10,490 research outputs found

    Near-optimal distributed edge coloring

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    Distributed (Δ+1)(\Delta+1)-Coloring in Sublogarithmic Rounds

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    We give a new randomized distributed algorithm for (Δ+1)(\Delta+1)-coloring in the LOCAL model, running in O(logΔ)+2O(loglogn)O(\sqrt{\log \Delta})+ 2^{O(\sqrt{\log \log n})} rounds in a graph of maximum degree~Δ\Delta. This implies that the (Δ+1)(\Delta+1)-coloring problem is easier than the maximal independent set problem and the maximal matching problem, due to their lower bounds of Ω(min(lognloglogn,logΔloglogΔ))\Omega \left( \min \left( \sqrt{\frac{\log n}{\log \log n}}, \frac{\log \Delta}{\log \log \Delta} \right) \right) by Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Our algorithm also extends to list-coloring where the palette of each node contains Δ+1\Delta+1 colors. We extend the set of distributed symmetry-breaking techniques by performing a decomposition of graphs into dense and sparse parts

    Brief Announcement: Streaming and Massively Parallel Algorithms for Edge Coloring

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    A valid edge-coloring of a graph is an assignment of "colors" to its edges such that no two incident edges receive the same color. The goal is to find a proper coloring that uses few colors. In this paper, we revisit this problem in two models of computation specific to massive graphs, the Massively Parallel Computations (MPC) model and the Graph Streaming model: Massively Parallel Computation. We give a randomized MPC algorithm that w.h.p., returns a (1+o(1))Delta edge coloring in O(1) rounds using O~(n) space per machine and O(m) total space. The space per machine can also be further improved to n^{1-Omega(1)} if Delta = n^{Omega(1)}. This is, to our knowledge, the first constant round algorithm for a natural graph problem in the strongly sublinear regime of MPC. Our algorithm improves a previous result of Harvey et al. [SPAA 2018] which required n^{1+Omega(1)} space to achieve the same result. Graph Streaming. Since the output of edge-coloring is as large as its input, we consider a standard variant of the streaming model where the output is also reported in a streaming fashion. The main challenge is that the algorithm cannot "remember" all the reported edge colors, yet has to output a proper edge coloring using few colors. We give a one-pass O~(n)-space streaming algorithm that always returns a valid coloring and uses 5.44 Delta colors w.h.p., if the edges arrive in a random order. For adversarial order streams, we give another one-pass O~(n)-space algorithm that requires O(Delta^2) colors

    Distributed Approximation of Maximum Independent Set and Maximum Matching

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    We present a simple distributed Δ\Delta-approximation algorithm for maximum weight independent set (MaxIS) in the CONGEST\mathsf{CONGEST} model which completes in O(MIS(G)logW)O(\texttt{MIS}(G)\cdot \log W) rounds, where Δ\Delta is the maximum degree, MIS(G)\texttt{MIS}(G) is the number of rounds needed to compute a maximal independent set (MIS) on GG, and WW is the maximum weight of a node. %Whether our algorithm is randomized or deterministic depends on the \texttt{MIS} algorithm used as a black-box. Plugging in the best known algorithm for MIS gives a randomized solution in O(lognlogW)O(\log n \log W) rounds, where nn is the number of nodes. We also present a deterministic O(Δ+logn)O(\Delta +\log^* n)-round algorithm based on coloring. We then show how to use our MaxIS approximation algorithms to compute a 22-approximation for maximum weight matching without incurring any additional round penalty in the CONGEST\mathsf{CONGEST} model. We use a known reduction for simulating algorithms on the line graph while incurring congestion, but we show our algorithm is part of a broad family of \emph{local aggregation algorithms} for which we describe a mechanism that allows the simulation to run in the CONGEST\mathsf{CONGEST} model without an additional overhead. Next, we show that for maximum weight matching, relaxing the approximation factor to (2+ε2+\varepsilon) allows us to devise a distributed algorithm requiring O(logΔloglogΔ)O(\frac{\log \Delta}{\log\log\Delta}) rounds for any constant ε>0\varepsilon>0. For the unweighted case, we can even obtain a (1+ε)(1+\varepsilon)-approximation in this number of rounds. These algorithms are the first to achieve the provably optimal round complexity with respect to dependency on Δ\Delta

    Communication Primitives in Cognitive Radio Networks

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    Cognitive radio networks are a new type of multi-channel wireless network in which different nodes can have access to different sets of channels. By providing multiple channels, they improve the efficiency and reliability of wireless communication. However, the heterogeneous nature of cognitive radio networks also brings new challenges to the design and analysis of distributed algorithms. In this paper, we focus on two fundamental problems in cognitive radio networks: neighbor discovery, and global broadcast. We consider a network containing nn nodes, each of which has access to cc channels. We assume the network has diameter DD, and each pair of neighbors have at least k1k\geq 1, and at most kmaxck_{max}\leq c, shared channels. We also assume each node has at most Δ\Delta neighbors. For the neighbor discovery problem, we design a randomized algorithm CSeek which has time complexity O~((c2/k)+(kmax/k)Δ)\tilde{O}((c^2/k)+(k_{max}/k)\cdot\Delta). CSeek is flexible and robust, which allows us to use it as a generic "filter" to find "well-connected" neighbors with an even shorter running time. We then move on to the global broadcast problem, and propose CGCast, a randomized algorithm which takes O~((c2/k)+(kmax/k)Δ+DΔ)\tilde{O}((c^2/k)+(k_{max}/k)\cdot\Delta+D\cdot\Delta) time. CGCast uses CSeek to achieve communication among neighbors, and uses edge coloring to establish an efficient schedule for fast message dissemination. Towards the end of the paper, we give lower bounds for solving the two problems. These lower bounds demonstrate that in many situations, CSeek and CGCast are near optimal

    Extremal Optimization at the Phase Transition of the 3-Coloring Problem

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    We investigate the phase transition of the 3-coloring problem on random graphs, using the extremal optimization heuristic. 3-coloring is among the hardest combinatorial optimization problems and is closely related to a 3-state anti-ferromagnetic Potts model. Like many other such optimization problems, it has been shown to exhibit a phase transition in its ground state behavior under variation of a system parameter: the graph's mean vertex degree. This phase transition is often associated with the instances of highest complexity. We use extremal optimization to measure the ground state cost and the ``backbone'', an order parameter related to ground state overlap, averaged over a large number of instances near the transition for random graphs of size nn up to 512. For graphs up to this size, benchmarks show that extremal optimization reaches ground states and explores a sufficient number of them to give the correct backbone value after about O(n3.5)O(n^{3.5}) update steps. Finite size scaling gives a critical mean degree value αc=4.703(28)\alpha_{\rm c}=4.703(28). Furthermore, the exploration of the degenerate ground states indicates that the backbone order parameter, measuring the constrainedness of the problem, exhibits a first-order phase transition.Comment: RevTex4, 8 pages, 4 postscript figures, related information available at http://www.physics.emory.edu/faculty/boettcher

    Streaming and Massively Parallel Algorithms for Edge Coloring

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    A valid edge-coloring of a graph is an assignment of "colors" to its edges such that no two incident edges receive the same color. The goal is to find a proper coloring that uses few colors. (Note that the maximum degree, Delta, is a trivial lower bound.) In this paper, we revisit this fundamental problem in two models of computation specific to massive graphs, the Massively Parallel Computations (MPC) model and the Graph Streaming model: - Massively Parallel Computation: We give a randomized MPC algorithm that with high probability returns a Delta+O~(Delta^(3/4)) edge coloring in O(1) rounds using O(n) space per machine and O(m) total space. The space per machine can also be further improved to n^(1-Omega(1)) if Delta = n^Omega(1). Our algorithm improves upon a previous result of Harvey et al. [SPAA 2018]. - Graph Streaming: Since the output of edge-coloring is as large as its input, we consider a standard variant of the streaming model where the output is also reported in a streaming fashion. The main challenge is that the algorithm cannot "remember" all the reported edge colors, yet has to output a proper edge coloring using few colors. We give a one-pass O~(n)-space streaming algorithm that always returns a valid coloring and uses 5.44 Delta colors with high probability if the edges arrive in a random order. For adversarial order streams, we give another one-pass O~(n)-space algorithm that requires O(Delta^2) colors
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