before placing $\\cp(C_i)$.\nWe identify the first maximal clique $C_j, j<i$ that is\nplaced to the right of all points in $\\Reg(C_i)$. Since $\\cp(C_j)\\notin \\Reg(C_i)$, \nwe have that $\\Pre(C_i)\\ne\\Pre(C_j)$.\nBy Lemma~\\ref{lem:singleGap}, the maximal clique\n$C_j$ has a gap $J$ with $\\Reg(C_i)\\subseteq J$, see Fig.~\\ref{fig:nhca_char}.\n    \nConsider the neighboring island $I$ of $C_j$ to the left of $J$. Since\n$\\cp(C_j)$ was not placed on $I$, the clique point $\\cp(C_{j-1})$ has\nbeen placed to the right of $I$. By the choice of $C_j$, we have that\n$\\cp(C_{j-1})$ is not placed to the right of $J$ and thus $\\cp(C_{j-1})\\in J$. \nSince $\\cp(C_j)$ has been placed to the right of $\\Reg(C_i)$ and thus to\nthe right of $J$, we have $p_D\\not\\in J$ and thus $D\\not\\in S_J$.\nWith $D<C_{j-1}< C_j< C_i$, where $C_{j-1}, C_i\\in S_J$ and $D,C_j\\not\\in S_J$\nwe get a contradiction with the property~\\ref{itm:linCh4}, since $S_J$ is not consecutive in $[<]$.\n\nFrom the placement of clique points, \nwe obtain a NHCA-extension of $\\calR'$ so that for every not yet represented \nvertex $u \\notin V(G')$ we choose $R(u)$ to be the minimal arc containing \nexactly the clique points of the maximal cliques from $M_u$.\n\nBy the proof of Lemma~\\ref{lem:nhcaG}, we obtain from\nproperties~\\ref{itm:linCh2},~\\ref{itm:linCh4} that this results in an NHCA representation of $G$\nextending $\\calR'$, since the maximal clique points are placed\ncorrectly with regards to both predrawn arcs and new arcs, and moreover, two arcs \nintersect if and only if they are predrawn or share a maximal clique.\n\\end{proof}\n\n\\begin{algorithm}[t!]\n\\KwData{A graph $G$ and a partial representation $\\calR'$.}\n\\KwResult{A \\hbox{\\rm \\sffamily NHCA}\\xspace{} representation $\\calR$ of $G$ extending $\\calR'$ or a message that none exists.}\n\\Begin{\n\\lIf{$G$ has a universal vertex}{resolve ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily INTR})$ instead}\n\\Else{\nDetermine the set of maximal cliques $\\calC$ of $G$\\;\n\\ForEach{$C\\in \\calC$}{\nDetermine $\\Reg(C)$\\;\n\\lIf{$\\Reg(C)=\\emptyset$}{\\Return{$\\calR'$ has no extension}}\n}\n$\\varepsilon:=\\frac{1}{2n+1}\\min\\{|I|,I \\text{ is a non-trivial island}\\}$\\;\nFind $C_1\\in\\calC$ with a single island\\;\n\\lIf{such $C_1$ does not exist}{\\Return{$\\calR'$ has no extension}}\n Set $\\cp(C_1)$ to the middle of $\\Reg(C_1)$\\;\nDetermine the partial order $\\prec$\\;\nBuild a PC tree $T$ on $\\calC$ capturing the constraints stated in  Theorem~\\ref{the:NHCArepextChar}\\;\n\\lIf{such $T$ does not exist}{\\Return{$\\calR'$ has no extension}}\nSolve ${\\textsc{Reorder}}(T, C_1, \\prec)$ to get the order $C_1<C_2<\\dots<C_k$\\;\n\\lIf{such order $<$ does not exist}{\\Return{$\\calR'$ has no extension}}\n\\For{$i=2$ \\KwTo $k$}{ \n $P:=\\Reg(C_i) \\cap \\{p: p> \\cp(C_{i-1})+\\varepsilon\\}$\\;\n \\lIf {$\\min P$ exists}{$\\cp(C_i):=\\min P$}\n \\lElse{$\\cp(C_i):=\\inf(P)+\\varepsilon$}\n}\n\\lForEach{$u \\notin V(G')$}{draw $R(u)$ to cover exactly $\\{\\cp(C): C\\in M_u\\}$} \n\\Return{$\\calR$}\n}\n}\n\\caption{The algorithm for the ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NHCAR})$ problem.}\\label{alg:nhca}\n\\end{algorithm}\n\n\\nhcapoly*\n\\begin{proof}\nThe correctness of Algorithm~\\ref{alg:nhca} follows from the already mentioned arguments.\nFor the computational complexity of the more complex steps note that:\n\\begin{compactitem}\n\\item\nLine 2: ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily INTR})$ can be solved in linear time~\\cite{klavik2013linear}.\n\\item\nLine 4: We run the linear time recognition algorithm for Helly circular-arc graphs~\\cite{lin2006characterizations} on $G$ and read its at most $n$ maximal cliques from any of its representation.\n\n\\item\nLines 5--8: The regions can be obtained in time $\\calO(n)$ by traversing the circle once.\nDuring the traversal two consecutive predrawn endpoints specify possible islands which can be assigned to appropriate maximal cliques.\n\\item\nLine 12: The comparable pairs of the partial order $\\prec$ can also be determined during the traversal in Steps 5--8.\n\\item\n  Line 13: The $\\calO(n^2)$ constraints can be computed in $O(n^3)$ time. The\n  construction of the PC-tree from~\\cite{hsu2003pc} also runs in $\\calO(n^3)$\n  time.\n\\item\nLine 16: The ${\\textsc{Reorder}}(T, C_1, \\prec)$ problem can be solved in $\\calO(n^2)$ time by Lemma~\\ref{lem:reordering_linear}.\n\\end{compactitem}\n\\end{proof}\n\nIn light of the hardness result from Theorem~\\ref{thm:hca_npc}, it is unlikely that this\nresult can be generalized to ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily HCAR})$. However, the hardness result for\nHelly representations crucially relies on predrawn arcs sharing endpoints.\nIndeed, if all predrawn arcs have distinct endpoints, the problem can be solved\nin a similar fashion.\n\n\\hcapoly*\n\\begin{proof}[Proof sketch]\n  We characterize extendable $\\hbox{\\rm \\sffamily HCA}\\xspace$ instances as in  \n  Theorem~\\ref{the:NHCArepextChar} (see Theorem~\\ref{the:HCArepextChar})\n  with Lemma~\\ref{lem:gavril_hca} instead of Lemma~\\ref{lem:nhcaG}.  Since the\n  predrawn arcs have pairwise distinct endpoints, we have no islands consisting\n  of a single point.\n  %\n  Note that Lemma~\\ref{lem:singleIsland} no longer applies and thus the\n  placement of $p_D$ cannot be chosen freely.  Instead, observe that\n  every maximal clique $D$ in a Helly circular-arc representation of $\\calR'$\n  has a clique point $\\cp(D)$ that can be chosen as~$p_D$.\n  %\n  Thus, we choose an arbitrary clique as $C_1$ and apply the remaining\n  procedure for every island $I$ of $C_1$, choosing $\\cp(C_1)\\in I$.\n  In contrast to our method for ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NHCAR})$\n  we have no special procedure for universal vertices.\n\\end{proof}\n\nFor details see Section~\\ref{sec:HCA}.\n\n\n\n\n\n\\section{Details fo PC-trees and The Reordering Problem}\n\\label{sec:PQtree}\n\n \\reorderingLinear*\n \\reorderingLinearProof\n\n \\section{Detailed Hardness Proofs}\n \\subsection{${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily CAR})$ and ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily HCAR})$}\n \\label{ssec:CARnp}\n\n\\hcanpc*\n\\begin{proof}\nWe reduce the $3$-{\\textsc{Partition}}\\ problem~\\cite{garey1975complexity}. \nLet $S=\\{s_1,\\dots, s_{3n}\\}$ be its instance, i.e., a collection of $3n$\nintegers summing up to $nt$ for some positive integer $t$\nsuch that each satisfies $\\frac{t}4<s_i<\\frac{t}2$, where the goal is to\npartition $S$ into $n$ disjoint subsets whose sum is always $t$. Note that the\nsize constraints on $s_i$'s ensure that every subset, which sums exactly to\n$t$, is a suitable triple.\n\nFrom $S$ and $t$ we construct a graph $G$ as follows: For the set of vertices\nwe choose\n$V(G)=\\{u_1,\\dots,u_{(t+1)n},v_1,\\dots,v_n,w_1,\\dots,w_{tn},z_1,\\dots,z_{3n}\\}$,\nwhere $u_1,\\dots,u_{(t+1)n}$ are universal vertices; vertices\n$v_1,\\dots,v_n,w_1,\\dots,w_{tn}$ form an independent set; and each $z_i$ is\nconnected to all of $u_1,\\dots,u_{(t+1)n}$ and    $s_i$ private vertices among\n$w_1,\\dots,w_{tn}$.\n\n\n\nObserve that this graph has exactly $(t+1)n$ maximal cliques, each identified\nby a vertex from $v_1,\\dots,v_n,w_1,\\dots,w_{tn}$. Moreover, each $z_i$ belongs\nto exactly $s_i$ maximal cliques.\n\nLet $p_1,\\dots, p_{(t+1)n}$ be distinct points of a circle ordered in a\nclockwise direction.\nFor each $u_i$ let $R'(u_i)=[p_i,p_{i-1}]$. For each $v_i$, let\n$R'(v_i)=[p_{(t+1)i},p_{(t+1)i}]$, (alternatively these degenerate intervals\ncould be extended to nondegenerate by expanding them by $\\frac{1}{3}$ distance\nof two nearest points $p_i,p_{i+1}$). The partial representation $\\calR'$ is\nthe just described representation of the subgraph induced by\n$\\{u_1,\\dots,u_{(t+1)n},v_1,\\dots,v_n\\}$, see Fig.~\\ref{fig:hca_npc} for an\nexample.\n\nWhen $G$ allows a representation $\\calR$ extending $\\calR'$, then each maximal\nclique with some $w_i$ will be placed on some point $p_j$ where $j\\not\\equiv 0\n\\pmod{t+1}$. Moreover clique points corresponding to some $z_i$ must be\nconsecutive and do not correspond to any other $z_{i'}$ or any $v_j$.\n\nTherefore the set system $\\{\\{s_i:\nR(z_i)\\cap\\{p_{(t+1)j+1},\\dots,p_{(t+1)j+t}\\}\\ne \\emptyset \\}:\nj\\in\\{1,\\dots,n\\}\\}$ yields the desired partition of $S$. A construction of the\nrepresentation from the set system is straightforward.\n\nTo argue that this reduction works also for \\hbox{\\rm \\sffamily NP}-hardness of ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily CAR})$\nobserve that we did not put any condition on the representation to enforce the\nHelly property, it is guaranteed by the partial representation and the\nconstruction itself. \n\nThis \\hbox{\\rm \\sffamily NP}-hardness reduction for ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily CAR})$ can be altered to the situation \nwhen the predrawn arcs are required to have pairwise distinct endpoints just by \nshrinking each arc $R'(t_i)$ by moving its tail by $\\frac{1}{5}$ distance of\nthe two nearest points $p_i,p_{i+1}$. The argument is the same, only to\nguarantee that each $w_i$ is adjacent to all universal vertices $u_1,\\dots,\nu_{(t+1)n}$, the corresponding arc $R(w_i)$ would have to cover not only a\nsingle point $p_j$ for some $j$, but instead the whole arc to the nearest tail\nin the clockwise direction, formally the arc $[p_j,R'(u_j)_t]$. Observe that by\nthis modofication we loose the Helly property as the universal vertices share\nno common point.\n\\end{proof}\n\n\n\n\n\n\\subsection{${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily UCAR})$}\n\nWe first argue the \\hbox{\\rm \\sffamily NP}-hardness of Theorem~\\ref{thm:uca_npc} by showing a\nreduction from the $3$-${\\textsc{Partition}}$ problem defined already in the proof of\nTheorem~\\ref{thm:hca_npc}.\n\nThen we provide a linear programming argument for the membership in the class\n\\hbox{\\rm \\sffamily NP}.\n\n\\ucanpc*\n\\begin{proof}\nFor a given instance of $3$-${\\textsc{Partition}}$, we construct a unit circular-arc\ngraph $G$ and its partial representation $\\calR'$. For technical reasons, we\nassume that $t \\geq 8$.\n\nLet $P_{2\\ell}$ be a path of length $2\\ell$. There exists a unit circular-arc\nrepresentation $P_{2\\ell}$ such that it spans $\\ell + \\varepsilon$ units, for some\n$\\varepsilon > 0$. To see this, note that $P_{2\\ell}$ has two independent sets of size\n$\\ell$ and each of this independent sets needs at least $\\ell + \\varepsilon$. Let $a,\nb, c$ be positive integers such that $a + b + c = t$. It follows that the\ndisjoint union of $P_{2a}$, $P_{2b}$, and $P_{2c}$ has a representation such\nthat it spans $t + \\varepsilon$ units, for some $\\varepsilon > 0$, and therefore, it can be\nfit into $t+1$ units.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.85]{uint_proof}\n\\caption{Example of packing of three paths corresponding to the triple\n$5+3+2=10$. The predrawn arcs are depicted in bold.}\n\\label{fig:unit_proof}\n\\end{figure}\n\nLet $x_0, \\dots, x_{n(t+2)-1}$ be points of the circle that divide it into\n$n(t+2)$ equal parts, i.e., vertices of a regular $n(t+2)$-gon.  The graph $G$\nis a disconnected graph consisting of $4n$ connected components.  For each\n$s_i$, we take the path $P_{2s_i}$. We further add an isolated vertex $v_j$,\nfor $j = 0, \\dots, n-1$. The partial representation $\\calR'$ is the collection\n$\\{R(v_j) : j = 0,\\dots, n-1\\}$, where $R(v_j)$ is the arc of the circle from\n$x_{j(t+2)}$ to $x_{j(t+2)+1}$ in the clockwise direction.\n\nThe predrawn arcs $R(v_0), \\dots, R(v_{n-1})$ split the circle into $n$ gaps,\nwhere each gap has exactly $t+1$ units. By the discussion above, if the $s_i$'s\ncan be partitioned into $n$ triples such that each triple sums to $t$, then a\nrepresentation of the disjoint union of the paths corresponding to a triple can\nbe placed in one of the $n$ gaps, see Fig.~\\ref{fig:unit_proof}. If the partial\nrepresentation $\\calR'$ can be extended, then we a have partition of the\n$s_i$'s into $n$ triples such that each triple sums to $t$.\n\nThe certificate for the membership in \\hbox{\\rm \\sffamily NP} of an instance $G$, $\\calR'$ is a\nlinear order of $\\prec$ on $V_G$ corresponding to the cyclic ordering of the\nintervals together with a subset of edges $E'\\subseteq E_G$.\nWithout loss of generality we assume that the predrawn arcs are\n$R(v_1),\\dots,R(v_k)$.\n\nThe following linear program, where for each $i\\in\\{1,\\dots,n\\}$ the variable\n$x_i$ corresponds to the tail coordinate $R(v_i)_t$ has a feasible solution if\nand only if $\\calR'$ can be extended to a unit interval representation $\\calR$\nwhere its cyclic ordering of intervals (as the equivalence class) contains\n$\\prec$, and where the set $E'$ specifies pairs of adjacent vertices with tail\ncoordinates at least $\\ell-1$ apart. \n\n$$\n\\begin{array}{rcll}\nx_i &=& R(v_i)_t & \\text{ for all } i\\in\\{1,\\dots,k\\} \\\\\nx_i &\\ge& 0 &  \\text{ for all } i\\in\\{k+1,\\dots,n\\} \\\\\nx_i &<& \\ell & \\text{ for all } i\\in\\{k+1,\\dots,n\\} \\\\\nx_j - x_i &\\ge& 0 & \\text{ if } i\\prec j \\text{ and } (v_i,v_j)\\in E_G \\setminus E' \\\\\nx_j - x_i &\\le& 1 &\\text{ if } i\\prec j \\text{ and } (v_i,v_j)\\in E_G \\setminus E' \\\\\nx_j - x_i &\\ge& \\ell-1 &\\text{ if } i\\prec j \\text{ and } (v_i,v_j)\\in E' \\\\\nx_j - x_i &>& 1 &\\text{ if } i\\prec j \\text{ and } (v_i,v_j)\\notin E_G \\\\\nx_j - x_i &<& \\ell-1 &\\text{ if } i\\prec j \\text{ and } (v_i,v_j)\\notin E_G \\\\\n\\end{array}\n$$\n\n\\relax\n\\end{proof}\n\n\n \\section{Additional Details for Normal Proper Helly Circular Arc Graphs}\n  \\label{sec:DetailsNPHCA}\n\n  \\nphcaChar*\n  \\nphcaCharProof\n\n  \\section{Proper Helly Circular-Arc Graphs}\n  \\label{sec:PHCA}\nNote that even though all proper Helly circular-arc graphs allow also a normal\nproper Helly circular-arc representation~\\cite{lin2013normal}, we cannot use\nAlgorithm~\\ref{alg:nphca} directly, as the given partial representation may not\nadmit a normal extension.  We can, however, still use the same machinery.\n\n\\begin{lemma}[{\\cite[Lemma~1]{lin2013normal}}]\n  \\label{lem:non-normal-universal}\n  Let $\\calR$ be a proper circular-arc representation of a graph $G$. Any two\n  vertices whose arcs are in non-normal position are both universal, i.e.\n  adjacent to all vertices of $G$.\n\\end{lemma}\n\\begin{proof}\n  Let $u$, $v$ be two vertices with arcs in non-normal position.  Then a\n  non-neighbor of $u$ would have to be represented as a proper sub-arc of\n  $R(v)$ (and vice-versa) in contradiction to $\\calR$ being proper.\n\\end{proof}\n\n\\begin{lemma}\n  \\label{lem:phca-nphca}\n  A yes-instance of ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily PHCAR})$ is a yes-instance of ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NPHCAR})$\n  if and only if no two predrawn arcs are in non-normal position.\n\\end{lemma}\n\n\\begin{proof}\n  Clearly, if the partial representation $\\calR'$ contains two\n  predrawn arcs in non-normal position, then there is no normal\n  extension of $\\calR'$.\n\n  Conversely, assume that there is no such pair and consider a\n  \\hbox{\\rm \\sffamily PHCAR}-extension $\\calR$ of $\\calR'$.\n  By Lemma~\\ref{lem:non-normal-universal}, only universal vertices can have arcs in\n  non-normal position.\n  If $\\calR'$ prescribes no universal vertex, we modify~$\\calR$ so\n  that all universal vertices are represented by the same arc\n  $R(v)$ for some arbitrary universal vertex $v$.  If $\\cal R'$\n  prescribes a universal vertex $v$, we modify~$\\calR$ by representing\n  all unprescribed universal vertices by the same arc $R(v)$.\n  \n  After this modification, if there still exists a non-normal pair of\n  arcs, there also exists a non-normal pair of prescribed arcs.  Since\n  such a pair does not exist by assumption, it follows that the\n  modified representation is an $\\hbox{\\rm \\sffamily NPHCAR}$-extension of $\\calR'$.\n\\end{proof}\n\n\n\\phcared*\n\\begin{proof}\n  Without loss of generality we assume that $G$ is not complete as otherwise \n  an extension can always be obtained by duplication of any predrawn arc. \n\n  Let $\\calR'$ be a partial representation of a graph $G\\in \\hbox{\\rm \\sffamily PHCA}\\xspace$.\n  Without loss of generality we also assume that predrawn arcs are distinct\n  as otherwise the identical arcs must correspond to twins and \n  it suffices to keep only one.\n\n  If there is no pair of prescribed arcs in non-normal position,\n  Lemma~\\ref{lem:phca-nphca} implies that the problem can be solved\n  with Theorem~\\ref{thm:nphca_poly}.\n  Hence assume that there exists a pair $R'(u),R'(v)$ of prescribed arcs in\n  non-normal position that by Lemma~\\ref{lem:non-normal-universal} must\n  correspond to universal vertices.\n  \nDenote by $A$ and $B$ the two disjoint circular arcs whose union is the\nintersection of $R'(u)$ and $R'(v)$ and by $C$ and $D$, resp., the arcs formed\nby the points of the circle that belong only to $R'(u)$ but not $R'(v)$ and\nvice-versa. The two closed arcs $A$, $B$ together with the two open $C$ and $D$\ncover the whole circle, see Fig.~\\ref{fig:phca}a.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=1,page=1]{phca.pdf}\n\\caption{(a) Partition of the circle into sections $A, B, C$ and $D$. (b) Violating\nthe Helly property by vertices for $v, w_A$ and $w_B$. }\n\\label{fig:phca}\n\\end{figure}\n\nIn order to characterize all graphs that allow a representation extension,\nconsider now any such graph $G$ together with its proper Helly representation\n$\\calR$. \n\nSince $\\calR$ is proper, any arc $R(w)\\in\\calR$ distinct from $R(v)$ \nmust intersect $C$ to guarantee $R(w)\\not\\subset R(v)$.\nWith an analogous condition on $D$ we get that either \n$A\\subseteq R(w)$ or $B\\subseteq R(w)$.\n\nThen let $V_A=\\{w_A\\in V(G): A\\subseteq R(w_A)\\}$ and $V_B=\\{w_B\\in V(G): B\\subseteq\nR(w_B)\\}$ be the sets of vertices whose arcs contain $A$ and $B$, respectively.\nBy the definition $V_A$ and $V_B$ are two cliques covering $V(G)$.\nSince $G$ is not complete, both $V_A$ and $V_B$ contain a non-universal vertex.\nAny arc intersecting $A$ and $B$ contains either $C$ or $D$, thus due to proper\nand distinct positions of arcs we have $V_A \\cap V_B = \\{u,v\\}$.\n\nObserve that when the arc representing some $w_A\\in V_A$ intersects $B$ (or vice\nversa), then $w_A$ is universal --- it would intersect all vertices from $V_B$ on\n$R(w_A)\\cap B$. On the other hand, we claim that every arc of a universal\nvertex  intersects both $A$ and $B$. Assume by a contradiction that such arc\n$R(w_A)$ would intersect only $A$.\nBy Lemma~\\ref{lem:non-normal-universal}, it is in normal position with respect\nto the arc $R(w_B)$ of some non-universal vertex $w_B\\in V_B$. Without loss of\ngenerality the intersection \nof these two arcs is in $C$. Then as depicted in Fig.~\\ref{fig:phca}b the arcs\nof $w_A,w_B$ and $v$ violate the Helly property, a contradiction. In summary,\neach universal arc  includes one of $A$ or $B$, one of $C$ or $D$ and\nintersects the remaining two.\n\nBy identical arguments in slightly altered context we show that after pruning\nall universal vertices from $G$ the resulting graph $G'$ is disjoint union of\ntwo cliques $V_A'=V_A\\setminus V_B$ and $V_B'=V_B\\setminus V_A$.\nAssume for contradiction that $G'$ contains two adjacent vertices \n$w_A\\in V_A'$ and $w_B\\in V_B'$.\nSince $w_A$, $w_B$ are not universal, $R(w_A)$, $R(w_B)$ must be in normal position by\nLemma~\\ref{lem:non-normal-universal}.\nTherefore, the intersection $R(w_A)\\cap R(w_B)$ is fully contained either in $C$ or in\n$D$. In the first case we get a contradiction as the arcs $R(v)$, $R(w_A)$,\n$R(w_B)$ violate the Helly property, see again Fig.~\\ref{fig:phca}b. In the\nother case, the arcs $R(u)$, $R(w_A)$ and $R(w_B)$ also violate the Helly\nproperty. \n\nBy Lemma~\\ref{lem:non-normal-universal}, any \narc $R(w_A)$ for a non universal vertex $w_A\\in V_A'$ contains exactly one\nendpoint of the each arc $R(x)$ representing a universal vertex.\nThe other endpoint of $R(x)$ must then belong to $R(w_B)$ of each $w_B\\in V_B'$. \nThis way we get a partition of the endpoints of universal arcs into two consecutive sets.\nWe indeed describe this partition more precisely: The arc $R(w_A)$ contains \n$\\{h_i: C\\subset R(u_i)\\} \\cup \\{t_i: D\\subset R(u_i)\\}  $, \nwhile  \n$\\{t_i: C\\subset R(u_i)\\} \\cup \\{h_i: D\\subset R(u_i)\\} \\subset R(w_B)$.\nWhen $C\\subset R(u_i)$ then $h_i$ belongs either to $A$ or $D$. The first case\nis enforced immediately and in the latter $t_i\\in B$. Hence we also have to put\n$t_i$ into $R(w_A)$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=1,page=2]{phca.pdf}\n\\caption{Position of $R(w_A)_t$ and $R(w_B)_h$ with respect to \nuniversal arcs containig $D$.}\n\\label{fig:phca-2}\n\\end{figure}\n\nThis necessary condition is also sufficient, namely a partial representation\n$\\calR'$ of a non-complete graph proper-Helly circular graph $G$ with arcs\n$R(u),R(v)$ in not-normal position allows an extension if and only if the sets\n$\\{h_i: C\\subset R'(u_i)\\} \\cup \\{t_i: D\\subset R'(u_i)\\}$, \nand   \n$\\{t_i: C\\subset R'(u_i)\\} \\cup \\{h_i: D\\subset R'(u_i)\\}$ are consecutive ---\nif no $w_A\\in V_A'$ is pre-drawn, we choose $R(w_A)$ to be the shortest arc\ncontaining the first of these two sets within $A\\cup C\\cup D$ and analogously\nfor any $w_B\\in V_B'$, see Fig.~\\ref{fig:phca-2}.\nThe remaining arcs could be obtained by replication.\n\nOur arguments convert directly to the following algorithm:\nAccept if $G$ is complete. If the partial proper Helly\nrepresentation has no pair of arcs in non-normal position, use\nAlgorithm~\\ref{alg:nphca}. \nOtherwise check whether $G'$ is isomorphic to the\nunion of two disjoint cliques; if not, reject. \nFinally, check whether the endpoints of the predrawn arcs of universal vertices\ncould be split into two consecutive sets described above and if the predrawn\narcs non-universal ones cover these sets appropriately. If not, reject.\n\nIn the affirmative case, a representation could be constructed by replication\nof arcs of three representatives: a universal vertex, one vertex from $V_A'$\nand one from $V_B'$.\nA minor simplification could applied when a representative from $V_A'$ or\n$V_B'$ is already pre-drawn and recognized. \n\\end{proof}\n\n\n\\section{Illustrations for the Normal Helly Case}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics{lem16.pdf}\n\\caption{An example of a construction of a normal Helly circular-arc representation.}\n\\label{fig:lem16}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics{lem15.pdf}\n\\caption{The region of a maximal clique cannot intersect two gaps of another.}\n\\label{fig:lem15}\n\\end{figure}\n\n\n\n\\section{Helly Circular Arc Graphs}\n\\label{sec:HCA}\n\nSurprisingly, when the Helly property is required, we can solve\n${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily HCAR})$ similarly to ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NHCAR})$\nwhen the predrawn arcs have pairwise distinct endpoints. The most notable\ndifference is that Lemma~\\ref{lem:singleIsland} no longer applies and we have\nto test every island of clique $D$ for placing $p_D$.\n\n\\begin{theorem}\n  \\label{the:HCArepextChar}\n  Let $\\calR'$ be a partial $\\hbox{\\rm \\sffamily HCA}\\xspace$ representation of $G$ where the arcs have\n  pairwise distinct endpoints and let $D$ be a maximal clique of $G$.\n  %\n  There exists an $\\hbox{\\rm \\sffamily HCA}\\xspace$ representation of $G$ that extends\n  $\\calR'$ if and only if for some point $p_D\\in \\Reg(D)$ with the corresponding\n  partial order $\\prec$ as defined in Section~\\ref{sec:NHCA}\n  there exists a linear extension $<$ of $\\prec$ such\n  that:\n  \\begin{compactenum}\n    \\item \\label{itm:HlinCh1} For every vertex $v$, $M_v$ is consecutive in $[<]$. %\n    \\item \\label{itm:HlinCh4} For every gap $J$ of a $C\\in \\calC$, the set\n      $S_J$ is consecutive in $[<]$. \n  \\end{compactenum}\n\\end{theorem}\n\n\n\n\\begin{proof}\nWe first show that, if there is such an $\\hbox{\\rm \\sffamily HCA}\\xspace$-extension of $\\calR'$, then\nthese properties are satisfied.\n\nWe obtain $<$ as the linearization of a clique ordering of an extension of $\\calR'$\nstarting with $D$ where the clique point of $D$ is $p_D$. \nBy the construction of $\\prec$, the order $<$ is a linear extension of $\\prec$.\nBy Lemmas~\\ref{lem:gavril_hca} and \\ref{lem:gapConsecutivity}, we obtain\nproperties~\\ref{itm:HlinCh1} and~\\ref{itm:linCh4}.\n    \nFor the opposite implication, let $p_D\\in \\Reg(D)$ be on an island $I_1$ of $D$\nand let $< = C_1,C_2,\\dots,C_k$ be a linear extension of $\\prec$ such that\nproperties~\\ref{itm:HlinCh1} and~\\ref{itm:HlinCh4} are\nsatisfied. We show that each $C_i\\in \\calC$ can be assigned its clique point\n$\\cp(C_i)\\in \\Reg(C_i)$ such that $\\cp(C_j)<\\cp(C_i)$ whenever $j<i$.  We\nrelocate $p_D$ to the middle of island $I_1$.  Note that the\nproperties~\\ref{itm:HlinCh1} and~\\ref{itm:HlinCh4} remain\nsatisfied for the new ordering $\\prec$ since the only possible change is that\nrelations to cliques with island $I_1$ may get lost. \n\nLet $\\varepsilon>0$ be the $\\frac{1}{2n+1}$-fraction of the length of the\nshortest nontrivial island. This choice allows us to draw all new endpoints at\ndistance at least $\\varepsilon$ but still within any chosen island or side of\nisland $I_1$.\n\nFor $C_1 = D$ we place $\\cp(C_1)$ on $p_D$. In a greedy way, when the location\nof the clique points $\\cp(C_1),\\dots,\\cp(C_{i-1})$ is settled, we determine the\nset $P$ of feasible points for $\\cp(C_i)$ that is $P = \\Reg(C_i) \\cap \\{p: p>\n\\cp(C_{i-1})+\\varepsilon\\}$. If $P$ has minimum, we place $\\cp(C_i)$ there,\notherwise we put $\\cp(C_i)$ at $\\inf(P)+\\varepsilon$.\nWe argue that such choice always exists.\n\nAssume for a contradiction that $C_i$ is the first maximal clique in the order $<$ whose \nclique point $\\cp(C_i)$ cannot be properly placed. \nNote that a maximal clique $C\\ne C_i$ can never have an island $I_C$ consisting of a\nsingle point since this can only occur as intersection of two predrawn arcs\nending at that point. Hence, by the choice of\n$\\varepsilon$, we never place a clique $C$ at the very last point of $\\Reg(C)$.\nTherefore, one clique point must be placed to the right of $\\Reg(C_i)$ before placing $\\cp(C_i)$.\nWe identify the first maximal clique $C_j, j < i$ that is\nplaced to the right of all points in $\\Reg(C_i)$. Since $\\cp(C_j)\\notin \\Reg(C_i)$, \nwe have that $\\Pre(C_i)\\ne\\Pre(C_j)$.\nBy Lemma~\\ref{lem:singleGap}, the maximal clique\n$C_j$ has a gap $J$ with $\\Reg(C_i)\\subseteq J$, see Fig.~\\ref{fig:nhca_char}.\n    \nConsider the neighboring island $I$ of $C_j$ to the left of $J$. Since\n$\\cp(C_j)$ was not placed on $I$, the clique point $\\cp(C_{j-1})$ has\nbeen placed to the right of $I$. By the choice of $C_j$, we have that\n$\\cp(C_{j-1})$ is not placed to the right of $J$ and thus $\\cp(C_{j-1})\\in J$. \nSince $\\cp(C_j)$ has been placed to the right of $\\Reg(C_i)$ and thus to\nthe right of $J$, we have $p_D\\not\\in J$ and thus $D\\not\\in S_J$.\nWith $D<C_{j-1}< C_j< C_i$, where $C_{j-1}, C_i\\in S_J$ and $D,C_j\\not\\in S_J$\nwe get a contradiction with the property~\\ref{itm:HlinCh4}, since $S_J$ is not consecutive in $[<]$.\n\nFrom the placement of clique points, \nwe obtain an $\\hbox{\\rm \\sffamily HCA}\\xspace$-extension of $\\calR'$ so that for every not yet represented \nvertex $u \\notin V(G')$ we choose $R(u)$ to be a minimal arc containing \nexactly the clique points of the maximal cliques from $M_u$.\n\nBy the Lemma~\\ref{lem:gavril_hca}, we obtain from\nproperties~\\ref{itm:HlinCh1},~\\ref{itm:HlinCh4} that this results in an $\\hbox{\\rm \\sffamily HCA}\\xspace$ representation of $G$\nextending $\\calR'$, since the maximal clique points are placed\ncorrectly with regards to both predrawn arcs and new arcs, and moreover, two arcs \nintersect if and only if they are predrawn or share a maximal clique.\nNote that we can perturbate the endpoints of the non-predrawn arcs to avoid\nshared endpoints.\n\\end{proof}\n\n\n\n\n\\begin{algorithm}[t!]\n  \\SetKw{Continue}{continue} \n  \\KwData{A graph $G$ and a partial representation $\\calR'$ consisting of arcs with pairwise distinct ends.}\n  \\KwResult{An \\hbox{\\rm \\sffamily HCA}\\xspace{} representation $\\calR$ of $G$ extending $\\calR'$ or a message that it does not exist.}\n  \\Begin{\n    Determine the set of maximal cliques $\\calC$ of $G$\\;\n    \\ForEach{$C\\in \\calC$}{\n      Determine $\\Reg(C)$\\;\n      \\lIf{$\\Reg(C)=\\emptyset$}{\\Return{$\\calR'$ has no extension}}\n    }\n    $\\varepsilon:=\\frac{1}{2n+1}\\min\\{|I|,I \\text{ is an island}\\}$\\;\n    Choose any $C_1\\in\\calC$\\;\n    \\ForEach{island $I$ of $C_1$}{\n      Set $\\cp(C_1)$ to the middle of $I$\\;\n      Determine the partial order $\\prec$\\;\n      Build a PC tree $T$ on $\\calC$ capturing the constraints stated in  Theorem~\\ref{the:HCArepextChar}\\;\n      \\lIf{such $T$ does not exist}{\\Continue}\n      Solve ${\\textsc{Reorder}}(T, C_1, \\prec)$ to get the order $C_1<C_2<\\dots<C_k$\\;\n      \\lIf{such order $<$ exists}{\n        \\For{$i=2$ \\KwTo $k$}{ \n          $P:=\\Reg(C_i) \\cap \\{p: p> \\cp(C_{i-1})+\\varepsilon\\}$\\;\n          \\lIf {$\\min P$ exists}{$\\cp(C_i):=\\min P$}\n          \\lElse{$\\cp(C_i):=\\inf(P)+\\varepsilon$}\n        }\n        \\lForEach{$u \\notin V(G')$}{draw $R(u)$ to cover exactly $\\{\\cp(C): C\\in M_u\\}$} \n        \\Return{$\\calR$}\n      }\n    }\n  }\n  \\Return{$\\calR'$ has no extension}\\;\n  \\caption{The algorithm for the ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily HCAR})$ problem.}\\label{alg:hca}\n\\end{algorithm}\n\n\\hcapoly*\n\\begin{proof}\n  The correctness of Algorithm~\\ref{alg:hca} follows from the already mentioned arguments.\n  For the computational complexity of the more complex steps note that:\n  \\begin{compactitem}\n    \\item\n      Line 2:We run the linear time recognition algorithm for Helly\n      circular-arc graphs~\\cite{lin2006characterizations} on $G$ and read its\n      at most $n$ maximal cliques from any of its representation.\n    \\item\n      Lines 3--5: The regions can be obtained in time $\\calO(n)$ by traversing\n      the circle once.  During the traversal two consecutive predrawn\n      endpoints specify possible islands which can be assigned to appropriate\n      maximal cliques.\n    \\item\n      Line 8: There are at most $O(n)$ islands for $C_1$.\n    \\item \n      Line 10: The comparable pairs of the partial order $\\prec$ can also be\n      determined by a traversal as in Steps 3--5.\n    \\item\n      Line 11: The construction of the PC-tree follows the standard approach\n      from the recognition of Helly circular-arc graphs~\\cite{hsu2003pc}. \n      Each arc is at the left start of at most one gap and each of the $O(n)$\n      maximal cliques has at most one gap without an arc inside. Hence, there\n      are $O(n)$ gaps and thus $O(n)$ constraints. $T$ can thus be constructed\n      in $O(n^2)$ time~\\cite{hsu2003pc}.\n    \n    \n    \n    \\item\n      Line 13: The ${\\textsc{Reorder}}(T, C_1, \\prec)$ problem can be solved in\n      $\\calO(n^2)$ time by Lemma~\\ref{lem:reordering_linear}.\n  \\end{compactitem}\n\\end{proof}\n\n\n \n\n\n\n\\section{Conclusions and Open Problems}\n\\label{sec:conclusions}\n\nOur study of the ${\\textsc{RepExt}}$ problem has been restricted in two ways:\n\nFirst, we have considered mostly representations satisfying Helly property as\nthis allows us to consider the clique points of maximal cliques. For\nrepresentations that do not have this property one would involve a completely\ndifferent approach.\n\nSecondly, for the recognition problem it is irrelevant whether arcs are closed\nor open, but this is not the case for the representation extension. Observe\nthat touching intervals in ${\\textsc{RepExt}}(\\hbox{\\rm \\sffamily NPHCAR})$ in Lemma~\\ref{lem:nphca_char}\nimply constraints on the ordering. For the sake of completeness it might be\nworth to check whether use of open or semi-open intervals would yield\nsignificant impact on the computational complexity.\n\n\n\\paragraph*{Acknowledgement.}\n\\label{sec:acknowledgement}\nWe thank Bartosz Walczak for inspiring comments, in particular for his hint to\nextend Theorem~\\ref{thm:hca_npc} to the case of $\\hbox{\\rm \\sffamily CAR}$ with distinct endpoints.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro}\n\nIn this paper we prove the following identity for a particular sum\nover two $9j$ and one $6j$ symbol which, to the best of our knowledge,\nseems not to appear previously as a distinct entity in the\nliterature:\n\\begin{eqnarray}\n\\lefteqn{\n\\sixj{j_1}{j_2}{j_3}{j_4}{j_5}{j_6}\n\\sixj{j_3}{j_4}{j_5}{\\Delta}{j^\\prime_5}{j^\\prime_4}\n\\sixj{j^\\prime_1}{j^\\prime_2}{j_3}{j^\\prime_4}{j^\\prime_5}{j^\\prime_6}\n}\n&& \\nonumber \\\\\n&=& \\sum_{k,\\ell} [k][\\ell] (-1)^\\Phi \n\\ninej{j_1}{j^\\prime_1}{k}{j_6}{j^\\prime_6}{\\ell}{j_5}{j^\\prime_5}\n{\\Delta}\n\\ninej{j_2}{j^\\prime_2}{k}{j_6}{j^\\prime_6}{\\ell}{j_4}{j^\\prime_4}\n{\\Delta}\n\\sixj{j_1}{j_2}{j_3}{j^\\prime_2}{j^\\prime_1}{k}\n\\, , \\label{newident}\n\\end{eqnarray}\nwhere $[j] \\equiv 2j + 1$ is the multiplicity of the spin-$j$\nirreducible representation, and the argument of the phase $(-1)^\\Phi$\nis given by\n\\begin{equation} \\label{phasearg}\n\\Phi = j_1 + j_2 + j_6 - j^\\prime_4 - j^\\prime_5 - j^\\prime_6 + k -\n\\Delta \\ .\n\\end{equation}\n\nThis identity is a generalization of the well-known Biedenharn-Elliott\n(BE) sum rule [Eq.~(\\ref{BE}) below], to which (as we shall show) it\nreduces when $\\Delta$ is set to zero.\n\nThis identity arises when considering the angular momentum coupling\nscheme\n\\begin{equation} \\label{coupling}\n\\begin{array}{ccc}\n{\\bf j_3} = {\\bf j_1} + {\\bf j_2} , \\ &\n{\\bf j_3} = {\\bf j^\\prime_1} + {\\bf j^\\prime_2} , \\ &\n{\\bf j^\\prime_1} = {\\bf j_1} + {\\bf k} , \\\\\n{\\bf j_6} = {\\bf j_2} + {\\bf j_4} , \\ &\n{\\bf j^\\prime_6} = {\\bf j^\\prime_2} + {\\bf j^\\prime_4} , \\ &\n{\\bf j_2} = {\\bf j^\\prime_2} + {\\bf k} , \\\\\n{\\bf j_5} = {\\bf j_3} + {\\bf j_4} , \\ &\n{\\bf j^\\prime_5} = {\\bf j_3} + {\\bf j^\\prime_4} , \\ & \n{\\bf j^\\prime_4} = {\\bf j_4} + {\\bf \\Delta} \\\\\n{\\bf j_5} = {\\bf j_1} + {\\bf j_6} , \\ &\n{\\bf j^\\prime_5} = {\\bf j^\\prime_1} + {\\bf j^\\prime_6} , \\ &\n{\\bf j^\\prime_5} = {\\bf j_5} + {\\bf \\Delta} , \\\\\n{\\bf j^\\prime_6} = {\\bf j_6} + {\\xbf \\ell} , \\ &\n{\\bf \\Delta} = {\\bf k} + {\\xbf \\ell} .\n\\end{array}\n\\end{equation}\nThe unprimes and primes suggest (for example) initial- and final-state\nquantum numbers whose inequality is enforced by a ``spurion'' ${\\bf\n\\Delta}$.  Other physically useful coupling schemes may be obtained by\nglobally replacing any of these angular momenta {\\bf j} by their\ntime-reversed forms $\\tilde{\\bf j} \\equiv -{\\bf j}$, where $\\tilde{\\bf\nj}$ is the angular momentum operator whose eigenstates are related to\nthose $( \\left| j m \\right>)$ of ${\\bf j}$ by $(-1)^{j+m} \\left| j -m\n\\right>$.  This manipulation gives a well-defined meaning to the\nconcept of subtracting angular momentum operators~\\cite{Edmonds}.\n\nAs an explicit physical example using this coupling scheme, consider\nmeson-baryon scattering ($\\phi B \\to \\phi^\\prime B^\\prime$) in which\nan additional isoscalar meson $f$ is produced with total angular\nmomentum $J_f$ with respect to the other final-state particles: $\\phi\nB \\to \\phi^\\prime B^\\prime f$.  The specified observables are isospins\nand angular momenta: $I_{\\phi (\\phi^\\prime)}$, $J_{\\phi\n(\\phi^\\prime)}$ for the mesons (as usual, $J$ denotes spin and orbital\nangular momenta combined), and $I_{B (B^\\prime)}$, $S_{B (B^\\prime)}$\nfor the baryons.  The total $s$-channel quantum numbers are ${\\bf I_s}\n\\! \\equiv \\! {\\bf I}_\\phi \\! + {\\bf I}_B \\! = \\! {\\bf I}_{\\phi^\\prime}\n\\! + {\\bf I}_{B^\\prime}$, ${\\bf J_s} \\! = \\! {\\bf J}_\\phi \\! + {\\bf\nS}_B$, ${\\bf J^\\prime_s} \\!  = \\! {\\bf J}_{\\phi^\\prime} \\! + {\\bf\nS}_{B^\\prime}$, and ${\\bf J^\\prime_s} \\!  = \\! {\\bf J_s} \\! - \\! {\\bf\nJ}_f$.  In a chiral soliton model or the $1\/N_c$ expansion of QCD, the\nvector sum of isospin and angular momentum for each particle assumes\nextra significance: The stable baryons (such as nucleons) are\nzero-eigenvalue states of the operators ${\\bf I}_B \\! + \\!  {\\bf S}_B$\nand ${\\bf I}_{B^\\prime} \\! + \\!  {\\bf S}_{B^\\prime}$, and the\nscattering is characterized by the ``grand spins'' ${\\bf K} \\! \\equiv\n\\! {\\bf I_s} \\! + \\! {\\bf J_s}$, ${\\bf K}^\\prime \\! \\equiv \\!  {\\bf\nI_s} \\! + \\! {\\bf J^\\prime_s}$.  The application of\nEq.~(\\ref{newident}) arises when one considers processes such as this\nnot in the $s$-channel but the $t$-channel~\\cite{cross}: Then ${\\bf\nI}_{\\phi^\\prime} \\! = \\! {\\bf I}_\\phi \\! + {\\bf I_t}$ and ${\\bf\nJ}_{\\phi^\\prime} \\! = \\! {\\bf J}_\\phi \\! + {\\bf J_t}$.  The full\nidentification using the notation of Eq.~(\\ref{coupling}) is ${\\bf\nj_1} \\! \\to \\!  {\\bf I}_\\phi$, ${\\bf j_2} \\! \\to \\! {\\bf I}_B$, ${\\bf\nj_3} \\! \\to \\!  {\\bf I_s}$, ${\\bf j_4} \\! \\to \\! {\\bf J_s}$, ${\\bf\nj_5} \\! \\to \\! {\\bf K}$, ${\\bf j_6} \\! \\to \\! {\\bf J}_\\phi$ (and\nanalogously for the primed ${\\bf j}$'s), and ${\\bf k} \\! \\to \\! {\\bf\nI_t}$, ${\\xbf \\ell} \\! \\to \\!  {\\bf J_t}$, and ${\\bf \\Delta} \\! \\to \\!\n-{\\bf J}_f$.  The transcription of Eq.~(\\ref{newident}) then reads\n\\begin{eqnarray}\n\\lefteqn{\n\\sixj{I_\\phi}{I_B}{I_s}{J_s}{K}{J_\\phi}\n\\sixj{I_s}{J_s}{K}{J_f}{K^\\prime}{J_s^\\prime}\n\\sixj{I_{\\phi^\\prime}}{I_{B^\\prime}}{I_s}{J_s^\\prime}{K^\\prime}\n{J_{\\phi^\\prime}}\n}\n&& \\nonumber \\\\\n&=& \\sum_{I_t,J_t} [I_t][J_t] (-1)^\\Phi \n\\ninej{I_\\phi}{I_{\\phi^\\prime}}{I_t}{J_\\phi}{J_{\\phi^\\prime}}{J_t}{K}\n{K^\\prime}{J_f}\n\\ninej{I_B}{I_{B^\\prime}}{I_t}{J_\\phi}{J_{\\phi^\\prime}}{J_t}{J_s}\n{J_s^\\prime}{J_f}\n\\sixj{I_\\phi}{I_B}{I_s}{I_{B^\\prime}}{I_{\\phi^\\prime}}{I_t}\n\\, , \\label{example}\n\\end{eqnarray}\nwhere now $\\Phi \\! = \\! I_\\phi \\! + \\! I_B \\! + \\! J_\\phi \\! - \\!\nJ_s^\\prime \\! - \\! K^\\prime \\! - \\! J_{\\phi^\\prime} \\! + \\! I_t \\! -\n\\! J_f$.  The three $6j$ symbols on the left-hand side (expressed\nsolely in terms of $s$-channel quantities) appear as coefficients in\nexpressions for partial-wave scattering amplitudes for the process\n$\\phi B \\to \\phi^\\prime B^\\prime f$ written in terms of underlying\n``reduced'' amplitudes labeled by $K$ values; for example, the case in\nwhich $f$ is absent (effectively, $J_f \\! = \\! 0$) has been studied\nfor quite some time~\\cite{MP}.  Expressing amplitudes in terms of\n$t$-channel quantities [as is manifest on the right-hand side of\nEq.~(\\ref{example})] is of particular interest because such amplitudes\nscale as $1\/N_c^{|I_t \\!  - \\!  J_t|}$~\\cite{cross}, thereby creating\na hierarchy of dominant and subdominant amplitudes in the $1\/N_c$\nexpansion of QCD.\n\nThe most illuminating proof of Eq.~(\\ref{newident}) uses diagrammatic\ntechniques, an approach we summarize in Sec.~\\ref{diagrammatic}.  We\npresent the diagrammatic proof in Sec.~\\ref{proof}, and finally an\nalgebraic proof, using standard SU(2) identities, in\nSec.~\\ref{algebraic}.\n\n\\section{Diagrammatic Method for Coupling Angular Momenta}\n\n\\label{diagrammatic}\n\n\\subsection{Notation}\n\nAlgebraic techniques for manipulating Clebsch-Gordan coefficients\n(CGC) to obtain invariants (quantities independent of magnetic quantum\nnumbers $m$, such as $6j$ and $9j$ symbols) are certainly\nstraightforward and appear in all standard treatments of the\ntopic~\\cite{Edmonds}.  However, at a certain point of complexity these\ntechniques become particularly cumbersome, and the bookkeeping\nnecessary to impose the required identities for simplifying such\nexpressions [particularly with regard to the numerous phases $(-1)^n$\nthat arise] becomes increasingly onerous.  A much cleaner strategy is\nto use diagrammatic techniques introduced originally by Jucys\n(alternate spellings {\\it Yutsis}, {\\it Iutsis}), Levinson, and\nVanagas (JLV)~\\cite{JLV}.  In this method, each angular momentum $j$\nis represented as a line, and each vertex represents a $3j$ symbol (or\nCGC).  The quantum number $m$ corresponding to $j$ is summed if and\nonly if line $j$ is internal to the diagram.\n\nThe diagrammatic technique is particularly valuable because of two\nfeatures: First, the identities involved in combining large complexes\nof angular momenta become topological in nature, and the ability to\nidentify them reduces to one's cunning in picturing how to connect the\nlines.  Second, the phases endemic to CGC are incorporated in the\ndiagrams very neatly (as too are factors of $[j]$, but we do not need\nthem here), appearing as either signs at the vertices or arrows on the\nlines, in the manner described below.  The JLV technique is laid out\npedagogically in the text by Lindgren \\& Morrison (LM)~\\cite{LM} or\nthe review by Wormer and Paldus~\\cite{WP}.  Here we list only the\nfeatures essential for this paper.\n\nFor starters, the vertex in Fig.~\\ref{fig:3j} represents a $3j$\nsymbol:\n\\begin{eqnarray}\n\\left(\\begin{array}{ccc}\nj_1 & j_2 & j_3 \\\\\nm_1 & m_2 & -m_3\n\\end{array}\n\\right)(-1)^{j_3-m_3},\n\\end{eqnarray}\n\\begin{figure}[ht]\n\\epsfxsize 1.8 in \\epsfbox{3j.eps}\n\\caption{Graphical representation of the $3j$ symbol.}\n\\label{fig:3j}\n\\end{figure}\nwith the specific ordering of the $j$'s and $m$'s as\n(counter)clockwise defining the vertex orientation as (positive)\nnegative, as indicated by a sign at the vertex in the diagram.  The\narrow on $j_3$ introduces the phase $(-1)^{j_3-m_3}$; were it pointing\ntoward the vertex, the phase introduced would be $(-1)^{j_3+m_3}$.\nNote that we follow the LM arrow convention, which is opposite that of\nJLV convention, since as argued in Ref.~\\cite{LM} it is more closely\nanalogous to the flow of momentum in scattering diagrams.\n\nSeveral manipulations help simplify such calculations:\n\\begin{enumerate}\n\\item Two arrows pointing in opposite direction on the same line can be\nremoved.\n\\item Reversing the direction of an arrow introduces an additional\n$(-1)^{2j}$.\n\\item Reversing the orientation of the vertex (changing the sign symbol)\nintroduces an additional $(-1)^{j_1+j_2+j_3}$.\n\\item Introducing three arrows all pointing inward or outward at a\nvertex does not change the value of the diagram.\n\\end{enumerate}\n\n\\subsection{Invariants}\n\nThe combination of several vertices (with no external lines) forms a\ndiagram representing a higher-order $3nj$ symbol; for example, the\nirreducible combination of 4 vertices, as depicted in\nFig.~\\ref{fig:6j}, forms the $6j$ symbol,\n\\begin{equation}\n\\sixj{j_1}{j_2}{j_3}{j_4}{j_5}{j_6} ,\n\\end{equation}\n\\begin{figure}[ht]\n\\epsfxsize 1.8 in \\epsfbox{6j.eps}\n\\caption{Graphical representation of the $6j$ symbol.}\n\\label{fig:6j}\n\\end{figure}\nand the irreducible combination of 6 vertices, as depicted in\nFig.~\\ref{fig:9j}, forms the $9j$ symbol,\n\\begin{equation}\n\\ninej{j_1}{j_2}{j_3}{j_4}{j_5}{j_6}{j_7}{j_8}{j_9} ,\n\\end{equation}\n\\begin{figure}[ht]\n\\epsfxsize 1.8 in \\epsfbox{9j.eps}\n\\caption{Graphical representation of the $9j$ symbol.}\n\\label{fig:9j}\n\\end{figure}\n\n\\subsection{Theorems}\n\nThe true power of the JLV approach arises through a serious of\ntheorems~\\cite{JLV,LM}, reminiscent of the factorization theorems of\nquantum field theory, that allow one to cut diagrams internally\nconnected only by a small number $n$ of lines.  Consider a diagram\nconsisting of two such blocks, $\\overline{\\alpha}$ and $\\beta$, such\nthat $\\overline{\\alpha}$ is {\\it closed\\\/} (no external lines) and in\n{\\it normal form\\\/} (every internal line on $\\overline{\\alpha}$\ncarries an arrow, and any arrows on the lines connecting with $\\beta$\nare pushed into the block $\\beta$).  Then one obtains a series of\ntheorems JLV$n$, $n \\! = \\! 1, 2, \\ldots$.  Of greatest interest to us\nhere are JLV3 (depicted in Fig.~\\ref{fig:JLV3}) and JLV4\n(Fig.~\\ref{fig:JLV4}).  JLV3, for example, applied to a system in\nwhich block $\\beta$ is empty, turns out to be none other than the\nWigner-Eckart theorem.\n\n\\begin{figure}[ht]\n\\epsfxsize 5.8 in \\epsfbox{JLV3.eps}\n\\caption{The JLV3 theorem.}\n\\label{fig:JLV3}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\epsfxsize 4.8 in \\epsfbox{JLV4.eps}\n\\caption{The JLV4 theorem.}\n\\label{fig:JLV4}\n\\end{figure}\n\n\\section{Proof of the Identity} \\label{proof}\n\nThe Biedenharn-Elliott sum rule~\\cite{Edmonds} reads\n\\begin{eqnarray} \\label{BE}\n\\lefteqn{\n\\sixj{j_1}{j_2}{j_3}{j_4}{j_5}{j_6}\n\\sixj{j^\\prime_1}{j^\\prime_2}{j_3}{j_4}{j_5}{j^\\prime_6}\n} \\nonumber \\\\ & = &\n\\sum_{\\cal J} (-1)^\\sigma [{\\cal J}]\n\\sixj{j_1}{j_6}{j_5}{j^\\prime_6}{j^\\prime_1}{\\cal J}\n\\sixj{j_2}{j_6}{j_4}{j^\\prime_6}{j^\\prime_2}{\\cal J}\n\\sixj{j_1}{j_2}{j_3}{j^\\prime_2}{j^\\prime_1}{\\cal J} ,\n\\end{eqnarray}\nwhere $\\sigma$ is the sum of the 9 distinct arguments on the left-hand\nside, plus ${\\cal J}$.  Here we wish to find the analog of the BE sum\nrule for the following three $6j$ symbols, the left-hand side of\nEq.~(\\ref{newident}), represented graphically in\nFig.~\\ref{fig:three_6j}:\n\\begin{equation} \\label{three6js}\n\\sixj{j_1}{j_2}{j_3}{j_4}{j_5}{j_6}\n\\sixj{j_3}{j_4}{j_5}{\\Delta}{j^\\prime_5}{j^\\prime_4}\n\\sixj{j^\\prime_1}{j^\\prime_2}{j_3}{j^\\prime_4}{j^\\prime_5}{j^\\prime_6}\n\\end{equation}\n\n\\begin{figure}[ht]\n\\epsfxsize 5.8 in \\epsfbox{three_6j.eps}\n\\caption{The three $6j$ symbols of Eq.~(\\ref{three6js}).}\n\\label{fig:three_6j}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\epsfxsize 4.0 in \\epsfbox{12j.eps}\n\\caption{Result of combining the three $6j$ symbols of\nEq.~(\\ref{three6js}) or Fig.~\\ref{fig:three_6j}.}\n\\label{fig:12j}\n\\end{figure}\n\n\n\\begin{figure}[ht]\n\\epsfxsize 4.8 in \\epsfbox{6j_12j.eps}\n\\caption{The result of applying JLV4 to Fig.~\\ref{fig:12j}.}\n\\label{fig:6j_12j}\n\\end{figure}\n\nBy means of the JLV3 theorem, the three $6j$ symbols of\nEq.~(\\ref{three6js}) combine into a diagram with 12 quantum numbers\nshown in Fig.~\\ref{fig:12j}.  In order to introduce the quantum number\n$k$ of Eq.~(\\ref{newident}), we cut the diagram on the four lines,\n$j_1$, $j^\\prime_1$, $j_2$, and $j^\\prime_2$ using the JLV4 theorem.\nThe summed quantum number in this diagram is indeed the desired $k$,\nand the result is a $6j$ symbol and a $12j$ {\\it symbol of the second\nkind\\\/}, as shown in Fig.~\\ref{fig:6j_12j}.  Since the latter object\nis surely obscure to most readers, we pause to point out that, while\nhigher $3nj$ symbols are not difficult to generate and manipulate\nusing the diagrammatic approach, and while they possess remarkable and\nintricate symmetry properties, they may always be reduced to products\nover convenient sums of $6j$ and $9j$ symbols using the JLV\ntheorems~\\cite{BC,MB}.  One other fine point is that the attractive\nsquare diagram in Fig.~\\ref{fig:6j_12j} actually differs from the true\n$12j$ symbol by a phase $(-1)^{j_1 - j_2 + j^\\prime_4 - j^\\prime_5}$,\nbut this distinction is purely formal; like Eq.~(\\ref{newident}), the\ndiagram as depicted is symmetric upon exchange of primed and unprimed\nquantum numbers.\n\nTo introduce the quantum number $\\ell$, we make another four-line cut,\nhere on $j_6$, $j^\\prime_6$, $k$, and $\\Delta$ in the $12j$\nsymbol of Fig.~\\ref{fig:6j_12j}.  The result of this action is shown\nin Fig.~\\ref{fig:6j_two_9j}.  The hexagonal figures are none other\nthan standard $9j$ symbols in canonical JLV form.\n\n\\begin{figure}[ht]\n\\epsfxsize 6.45 in \\epsfbox{6j_two_9j.eps}\n\\caption{The result of applying JLV4 to Fig.~\\ref{fig:6j_12j},\nwhich is expressed algebraically in Eq.~(\\ref{newident}).}\n\\label{fig:6j_two_9j}\n\\end{figure}\nIn fact, we have expressed a particular product of three $6j$ symbols\n[Eq.~(\\ref{three6js})] as a $6j$ and two $9j$ symbols summed over two\nnew quantum numbers, $k$ and $\\ell$.  To repeat Eq.~(\\ref{newident}),\n\\begin{eqnarray}\n\\lefteqn{\n\\sixj{j_1}{j_2}{j_3}{j_4}{j_5}{j_6}\n\\sixj{j_3}{j_4}{j_5}{\\Delta}{j^\\prime_5}{j^\\prime_4}\n\\sixj{j^\\prime_1}{j^\\prime_2}{j_3}{j^\\prime_4}{j^\\prime_5}{j^\\prime_6}\n}\n&& \\nonumber \\\\\n&=& \\sum_{k,\\ell} [k][\\ell] (-1)^\\Phi \n\\ninej{j_1}{j^\\prime_1}{k}{j_6}{j^\\prime_6}{\\ell}{j_5}{j^\\prime_5}\n{\\Delta}\n\\ninej{j_2}{j^\\prime_2}{k}{j_6}{j^\\prime_6}{\\ell}{j_4}{j^\\prime_4}\n{\\Delta}\n\\sixj{j_1}{j_2}{j_3}{j^\\prime_2}{j^\\prime_1}{k}\n\\, ,\n\\end{eqnarray}\nwith the phase given by\n\\begin{equation}\n\\Phi = j_1 + j_2 + j_6 - j^\\prime_4 - j^\\prime_5 - j^\\prime_6 + k -\n\\Delta \\ .\n\\end{equation}\nWhile $\\Phi$ is not primed-unprimed symmetric, neither are the two new\n$9j$ symbols, because switching their initial and final quantum\nnumbers requires column exchanges; when the necessary permutations are\ntaken into account, it is quite straightforward to show this explicit\nsymmetry.  To the best of our knowledge, Eq.~(\\ref{newident}) actually\nrepresents a new SU(2) identity~\\cite{why_new}, one that reduces for\n$\\Delta \\! = \\! 0$ to the BE sum rule.  This reduction becomes\napparent when one simplifies using special cases\n\\begin{equation}\n\\sixj{j_3}{j_4}{j_5}{0}{j^\\prime_5}{j^\\prime_4} =\n\\frac{(-1)^{j_3 + j_4 + j_5}}{\\sqrt{[j_4][j_5]}}\n\\delta_{j^{\\vphantom\\dagger}_4 j^\\prime_4}\n\\delta_{j^{\\vphantom\\dagger}_5 j^\\prime_5} ,\n\\end{equation}\n\\begin{equation}\n\\ninej{j_1}{j^\\prime_1}{k}{j_6}{j^\\prime_6}{\\ell}{j_5}{j^\\prime_5}{0}\n= \\frac{(-1)^{j^\\prime_1 + j_6 + j_5 + k}}{\\sqrt{[k][j_5]}}\n\\delta_{j^{\\vphantom\\dagger}_5 j^\\prime_5}\n\\delta_{k^{\\vphantom\\dagger}_{\\vphantom\\bullet}\n\\ell^{\\vphantom\\dagger}_{\\vphantom\\bullet}}\n\\sixj{j_1}{j_6}{j_5}{j^\\prime_6}{j^\\prime_1}{k} ,\n\\end{equation}\nand\n\\begin{equation}\n\\ninej{j_2}{j^\\prime_2}{k}{j_6}{j^\\prime_6}{\\ell}{j_4}{j^\\prime_4}{0}\n= \\frac{(-1)^{j^\\prime_2 + j_6 + j_4 + k}}{\\sqrt{[k][j_4]}}\n\\delta_{j^{\\vphantom\\dagger}_4 j^\\prime_4}\n\\delta_{k^{\\vphantom\\dagger}_{\\vphantom\\bullet}\n\\ell^{\\vphantom\\dagger}_{\\vphantom\\bullet}}\n\\sixj{j_2}{j_6}{j_4}{j^\\prime_6}{j^\\prime_2}{k} ,\n\\end{equation}\nin which case both $k$ and $\\ell$ reduce to ${\\cal J}$ of\nEq.~(\\ref{BE}).\n\n\\section{Algebraic Proof} \\label{algebraic}\n\nEquation~(\\ref{newident}) is also fairly straightforward to verify\nalgebraically, once the right-hand side is known.  Here we reduce\nthis side of the equation.  We use the symmetry properties of $6j$\nsymbols and $9j$ symbols: $6j$ symbols are invariant under the\npermutation of any two columns or under the exchange of upper and\nlower entries for any two columns, while $9j$ symbols are invariant\nunder even permutations of any two rows or columns.  We also use the\nBE sum rule, Eq.~(\\ref{BE}).  Furthermore, $9j$ symbols may be\nexpanded in terms of $6j$ symbols using the standard\nidentity~\\cite{Edmonds}\n\\begin{equation} \\label{9jexpand}\n\\ninej{j_6}{j^\\prime_6}{\\ell}{j_4}{j^\\prime_4}{\\Delta}{j_2}\n{j^\\prime_2}{k}\n= \\sum_x (-1)^{2x} [x] \\sixj{j_6}{j_4}{j_2}{j^\\prime_2}{k}{x}\n\\sixj{j^\\prime_6}{j^\\prime_4}{j^\\prime_2}{j_4}{x}{\\Delta}\n\\sixj{\\ell}{\\Delta}{k}{x}{j_6}{j^\\prime_6} ,\n\\end{equation}\nwhere the $9j$ symbol is equivalent to the second one in\nEq.~(\\ref{newident}).  The first $9j$ symbol of Eq.~(\\ref{newident})\nand the last $6j$ symbol of Eq.~(\\ref{9jexpand}) (arguments\nrearranged), along with their sum over $\\ell$, may be re-expressed\nusing another standard identity~\\cite{Edmonds}:\n\\begin{equation} \\label{Edmonds6.4.8}\n\\sum_{\\ell} [\\ell]\n\\ninej{j_6}{j^\\prime_6}{\\ell}{j_5}{j^\\prime_5}{\\Delta}{j_1}\n{j^\\prime_1}{k} \\sixj{j_6}{j^\\prime_6}{\\ell}{\\Delta}{k}{x} = (-1)^{2x}\n\\sixj{j_5}{j^\\prime_5}{\\Delta}{j^\\prime_6}{x}{j^\\prime_1}\n\\sixj{j_1}{j^\\prime_1}{k}{x}{j_6}{j_5} .\n\\end{equation}\nThe phase $(-1)^{2x}$ cancels between Eqs.~(\\ref{9jexpand}) and\n(\\ref{Edmonds6.4.8}), and the remaining expression reads\n\\begin{equation} \\label{preBE}\n\\sum_x [x] \\sum_k [k] (-1)^\\Phi \n\\sixj{j^\\prime_6}{j^\\prime_4}{j^\\prime_2}{j_4}{x}{\\Delta}\n\\sixj{j_5}{j^\\prime_5}{\\Delta}{j^\\prime_6}{x}{j^\\prime_1}\n\\sixj{j_6}{j_4}{j_2}{j^\\prime_2}{k}{x}\n\\sixj{j_1}{j^\\prime_1}{k}{x}{j_6}{j_5}\n\\sixj{j_1}{j_2}{j_3}{j^\\prime_2}{j^\\prime_1}{k} .\n\\end{equation}\n\nThe summed angular momentum $k$ appears only in the last three of\nthese $6j$ symbols, and the phase argument $\\Phi$\n[Eq.~(\\ref{phasearg})] conveniently contains a factor of $k$,\nsuggesting that they can be simplified using the BE sum rule.  Indeed,\nwriting the sum of the ten angular momenta in the last three $6j$\nsymbols as $\\sigma$, one has\n\\begin{equation} \\label{firstBE}\n\\sum_k (-1)^\\sigma [k]\n\\sixj{j_2}{j_6}{j_4}{x}{j^\\prime_2}{k}\n\\sixj{j_6}{j_5}{j_1}{j^\\prime_1}{k}{x}\n\\sixj{j_2}{j_1}{j_3}{j^\\prime_1}{j^\\prime_2}{k} =\n\\sixj{j_1}{j_2}{j_3}{j_4}{j_5}{j_6}\n\\sixj{j_4}{j_5}{j_3}{j^\\prime_1}{j^\\prime_2}{x} .\n\\end{equation}\nThe first $6j$ symbol appears on the left-hand side of\nEq.~(\\ref{newident}).  The remaining factors to be simplified (those\ncontaining $x$) are the first two $6j$ symbols in Eq.~(\\ref{preBE}),\nthe second in Eq.~(\\ref{firstBE}), and the phase $(-1)^{\\Phi-\\sigma}\n\\! = \\!  (-1)^{\\sigma - \\Phi}$ (since $\\Phi$ and $\\sigma$ are integers)\n$\\equiv \\! (-1)^{\\tilde\\sigma}$.  The BE sum rule again simplifies the\nexpression:\n\\begin{equation} \\label{secondBE}\n\\sum_x (-1)^{\\tilde\\sigma} [x]\n\\sixj{j_4}{\\Delta}{j^\\prime_4}{j^\\prime_6}{j^\\prime_2}{x}\n\\sixj{\\Delta}{j^\\prime_5}{j_5}{j^\\prime_1}{x}{j^\\prime_6}\n\\sixj{j_4}{j_5}{j_3}{j^\\prime_1}{j^\\prime_2}{x} =\n\\sixj{j_3}{j_4}{j_5}{\\Delta}{j^\\prime_5}{j^\\prime_4}\n\\sixj{j^\\prime_1}{j^\\prime_2}{j_3}{j^\\prime_4}{j^\\prime_5}{j^\\prime_6}\n,\n\\end{equation} \nsince $\\tilde\\sigma$ is easily shown to be the sum of the ten angular\nmomenta on the left-hand side.  These two $6j$ symbols, along with the\nfirst from Eq.~(\\ref{firstBE}), complete the left-hand side of\nEq.~(\\ref{newident}), and hence complete the proof.\n\n\\section*{Acknowledgments}\nThis work was supported by the NSF under Grant No.\\ PHY-0456520.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\uppercase{Introduction}}\\label{sec:introduction}\nIn last decade, Government of India has taken several e-Governance initiatives such as a unique digital identity (referred to as \\textit{Aadhaar} \\cite{aadhaar}) for every resident, online Aadhaar based authentication and several online citizen centric services such as \\textit{eKYC}, \\textit{eSign}, and \\textit{DigiLocker}. At present, most of these services are built using traditional Public Key Infrastructure (PKI) with limited data privacy in which specifying authorized entities beforehand which are permitted to access data may not be possible and even if possible, the solution may not scale.\n\nIn \\textit{DigiLocker} \\cite{digilocker}, documents of subscribers are hosted on public cloud which is assumed to be a trusted entity. However, cloud storage may not be trustworthy and may be susceptible to insider attacks. Moreover, instead of providing a \\textit{reactive access authorization to a single requester} (using OAuth2 \\cite{oauth2}), a\n subscriber may want to provide a \\textit{proactive access authorization to multiple requester} meeting certain criteria of attributes.\n\nCiphertext-Policy Attribute-Based Encryption (CP-ABE) \\cite{bethencourt2007ciphertext} is a recent cryptographic mechanism which can improve data privacy, but the right implementation and efficiency of it are still some of the major concerns for its wide deployment. This paper presents a scheme to implement privacy enhanced DigiLocker based on CP-ABE.\n\n\\section{\\uppercase{Related Work}}\n\\noindent Recent developments in cryptography have introduced Attribute-Based Encryption (ABE) \\cite{goyal2006attribute} in which encryption is done under a set of attributes. ABE is classified in Key-Policy ABE (KP-ABE) \\cite{goyal2006attribute} and CP-ABE. In KP-ABE, access policy is encoded in subscriber's private key and a set of attributes are encoded in ciphertext. In CP-ABE, access policy is encoded in ciphertext and a set of attributes are encoded in subscriber's private key. In CP-ABE, only if the set of required attributes encoded in receiver's private key satisfies the access policy encoded in received ciphertext, will the receiver be able to decrypt the ciphertext. Since the introduction of CP-ABE, researchers have proposed innovative mechanisms to use it to improve data privacy \\cite{zhou2012efficient}, \\cite{ji2014privacy}.\n\n\\section {\\uppercase{Digital Lockers in India}}\n\\noindent \\textit{DigiLocker} is an Aadhaar based online service which facilitates \\textit{subscribers} to store e-documents, \\textit{issuer} agencies to provide e-documents and \\textit{requester} applications to get access to e-documents. An \\textit{e-document} is a digitally signed electronic document. \\textit{Repositories} are provided by issuers to host collection of e-documents. \\textit{Digital Locker} is a storage space provided to each subscriber to store e-documents. \\textit{Requester} is an application which seeks access to some e-document. All participating entities must adhere to \\textit{Digital Locker Technology Specification (DLTS)} \\cite{dlts}. \n\nAn e-document is uniquely identified by a \\textit{Unique Resource Identifier (URI)} which is a triplet of the form $\\mathrm{\\langle{IssuerID::DocType::DocID}\\rangle}$, where $IssuerID$ is a unique identifier ot the issuer, e.g., $\\mathtt{CBSE}$, for Central Board of Secondary Education. $DocType$ is a classification of e-documents as defined by the issuer. For example, CBSE may classify certificates into $\\mathtt{MSTN}$ for 10th mark sheet and $\\mathtt{KVYP}$ for certificates issued to KVPY scholarship fellow. $DocType$ also helps issuers to use different repositories for different types of e-documents. $DocID$ is an issuer defined unique identifier (an alphanumeric string) of the e-document within a document type. Some hypothetical examples of e-document URI are $\\langle{\\mathtt{CBSE::MSTN::22636726}}\\rangle$, $\\langle{\\mathtt{DLSSB::HSMS::GJSGEJXS}\\rangle}$. DigiLocker ensures data integrity of e-documents by mandating that all e-documents are digitally signed by issuers.\n\nWhen an issuer is registered, it provides two APIs, namely, $\\mathtt{PullDoc}$ to pull an e-document based on a given URI and $\\mathtt{PullUri}$ to pull all URIs meeting a given search criteria. When a requester application is registered, it is given a unique requester identifier, a secret key which is shared between DigiLocker and requester application and a $\\mathtt{FetchDoc}$ API is given to access e-documents based on URI. Based on the URI, $\\mathtt{FetchDoc}$ forwards the request to appropriate issuers to retrieve the e-document. DigiLocker ensures secure data access of e-documents by API license keys, secure transport, an explicit authentication (if required by \\textit{DocType}) and all requests and responses to be digitally signed.\n\n\\section{\\uppercase{Preliminaries}}\n\\noindent This section briefly describes some of the necessary background.\n\n\\subsection{Bilinear pairings \\cite{zhang2004efficient}}\nLet $\\mathbb{G}_1$ and $\\mathbb{G}_2$ are elliptic groups of order $\\mathrm{p}$, $\\mathbb{G}_{\\mathrm{T}}$ is a multiplicative group of order $\\mathrm{p}$, $\\mathrm{g_1}$ is a generator of $\\mathbb{G}_{1}$, $\\mathrm{g_2}$ is a generator of $\\mathbb{G}_2$, $\\mathrm{P} \\in \\mathbb{G}_1$, $\\mathrm{Q}\\in \\mathbb{G}_2$ and $\\mathrm{a}$, $\\mathrm{b}$ $\\in \\mathbb{Z}_p$, then a bilinear pairing is a map $\\mathrm{e}: \\mathbb{G}_1 \\times \\mathbb{G}_2 \\rightarrow \\mathbb{G}_{\\mathrm{T}}$ that satisfies the following three properties.\n\\begin{enumerate}\n\t\\item [1] Bilinearity: $\\mathrm{e(P^a, Q^b) = e(P, Q)^{ab}}$\n\t\\item [2] Non-Degeneracy: $\\mathrm{e(g_1, g_2) \\neq 1}$\n\t\\item [3] Computability: $\\mathrm{e(P, Q)}$ can be computed efficiently.\n\\end{enumerate}\n\n\\subsection {Decision Bilinear Diffie-Hellman (DBDH) assumption \\cite{yacobi2002note}}\nLet $\\mathbb{G}$, $\\mathbb{G}_T$ are cyclic groups of prime order $\\mathrm{p} > 2^{\\lambda}$ where $\\lambda \\in \\mathbb{N}$, $\\mathrm{g}$ is the generator of $\\mathbb{G}$, $\\mathrm{e}: \\mathbb{G} \\times \\mathbb{G} \\rightarrow \\mathbb{G}_{\\mathrm{T}}$ is an efficiently computable symmetric bilinear pairing map and $\\mathrm{a, b, c, z} \\in \\mathbb{Z}_{\\mathrm{p}}$ are random numbers. The DBDH assumption states that no probabilistic polynomial time algorithm can distinguish between $\\mathrm{\\langle{{g},{g^a},{g^b},{g^c},{e(g,g)^{abc}}}\\rangle}$ and $\\mathrm{\\langle{{g},{g^a},{g^b},{g^c},{e(g,g)^{z}}}\\rangle}$ with more than a negligible advantage.\n\n\\subsection {Security Model}\nThe security model of proposed scheme is based on the following IND-sAtt-CPA game \\cite{ibraimi2009efficient} between a challenger and an adversary $\\mathcal{A}$. \n\\begin{enumerate}[leftmargin=0pt]\n\t\\item []\\textit{Init Phase}: Adversary $\\mathcal{A}$ chooses a challenge access tree $\\mathcal{T}^*$ and gives it to challenger.\n\t\n\t\\item []\\textit{Setup Phase}: Challenger runs a \\textit{setup} procedure to generate $\\mathrm{\\langle{ASK, APK}\\rangle}$ and gives the public key $\\mathrm{APK}$ to adversary $\\mathcal{A}$.\n\t\n\t\\item []\\textit{Phase I}: Adversary $\\mathcal{A}$ makes an attribute- based private key request to the key generation oracle for any attribute set with the restriction that the attribute set should not include any attribute which is part of $\\mathcal{T}^*$. Challenger generates the key as described in section \\ref{key_generation} and returns the same to adversary $\\mathcal{A}$.\n\t\n\t\\item []\\textit{Challenge Phase}: Adversary $\\mathcal{A}$ sends two equal length messages $\\mathrm{m_0}$ and $\\mathrm{m_1}$ to challenger. Challenger chooses a random number $\\mathrm{b} \\in_{\\mathrm{R}} \\{0, 1\\}$, encrypts $\\mathrm{m_b}$ using $\\mathcal{T}^*$ and $\\mathrm{APK}$ as is described in section \\ref{encryption}.\n\t\n\t\\item []\\textit{Phase II}: Adversary $\\mathcal{A}$ can send multiple requests to generate attribute-based private key with the same restriction as in Phase I.\n\t\n\t\\item []\\textit{Guess Phase}: Adversary $\\mathcal{A}$ outputs a guess $\\mathrm{b\\prime} \\in \\{0, 1\\}$.\n\\end{enumerate}\nThe advantage of adversary $\\mathcal{A}$ in this game is defined to be $\\epsilon = \\vert {\\mathrm{Pr[b\\prime = b]} - \\frac{1}{2}} \\vert$. Only if any polynomial time adversary $\\mathcal{A}$ has a negligible advantage, the scheme is considered secure against an adaptive chosen plaintext attack (CPA).\n\n\\subsection {Access tree} \\label{access_tree}\nAccess tree structure is a means to specify an access policy during encryption that must be satisfied by attribute-based private keys in order to decrypt. Let $\\mathcal{T}$ be a tree representing an access structure. Each non-leaf node of the tree represents a threshold gate, described by its children and a threshold value. If $\\mathrm{num_x}$ is the number of children and $\\mathrm{k_x}$ is the threshold value of a node $\\mathrm{x}$, then, $\\mathrm{k_x=1}$ represents an OR gate and $\\mathrm{k_x=num_x}$ represents an AND gate. Each leaf node $\\mathrm{x}$ of the tree is described by an attribute and a threshold value $\\mathrm{k_x = 1}$. \n\nLet $\\mathcal{T}_x$ denotes the subtree rooted at node $\\mathrm{x}$. If a set of attributes $\\lambda$ satisfies the subtree $\\mathcal{T}_x$, it is represented as $\\mathcal{T}_x(\\lambda) = 1$. $\\mathcal{T}_x(\\lambda)$ is computed recursively as follows. If $\\mathrm{x}$ is a non-leaf node, evaulate $\\mathcal{T}(y)$ for all children nodes $\\mathrm{y}$ of node $\\mathrm{x}$. $\\mathcal{T}_x(\\lambda)$ returns $1$ if and only if at least $\\mathrm{k_x}$ children return 1. If $\\mathrm{x}$ is a leaf node, then $\\mathcal{T}_x(\\lambda)$ returns $1$ if and only if $\\mathtt{attr(x)} \\in \\lambda$.\n\n\\section {\\uppercase{Our Construction}}\n\n\\noindent The proposed scheme introduces two new roles, namely, \\textit{Attribute Authority Manager (AAM)} and \\textit{Attribute Authority (AA)}. AAM is an entity which manages the universe of attributes and AA is an entity which manages a set of attributes (as assigned by AAM). DigiLocker is proposed to assume the role of AAM and individual issuers are proposed to assume the role of AA. A subscriber is assigned a set of attributes from each issuer which holds at least one e-document of the subscriber. Each requester application is assigned a set of attributes from DigiLocker based on certain criteria such as purpose of access, for how long the data is going to be used, etc. To create a privacy enhanced e-document for a subscriber, issuer and subscriber mutually creates an attribute-based token (which will be used later in encryption) for an access policy, generates a symmetric key, encrypts the document with symmetric key, encrypts the symmetric key with attribute-based token, creates an e-document enclosing both the encrypted symmetric key and the encrypted document, creates a URI for this e-document and pushes it to subscriber's digital locker using $\\mathtt{PushURI}$ API. When this e-document is shared with a requester application, the requester will be able to decrypt the encrypted symmetric key only if the requester is associated with a set of attributes which satisfies the access policy used to encrypt the symmetric key. Only when the requester obtains the symmetric key, will he be able to decrypt and retrieve the document.\n\nIn \\textit{Setup($\\kappa$)} procedure, AAM chose a cyclic group ${\\mathbb{G}_0}$ of large prime order $\\mathrm{p}$ ($\\kappa$ defines the size of group) on which discrete logarithm problem is assumed to be hard, generator $\\mathrm{g}$, a bilinear map $\\mathrm{e}:{\\mathbb{G}_0}\\times {\\mathbb{G}_0}\\rightarrow {\\mathbb{G}_1}$ for which bilinear diffie hellman problem is assumed to be hard, a hash function $\\mathrm{H}: \\{0, 1\\}^{*} \\rightarrow \\mathbb{G}_0$ which maps a binary string encoded attribute to a group element, chose random numbers $\\alpha, \\beta \\in_{\\mathrm{R}} \\mathbb{Z}_{\\mathrm{p}}$ and set its private key $\\mathrm{ASK}$ and public key $\\mathrm{APK}$ as below.\n\\begin{center}\n\t\\begin{tabular}{ l c l }\n\t\t$\\mathrm{ASK}$ & = & $\\{\\beta, \\mathrm{g}^{\\alpha} \\}$ \\\\ \n\t\t$\\mathrm{APK}$ & = & $\\{\\mathrm{g}^{\\beta}, \\mathrm{e(g, g)}^{\\alpha}, \\mathbb{G}_0, \\mathrm{g}\\}$ \\\\  \n\t\\end{tabular}\n\\end{center}\n\\subsection {Attribute Assignment}{\\label{attribute_assignment}}\nAn attribute can be any characteristic of a subscriber or requester and is represented by a binary string $\\{0,1\\}^*$. Attribute assignment to both subscribers and requesters is proposed to be done lazily in the background with the aim to keep the list of associated attributes in DigiLocker up to date.\n\nFor subscriber's attribute assignment and modification, two APIs are proposed to be introduced. First is  $\\mathtt{PullAttrs(ID_i)}$ which is provided by issuers and is consumed by DigiLocker to pull updated list of attributes of subscriber with Aadhaar number $\\mathrm{ID_i}$. Second is $\\mathtt{PushAttrs(ID_i, NewAttrs)}$ which is provided by DigiLocker and is consumed by issuer to push any change in attributes of subscriber with Aadhaar number $\\mathrm{ID_i}$. For requester applications, attributes are assigned and updated by DigiLocker.\n\nIt is important to take appropriate measures to handle load of a voluminous country like India. One such measure could be to prepone part of the encryption process. This preponed encryption process generates a token with mutual cooperation between subscriber and issuer. This token can be reused every time for a given subscriber and for a given access policy. \n\nA helper procedure $\\mathtt{encPartial(\\mathcal{T}, r)}$ is assumed to be present which works as follows. It chooses a polynomial $\\mathrm{q_x}$ for each node $\\mathrm{x}$ (including the leaves) in the tree $\\mathcal{T}$. These polynomials are chosen in the following way in a top-down manner, starting from the root node $\\mathrm{R}$. For each node $\\mathrm{x}$ in the tree, set the degree $\\mathrm{d_x}$ of the polynomial $\\mathrm{q_x}$ to be one less than the threshold value $\\mathrm{k_x}$ of that node, that is, $\\mathrm{d_x = k_x - 1}$. Starting with the root node $\\mathrm{R}$ the procedure chooses a random $\\mathrm{r}\\in_{\\mathrm{R}} \\mathbb{Z}_{\\mathrm{p}}$ and sets $\\mathrm{q_r(0)=r}$. Then, it chooses $\\mathrm{d_R}$ other points of the polynomial $\\mathrm{q_R}$ randomly to define it completely. For any other node $\\mathrm{x}$, it sets $\\mathrm{q_x(0)} = \\mathrm{q_{parent(x)}(index(x))}$ and chooses $\\mathrm{d_x}$ other points randomly to completely define $\\mathrm{q_x}$.\n\n\\subsection {Token Generation}{\\label{token_generation}}\n\\begin{center}\n\\begin{figure}\n\t\\caption{Example of an access policy tree}\n\t\\label{fig:AP}\n\t\\resizebox {0.9\\columnwidth}{!} {\n\t\t\\begin{tikzpicture} [level\tdistance = 50pt,\n\t\tsibling distance = 20pt, \n\t\tedge from parent\/.style = {\n\t\t\tdraw, \n\t\t\tedge from parent path = {(\\tikzparentnode) -- (\\tikzchildnode.north)}\n\t\t}\n\t\t]\n\t\t\\tikzset{every internal node\/.style\t= {draw, circle, black, font = \\Large}}\n\t\t\\tikzset{every leaf node\/.style\t\t= {draw, black, regular polygon, regular polygon sides = 3, inner sep = 1pt}}\n\t\t\n\t\t\\Tree [.\\node(1)[label={[label distance=0.25cm]10:\\LARGE $R$}]{$\\lor$};\n\t\t[.\\node(2)[label={[label distance=0.25cm]10:\\LARGE ${R_S}_{iv}$}]{$\\wedge$};\n\t\t\\node(3){${A_R}_1$};\n\t\t\\node(4){${A_R}_n$};\n\t\t] \n\t\t[.\\node(5)[label={[label distance=0.25cm]10:\\LARGE ${R_I}_{iv}$}]{$\\wedge$};\n\t\t\\node(6){${A_R}_1$};\n\t\t\\node(7){${A_R}_n$};\n\t\t] \n\t\t]\n\t\t\n\t\t\\node[draw, dashed, inner xsep=3mm, inner ysep=6mm, fit=(2)(3)(4)](r1){};\n\t\t\\node[draw, dashed, inner xsep=3mm, inner ysep=6mm, fit=(5)(6)(7)](r2){};\n\t\t\\node[draw, dashed, inner xsep=7mm, inner ysep=9mm, fit=(3)(1)(7)](r3){};\n\t\t\n\t\t\\node [above, inner sep=5pt] at (r1.south) {\\large ${\\mathcal{T}_S}_{iv}$};\n\t\t\\node [above, inner sep=5pt] at (r2.south) {\\large ${\\mathcal{T}_I}_{iv}$};\n\t\t\\node [above, inner sep=5pt] at (r3.south) {\\large ${\\mathcal{T}_{iv}}$};\n\t\n\t\t\n\t\t\n\t\t\\end{tikzpicture}\n\t}\n\\end{figure}\n\\end{center}\n\nAn access tree $\\mathcal{T}_{iv}$ is comprised of access subtree $\\mathcal{T}_{S_{iv}}$ from subscriber $\\mathrm{S_i}$ and access subtree $\\mathcal{T}_{I_{iv}}$ from issuer $\\mathrm{I_v}$ (refer figure \\ref{fig:AP}). If issuer $\\mathrm{I_v}$ needs to generate its part of token for subscriber $\\mathrm{S_i}$, for access tree $\\mathcal{T}_{iv}$, it generates a random number $\\mathrm{r_i} \\in_{\\mathrm{R}} \\mathbb{Z}_{\\mathrm{p}}$, and generates following partial-token using $\\mathrm{APK}$ and $\\mathtt{encPartial(\\mathcal{T}_{I_{iv}}, r_i)}$. Let $\\mathrm{Y_I}$ is the set of leaf nodes in $\\mathcal{T}_{I_{iv}}$.\n\n$\\mathrm{CTtok_{I_{iv}}} =\t\\begin{cases}\n\t\\quad \\mathcal{T}_{I_{iv}}\\\\\n\t\\quad \\mathrm{C1_I} = \\mathrm{e(g, g)}^{\\alpha \\mathrm{r_i}}\\\\\n\t\\quad \\mathrm{C2_I} = \\mathrm{g}^{\\beta \\mathrm{r_i}}\\\\\n\t\\begin{rcases}\n\t\t\\quad \\mathrm{C3_{I_y} = g^{q_y(0)}}\\\\\n\t\t\\quad \\mathrm{C4_{I_y} = H(\\mathtt{attr(y)})^{q_y(0)}}\\\\\n\t\\end{rcases} {\\!\\scriptscriptstyle\\mathrm{\\forall\\ y \\in Y_I}}\\\\\n\\end{cases}$\nIssuer notifies subscriber to provide its part of the token. Subscriber $\\mathrm{S_i}$ generates a random number $\\mathrm{r_s} \\in_{\\mathrm{R}} \\mathbb{Z}_{\\mathrm{p}}$ and generates following partial-token using $\\mathrm{APK}$ and $\\mathtt{encPartial(\\mathcal{T}_{S_{iv}})}$. Let $\\mathrm{Y_S}$ is the set of leaf nodes in $\\mathcal{T}_{S_{iv}}$.\n\n$\\mathrm{CTtok_{S_{iv}}} = \\begin{cases}\n\t\\quad \\mathcal{T}_{S_{iv}}\\\\\n\t\\quad \\mathrm{C1_S} = \\mathrm{e(g, g)}^{\\alpha \\mathrm{r_s}}\\\\\n\t\\quad \\mathrm{C2_S} = \\mathrm{g}^{\\beta \\mathrm{r_s}}\\\\\n\t\\begin{rcases}\n\t\t\\quad \\mathrm{C3_{S_y} = g^{q_y(0)}}\\\\\n\t\t\\quad \\mathrm{C4_{S_y} = H(\\mathtt{attr(y)})^{q_y(0)}}\\\\\n\t\\end{rcases} {\\!\\scriptscriptstyle\\mathrm{\\forall\\ y \\in Y_S}}\\\\\n\\end{cases}$\n\nSubscriber provides its part of partial-token to issuer. Issuer creates the final token by combining the two partial-tokens and keeps it securely with it.\n\n$\\mathrm{CTtok_{iv}} = \\begin{cases}\n\t\\quad \\mathcal{T}_{iv} = \\mathcal{T}_{S_{iv}} \\cup \\mathcal{T}_{I_{iv}}\\\\\n\t\\quad \\mathrm{C1} = \\mathrm{C1_S}.\\mathrm{C1_I}\\\\\n\t\\qquad\\ \\ = \\mathrm{e(g, g)}^{\\alpha \\mathrm{r_s}}\\mathrm{e(g, g)}^{\\alpha \\mathrm{r_i}}\\\\\n\t\\quad \\mathrm{C2} = \\mathrm{C2_S}.\\mathrm{C2_I} = \\mathrm{g}^{\\beta \\mathrm{r_s}}\\mathrm{g}^{\\beta \\mathrm{r_i}}\\\\\n\t\\begin{rcases}\n\t\t\\quad \\mathrm{C3 = C3_{S_y} \\cup C3_{I_y}}\\\\\n\t\t\\qquad\\ \\  = \\mathrm{g^{q_y(0)}}\\\\\n\t\t\\quad \\mathrm{C4 = C4_{S_y} \\cup C4_{I_y}}\\\\\n\t\t\\qquad\\ \\ = \\mathrm{H(\\mathtt{attr(y)})^{q_y(0)}}\\\\\n\t\\end{rcases} {\\!\\scriptscriptstyle\\forall\\ y \\in Y_S \\cup Y_I}\\\\\n\\end{cases}$\n\n\\subsection{Encryption} {\\label {encryption}}\nA new $DocType$ $\\mathtt{PRIV}$ is proposed to be introduced for privacy enhanced e-documents. To create a privacy enhanced e-document, issuer creates a URI $\\mathrm{\\langle{I_v::PRIV::D_w}\\rangle}$ where $\\mathrm{I_v}$ is the issuer identifier and $\\mathrm{D_w}$ is the document identifier within the document type $\\mathtt{PRIV}$. Now, issuer generates a random number $\\mathrm{r_{ie}} \\in_{\\mathrm{R}} \\mathbb{Z}_{\\mathrm{p}}$, generates a symmetric key $\\mathrm{SK_{ivw}}$, encrypts e-document $\\mathrm{m}$ with $\\mathrm{SK_{ivw}}$, encrypts $\\mathrm{SK_{ivw}}$ with $\\mathrm{CTtok_{I_{iv}}}$ and produces the following ciphertext.\n\n$\\mathrm{CT_{ivw}} = \\begin{cases}\n\t\\quad \\mathcal{T}_{iv} = \\mathcal{T}_{S_{iv}} \\cup \\mathcal{T}_{I_{iv}}\\\\\n\t\\quad \\mathrm{C1} = \\mathrm{e(g, g)}^{\\alpha \\mathrm{r_s r_{ie}}}\\mathrm{e(g, g)}^{\\alpha \\mathrm{r_i r_{ie}}}\\mathrm{SK_{ivw}}\\\\\n\t\\quad \\mathrm{C2} = \\mathrm{g}^{\\beta \\mathrm{r_s r_{ie}}}\\mathrm{g}^{\\beta \\mathrm{r_i r_{ie}}}\\\\\n\t\\begin{rcases}\n\t\t\\quad \\mathrm{C3_{y} = g^{r_{ie}(q_y(0))}}\\\\\n\t\t\\quad \\mathrm{C4_{y} = H(\\mathtt{attr(y)})^{q_y(0)}}\\\\\n\t\\end{rcases} {\\!\\scriptscriptstyle\\mathrm{\\forall\\ y \\in Y_{iv}}}\\\\\n\t\\quad \\mathrm{C5 = \\{m\\}_{SK_{ivw}}}\\\\\n\\end{cases}$\n\n\\subsection {Key Generation} {\\label{key_generation}\nA new API $\\mathtt{GenABPvtKey(ID_i, IS_j)}$ is proposed to be provided by DigiLocker to generate an attribute-based private key for a subscriber with Aadhaar identifier $\\mathrm{ID_i}$ and with attributes from issuers in the set $\\mathrm{IS_j}$. Let $\\mathrm{S_{ij}}$ is the set of all attributes assigned to $\\mathrm{S_i}$ by all issuers in set $\\mathrm{IS_j}$. DigiLocker generates random numbers $\\mathrm{r} \\in_{\\mathrm{R}} \\mathbb{Z}_{\\mathrm{p}}$, $\\mathrm{r_j} \\in_{\\mathrm{R}} \\mathbb{Z}_{\\mathrm{p}}$ for each attribute $\\mathrm{j}\\in_{\\mathrm{R}} \\mathrm{S_{ij}}$, computes attribute-based private key $\\mathrm{ASK_{ID_{i}IS_{j}}}$ as below and keeps this key securely with it.\n\n\n\t$\\mathrm{ASK_{{ID_i}{IS_j}}} = \\begin{cases}\n\t\t\\quad \\mathrm{D} = \\mathrm{g}^{(\\alpha+\\mathrm{r})\/\\beta}\\\\\n\t\t\\begin{rcases}\n\t\t\t\\quad \\mathrm{D_j} = \\mathrm{g^{r}.H(j)^{r_j}}\\\\\n\t\t\t\\quad \\mathrm{D_j\\prime} = \\mathrm{g^{r_j}}\\\\\n\t\t\\end{rcases} {\\!\\scriptscriptstyle\\mathrm{\\forall\\ j \\in S_{ij}}}\\\\\n\t\\end{cases}$\n\nNote that multiple attribute based private keys can exist for a subscriber for different set of attributes. If anyone issuer set $\\mathrm{IS_i}$ is a proper subset of another issuer set $\\mathrm{IS_j}$, the key corresponding to $\\mathrm{IS_i}$ is redundant and can be removed.\n\n\t\n\\subsection {Decryption} {\\label{decryption}}\nA new API $\\mathtt{FetchPrivDoc{URI}}$ is proposed to be provided by DigiLocker for decryption purpose. This API facilitates a requester with identifier $\\mathrm{ID_R}$ to retrieve ciphertext $\\mathrm{CT_{ivw}}$ of e-document from URI $\\mathrm{\\langle{I_v::PRIV::D_W}\\rangle}$ of subscriber $\\mathrm{S_i}$. DigiLocker extracts the set of attribute issuers $\\mathrm{IS_k}$ from $\\mathrm{CT_{ivw}}\\rightarrow\\mathcal{T}_{iv}$, retrieves $\\mathrm{ASK_{ID_RIS_k}}$ and calls $\\mathtt{Decrypt(CT_{ivw}, ASK_{ID_RIS_k})}$. A helper procedure $\\mathtt{DecryptNode(CT_{ivw}, ASK_{ID_RIS_K})}$ is defined as below. Let $\\mathrm{S_k}$ is the set of all attributes from issuers in set $\\mathrm{IS_k}$. If $\\mathrm{x}$ is a leaf node and if $\\mathtt{attr(x)} \\notin \\mathrm{S_k}$, then $\\mathtt{DecryptNode(CT_{ivw}, ASK_{{ID_R}{IS_k}}, x)} = \\perp$ else if $\\mathtt{attr(x)}\\in \\mathrm{S_k}$, then the procedure is defined as below.\n\\begin{tabular}{ m{1cm} l }\n\t\\multicolumn{2}{l}{ $\\mathtt{DecryptNode(CT_{ivw}, ASK_{{ID_R}{IS_k}}, x)}$ }\\\\\n\t& = $ \\mathrm{\\frac{e(D_x, C4_y)}{e(D_x\\prime, C5_y)}}$ \\\\ \n\t& = $ \\mathrm{\\frac{e(g^r.H(\\mathtt{attr(x)})^{r_j}, g^{r_{ie}q_y(0)})}{e(g^{r_j}, H(\\mathtt{attr(x)})^{q_x(0)})}}$ \\\\  \n\t& = $ \\mathrm{e(g, g)^{rr_{ie}q_x(0)}}$\n\\end{tabular}\n\nIf $\\mathrm{x}$ is a non-leaf node, the recursive procedure is defined as follows. For all children nodes $\\mathrm{z}$ of $\\mathrm{x}$, $\\mathtt{DecryptNode(CT_{ivw}, ASK_{i}, x)}$ is called and their output is stored in $\\mathrm{F_z}$. Let $\\mathrm{S_x}$ be an arbitrary $\\mathrm{k_x}$ sized set of child nodes $\\mathrm{z}$ such that $\\mathrm{F_z} \\neq \\perp$. If no such set exists then the node was not satisfied and the function returns $\\perp$. Otherwise, $\\mathrm{F_x}$ is computed as below.\n\n\\begin{tabular}{ m{0cm} l }\n\t\\multicolumn{2}{l}{$\\mathrm{F_x}$ = $  \\mathrm{\\displaystyle\\prod_{z\\in S_x} F_{z}^{\\Delta_{i, S_x\\prime}(0)}\\quad {\\tiny where \\{ {i=index(z)} \\atop  {S_x\\prime = \\{index(z): z \\in S_x} \\} }} $ }\\\\\n\t& = $ \\mathrm{\\displaystyle\\prod_{z\\in S_x} F_{z}^{\\Delta_{i, S_x\\prime}(0)}}$ \\\\ \n\t& = $ \\mathrm{\\displaystyle\\prod_{z\\in S_x} (e(g, g)^{r.r_{ie}.q_z(0)})^{\\Delta_{i, S_x\\prime}(0)}}$\\\\  \n\t& = $ \\mathrm{\\displaystyle\\prod_{z\\in S_x} (e(g, g)^{r.r_{ie}.q_{parent(z)}(index(z))})^{\\Delta_{i, S_x\\prime}(0)}}$\\\\\n\t& = $ \\mathrm{\\displaystyle\\prod_{z\\in S_x} e(g, g)^{r.r_{ie}.q_x(i)\\Delta_{i, S_x\\prime}(0)}}$\\\\\n\t& = $ \\mathrm{e(g, g)^{rr_{ie}q_x(0)}\\ (\\tiny{using\\ polynomial\\ interpolation})}$\n\\end{tabular}\n$\\mathtt{Decrypt}(\\mathtt{CT_{ivw}}, \\mathtt{ASK_{{ID_R}{IS_k}}})$ calls $\\mathtt{DecryptNode(}$ $\\mathtt{CT_{ivw}}, \\mathtt{ASK_{{ID_R}{IS_k}}}, \\mathtt{R})$ where $\\mathrm{R}$ is root node of $\\mathrm{{T}_{iv}}$. If the access tree is satisfied by attributes in $\\mathrm{S_k}$, set $\\mathrm{A =}$ $\\mathtt{DecryptNode(CT_{ivw}, ASK_{{ID_R}{IS_k}}, R)} =$ $\\mathrm{e(g, g)}^{\\mathrm{r}.\\mathrm{r_{ie}}.(\\mathrm{r_s}+\\mathrm{r_i})}$. Now the procedure obtains symmetric key $\\mathrm{SK_{ivw}}$ by computing \n\\begin{tabular}{ m{1cm} l }\n\t\\multicolumn{2}{l}{$\\mathrm{\\dfrac{C1}{\\dfrac{e(C2, D)}{A}}}$ = $\\dfrac{\\mathrm{e(g,g)}^{\\alpha \\mathrm{r_s r_{ie}}}\\mathrm{e(g,g)}^{\\alpha \\mathrm{r_i r_{ie}}}\\mathrm{SK_{ivw}}}{\\dfrac{\\mathrm{e}(\\mathrm{g}^{\\beta \\mathrm{r_s r_{ie}}}\\mathrm{g}^{\\beta \\mathrm{r_i r_{ie}}}, \\mathrm{g}^{(\\alpha+\\mathrm{r})\/\\beta})}{\\mathrm{e(g, g)}^{\\mathrm{r r_{ie} (r_s+r_i) )} }}}$}\\\\\n\t& = $\\dfrac{\\mathrm{e(g,g)}^{\\alpha \\mathrm{r_s r_{ie}}}\\mathrm{e(g,g)}^{\\alpha \\mathrm{r_i r_{ie}}}\\mathrm{SK_{ivw}}} {\\dfrac {\\mathrm{e}(\\mathrm{g}^{\\beta \\mathrm{r_{ie} (r_s+r_i)}}, \\mathrm{g}^{(\\alpha+\\mathrm{r})\/\\beta})} {\\mathrm{e(g, g)}^{\\mathrm{r r_{ie} (r_s+r_i))}}}}$ \\\\ \n\t& = $\\dfrac{\\mathrm{e(g,g)}^{\\alpha \\mathrm{r_{ie} (r_s + r_i)}}\\mathrm{SK_{ivw}}} {\\dfrac {\\mathrm{e(g, g)}^{\\mathrm{r_{ie} (r_s+r_i)} (\\alpha+\\mathrm{r})}} {\\mathrm{e(g, g)}^{\\mathrm{r r_{ie} (r_s+r_i))}}}}$\\\\  \n\t& = $\\mathrm{SK_{ivw}}$\\\\\n\\end{tabular}\n\nSymmetric key $\\mathrm{SK_{ivw}}$ is now used to decrypt the encrypted e-document.\n\n\\begin{tabular}{m{0cm} l}\n\t\\multicolumn{2}{l}{$\\mathrm{m = \\{CT_{ivw}\\rightarrow{C5}\\}_{SK_{ivw}}}$}\n\\end{tabular}\n\nDigiLocker returns the decrypted document $\\mathrm{m}$ to requester.\n\n\\section{\\uppercase{Security Analysis}}\n\\noindent If the proposed scheme is not secure than an adversary $\\mathcal{A}$ can win IND-sAtt-CPA game and solve the DBDH assumption with advantage $\\epsilon\/2$. If the DBDG assumption is solved by adversary $\\mathcal{A}$, a simulator $\\beta$ can be built which can solve DBDH assumption with advantage $\\epsilon\/2$. Challenger chose a group $\\mathbb{G}_0$, a generator $\\mathrm{g}$, a bilinear map $\\mathrm{e}$ and chose random numbers $\\mathrm{a, b, c,} \\theta \\in_{\\mathrm{R}} \\mathbb{Z}_p^*$. The challenger selects at random $\\mu \\in_{\\mathrm{R}} {0, 1}$ and sets $\\mathrm{Z}_\\mu$ as below.\n\\begin{align*}\n\t\\mathrm{Z}_\\mu = \\begin{cases}\n\t\\mathrm{(g, g)^{abc}}, &\\mathrm{if}\\ \\mu = 0\\\\\n\t\\mathrm{e(g, g)}^{\\theta}, &\\mathrm{if}\\ \\mu = 1\\\\\n\t\\end{cases}\t\n\\end{align*}\n\nChallenger provides DBDB challenge to the simulator: $\\langle{\\mathrm{g, A, B, C, Z}_{\\mu}}\\rangle$  $\\langle{\\mathrm{g, g^a, g^b, g^c, Z}_{\\mu}}\\rangle$. In IND-sAtt-CPA game, simulator $\\beta$ plays the role of challenger for adversary $\\mathcal{A}$.\n\t\n\\begin{itemize}[leftmargin=*]\n\t\t\\item [] \\textit{Init Phase}: The adversary chose the challenge access tree $\\mathcal{T}^{*}$ and gives it to simulator.\n\t\t\n\t\t\\item [] \\textit{Setup Phase}: The challenger chose a random number $\\mathrm{x\\prime} \\in \\mathbb{Z}_p$, sets $\\alpha = \\mathrm{ab + x\\prime}$ and computes $\\mathrm{y}$ as below.\n\t\t\\[\n\t\t\\mathrm{y = e(g, g)}^{\\alpha} = \\mathrm{e(g, g)^{ab}e(g, g)^{x\\prime}}\n\t\t\\]\n\t\tNow, challenger chose a random numbers $\\mathrm{r \\in_R \\mathbb{Z}_p}$ and $\\mathrm{r_i \\in_R \\mathbb{Z}_p}$ for ($\\mathrm{1 \\le i \\le \\vert{\\mathtt{U}}\\vert}$) and for all $\\mathrm{a_j \\in \\mathbb{U}}$, computes $\\mathrm{d_j}$ and $\\mathrm{d_j\\prime}$ as below.\n\t\t\\begin{align*}\n\t\t\t\\begin{rcases}\n\t\t\t\t\\mathrm{d_j} &= \\begin{cases}\n\t\t\t\t\t\\mathrm{g^{r\/b} H(j)^{r_j}} & \\mathrm{...if\\ a_j} \\notin \\mathcal{T}^*\\\\\n\t\t\t\t\t\\mathrm{g^{r} H(j)^{r_j}} & \\mathrm{...if\\ a_j} \\in \\mathcal{T}^*\\\\\n\t\t\t\t\t\\end{cases}\\\\\n\t\t\t\t\t\\mathrm{d_j\\prime} &= \\mathrm{g^{r_j}}\n\t\t\t\t\\end{rcases} \n\t\t\t\\mathrm{(1 \\le j \\le \\vert{U}\\vert)}\n\t\t\\end{align*}\n\t\tNow, challenger sends public parameters $\\mathrm{APK} = \\{\\mathrm{g}^\\beta, \\mathrm{e(g, g)}^\\alpha, \\mathbb{G}, \\mathrm{g}\\}$ to adversary $\\mathcal{A}$.\n\t\t\n\t\t\\item [] \\textit{Phase 1}: In this phase, adversary $\\mathcal{A}$ sends requests for private key for any set of attributes $w_j$ which does not contain any attribute in $\\mathcal{T}^*$.\n\t\t\\begin{align*}\n\t\t\t\\mathrm{w_j} = \\{\\mathrm{a_j} \\mid (\\mathrm{a_j} \\in \\mathbb{U} \\wedge \\mathrm{a_j} \\notin \\mathcal{T}^*)\\}\n\t\t\\end{align*}\n\t\tFor each query from adversary $\\mathcal{A}$, challenger chose a random number $\\mathrm{r\\prime \\in_R\\mathbb{Z}_p}$, sets $\\mathrm{r= - b(r\\prime + a)}$ and computes $\\mathrm{D}$ as below.\n\t\t\\begin{align*}\n\t\t\t\\mathrm{D} &= \\mathrm{g}^{(\\alpha+\\mathrm{r})\/\\beta} = (\\mathrm{g}^{(\\alpha+\\mathrm{r})})^{1\/\\beta} = (\\mathrm{g}^{((\\mathrm{ab+x\\prime})-\\mathrm{b(r\\prime+a)})})^{1\/\\beta}\\\\\n\t\t\t&= (\\mathrm{g^{x\\prime-br\\prime})}^{1\/\\beta} = (\\mathrm{g^{x\\prime}.(g^{b})^{-r\\prime})}^{1\/\\beta}\n\t\t\\end{align*}\n\t\tBecause of restriction $\\mathrm{a_j} \\notin \\mathcal{T}^*$ in this phase, $\\mathrm{D_j}$ can be computed as below.\n\t\t\\begin{align*}\n\t\t\t\\mathrm{D_j} &= \\mathrm{g^{r\/b}H(j)^{r_j} = g^{r\/b}H(j)^{r_j} = g^{-(r_\\prime+a)}H(j)^{r_j}}\\\\\n\t\t\t    &= \\mathrm{(g^{a})^{-1}g^{-r\\prime}H(j)^{r_j}}\n\t\t\\end{align*}\n\t\tNow, challenger sends private key $\\mathrm{ASK_{w_j}} = \\mathrm{D}, \\mathrm{(D_j, D_j\\prime)}\\mid\\forall \\mathrm{a_j\\in w_j}$ to adversary $\\mathcal{A}$\n\t\t\n\t\t\\item [] \\textit{Challenge Phase}: In this phase, adversary $\\mathcal{A}$ submits two plaintext messages $\\mathrm{m_0}$ and $\\mathrm{m_1}$ to the challenger. Challenger selects a random plaintext message $\\mathrm{m_b}$ from the two messages where $\\mathrm{b \\in_R \\{0, 1\\}}$, sets $\\mathrm{r_{ie}=1}$, chose random variables $\\mathrm{r_i}$ and $\\mathrm{r_s}$ such that $\\mathrm{c=r_i+r_c}$. Now, set value of root node $\\mathcal{T}^*$ to $\\mathrm{c}$ and assign values to leaf nodes of $\\mathcal{T}^*$ as described in section \\ref{access_tree} to arrive at $\\mathrm{C3_y}$ and $\\mathrm{C4_y}$. The final ciphertext $\\mathrm{CT}_{\\mathcal{T}^*}$ is computed as below. The ciphertext is returned to adversary $\\mathcal{A}$.\n\t\t\n\t\t$CT_{\\mathcal{T}^*} = \\begin{cases}\n\t\t\t\\quad \\mathcal{T}_{iv} = \\mathcal{T}^*\\\\\n\t\t\t\\quad \\mathrm{C1} = \\mathrm{e(g, g)}^{\\alpha \\mathrm{r_s}}\\mathrm{e(g, g)}^{\\alpha \\mathrm{r_i}}\\mathrm{m_b}\\\\\n\t\t\t\\qquad\\ \\  = \\mathrm{e(g, g)}^{\\alpha(\\mathrm{r_s+r_i})}\\mathrm{m_b}\\\\\n\t\t\t\\qquad\\ \\  = \\mathrm{e(g, g)^{c}m_b}\\\\\n\t\t\t\\quad \\mathrm{C2} = \\mathrm{g}^{\\beta \\mathrm{r_s}}\\mathrm{g}^{\\beta \\mathrm{r_i}} = \\mathrm{g}^{\\beta(\\mathrm{r_s+r_i})}\\\\\n\t\t\t\\qquad\\ \\  = \\mathrm{g}^{\\beta} \\mathrm{g^{c}}\\\\\n\t\t\t\\begin{rcases}\n\t\t\t\t\\quad \\mathrm{C3_{y} = g^{(q_y(0))}}\\\\\n\t\t\t\t\\quad \\mathrm{C4_{y} = H(\\mathtt{attr(y)})^{q_y(0)}}\\\\\n\t\t\t\\end{rcases} {\\mathrm{\\forall\\ y \\in Y_{iv}}}\\\\\n\t\t\\end{cases}$\n\t\t\n\t\t\\item [] \\textit{Phase 2}: In this phase, adversary $\\mathcal{A}$ can continue to send secret key generation requests with the same restriction as in $Phase 1$, i.e., $\\mathrm{a_j} \\notin \\mathcal{T}^*$.\n\t\t\n\t\t\\item [] \\textit{Guess Phase}: In this phase, adversary $\\mathcal{A}$ outputs a guess $\\mathrm{b\\prime} \\in \\{0, 1\\}$. If $\\mathrm{b\\prime} = \\mathrm{b}$, the simulator $\\beta$ will guess that $\\mu = 0$ and $\\mathrm{Z}_\\mu = \\mathrm{e(g, g)^{abc}}$, otherwise will guess that $\\mu = 1$ and $\\mathrm{Z}_\\mu = \\mathrm{e(g, g)}^{\\theta}$. When $\\mathrm{Z}_u = \\mathrm{e(g, g)^{abc}}$ the simulator $\\beta$ gives the perfect simulation and $\\mathrm{c}_{\\mathcal{T}^*}$ is a valid ciphertext. Therefore, the advantage of the adversary is \n\t\t\\begin{align*}\n\t\t\t\\mathrm{Pr} [ \\mathrm{b\\prime} = \\mathrm{b} \\mid \\mathrm{Z}_\\mu = \\mathrm{e(g, g)^{abc}} ] = \\frac{1}{2} + \\epsilon\n\t\t\\end{align*}\n\t\tIf $\\mu = 1$ then $\\mathrm{Z}_\\mu = \\mathrm{e(g, g)}^\\theta$ and $\\mathrm{c}_{\\mathcal{T}^*}$ is random ciphertext for the adversary, and the adversary does not gain information about $\\mathrm{m_b}$. Hence, we have\n\t\t\\begin{align*}\n\t\t\t\\mathrm{Pr} [ \\mathrm{b\\prime} \\neq \\mathrm{b} \\mid \\mathrm{Z}_\\mu = \\mathrm{e(g, g)}^{\\theta} ] = \\frac{1}{2}\n\t\t\\end{align*}\n\t\tSince the simulator $\\beta$ guesses $\\mu\\mathrm{\\prime} = 0$ when $\\mathrm{b\\prime} = \\mathrm{b}$ and $\\mu\\mathrm{\\prime} = 1$ when $\\mathrm{b\\prime} \\neq \\mathrm{b}$, the overall advantage of $\\beta$ to solve DBDH assumption is \n\t\t\\begin{align*}\n\t\t\\dfrac{1}{2} \\mathrm{Pr}[\\mu\\mathrm{\\prime} = \\mu \\mid \\mu = 0] + \\dfrac{1}{2} \\mathrm{Pr}[\\mu\\mathrm{\\prime} = \\mu \\mid \\mu = 1] - \\dfrac{1}{2} = \\dfrac{\\epsilon}{2}\n\t\t\\end{align*}\n\t\tIf the adversary $\\mathcal{A}$ has the above advantage $\\epsilon$ to win the IND-sAtt-CPA game, the challenger can solve the DBDH assumption problem with $\\epsilon\/2$ advantage with the help of adversary $\\mathcal{A}$. However, there are no effective polynomial algorithms which can solve the DBDH assumption problem with non-negligible advantage according to the DBDH assumption. Hence, the adversary cannot win the IND-sAtt-CPA game with the above advantage $\\epsilon$, namely the adversary having no advantage to break through the proposed scheme.\n\t\\end{itemize}\n\t\n\\section {\\uppercase{Conclusion}}\n\\noindent\nThis paper presented a scheme to improve data privacy in DigiLocker by using CP-ABE. The scheme also proposed to prepone part of the encryption process to increase performance. This preponed process creates a token which can be reused later. The proposed scheme is proved to be secure against IND-sAtt-CPA game. The proposed scheme can further be enhanced by using homomorphic encryption which allows processing on encrypted data and using post-quantum ABE schemes, for both of which, though schemes exist but are still non-trivial and not practical. \n\n\\bibliographystyle{apalike}\n{\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}