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Distributed Graph Coloring: Fundamentals and Recent Developments
Leonid Barenboim,Michael Elkin +1 more
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The objective of this monograph is to provide a treatise on theoretical foundations of distributed symmetry breaking in the message-passing model of distributed computing and to stimulate further progress in this exciting area.Abstract:ย
The focus of this monograph is on symmetry breaking problems in the message-passing model of distributed computing. In this model a communication network is represented by a n-vertex graph G = (V,E), whose vertices host autonomous processors. The processors communicate over the edges of G in discrete rounds. The goal is to devise algorithms that use as few rounds as possible. A typical symmetry-breaking problem is the problem of graph coloring. Denote by [delta] the maximum degree of G. While coloring G with [delta]+ 1 colors is trivial in the centralized setting, the problem becomes much more challenging in the distributed one. One can also compromise on the number of colors, if this allows for more efficient algorithms. Other typical symmetry-breaking problems are the problems of computing a maximal independent set (MIS) and a maximal matching (MM). The study of these problems dates back to the very early days of distributed computing. The founding fathers of distributed computing laid firm foundations for the area of distributed symmetry breaking already in the eighties. In particular, they showed that all these problems can be solved in randomized logarithmic time. Also, Linial showed that an O([delta]2)-coloring can be solved very efficiently deterministically. However, fundamental questions were left open for decades. In particular, it is not known if the MIS or the ([delta] + 1)-coloring can be solved in deterministic polylogarithmic time. Moreover, until recently it was not known if in deterministic polylogarithmic time one can color a graph with significantly fewer than [delta]2 colors. Additionally, it was open (and still open to some extent) if one can have sublogarithmic randomized algorithms for the symmetry breaking problems. Recently, significant progress was achieved in the study of these questions. More efficient deterministic and randomized ([delta] + 1)-coloring algorithms were achieved. Deterministic [delta]1 + o(1)-coloring algorithms with polylogarithmic running time were devised. Improved (and often sublogarithmic-time) randomized algorithms were devised. Drastically improved lower bounds were given. Wide families of graphs in which these problems are solvable much faster than on general graphs were identified. The objective of our monograph is to cover most of these developments, and as a result to provide a treatise on theoretical foundations of distributed symmetry breaking in the message-passing model. We hope that our monograph will stimulate further progress in this exciting area. Table of Contents: Acknowledgments / Introduction / Basics of Graph Theory / Basic Distributed Graph Coloring Algorithns / Lower Bounds / Forest-Decomposition Algorithms and Applications / Defective Coloring / Arbdefective Coloring / Edge-Coloring and Maximal Matching / Network Decompositions / Introduction to Distributed Randomized Algorithms / Conclusion and Open Questions / Bibliography / Authors' Biographiesread more
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Journal ArticleDOI
Distributed Optimization in Sensor Network for Scalable Multi-Robot Relative State Estimation
Tianyue Wu,Fei Gao +1 more
TL;DR: In this paper , the authors propose an efficient and scalable optimization algorithm with the classical block coordinate descent method as its backbone to solve each block update subproblem with a closed-form solution while ensuring convergence.
Locally-iterative $(\Delta+1)$-Coloring in Sublinear (in $\Delta$) Rounds
TL;DR: This paper gives the first locally-iterative (โ + 1) -coloring algorithm with sublinear-in- โ running time, and answers the main open question raised in a recent breakthrough.
Posted Content
Distributed Graph Coloring Made Easy
TL;DR: In this article, the authors presented a deterministic CONGEST algorithm to compute an O(k ฮ)$-vertex coloring in O( ฮ/k) + log โ n$ rounds, where ฮ is the maximum degree of the network graph and any vertex color can be freely chosen.
Proceedings ArticleDOI
Experimental Evaluation of Distributed Node Coloring Algorithms for Wireless Networks
TL;DR: Rand4DColor is very fast, computing a valid (4Degree)-coloring in less than one third of the time slots required for local broadcasting, where Degree is the maximum node degree in the network and the algorithm is robust even in networks with mobile nodes.
Distributed Edge Coloring in Time Polylogarithmic in $\Delta$
TL;DR: It is shown that a (2โ โ 1)-edge coloring can be computed in time poly log โ+ O (log โ n ) in the LOCAL model, which improves a result of Balliu, Kuhn, and Olivetti [PODC โ20], who gave an algorithm with a quasi-polylogarithmic dependency on โ.
References
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Book ChapterDOI
Reducibility Among Combinatorial Problems
TL;DR: The work of Dantzig, Fulkerson, Hoffman, Edmonds, Lawler and other pioneers on network flows, matching and matroids acquainted me with the elegant and efficient algorithms that were sometimes possible.
Reducibility Among Combinatorial Problems.
TL;DR: Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held, which made me aware of the importance of distinction between polynomial-time and superpolynomial-time solvability.