Distributed deterministic edge coloring using bounded neighborhood independence

Barenboim, Leonid; Elkin, Michael

doi:10.1145/1993806.1993825

Cited by 30 publications

(41 citation statements)

References 29 publications

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“…It was formally defined in mid-eighties by [19,20,36], but was implicitly studied already in the mid-sixties [58]. Quite recently defective coloring was shown to be very useful for computing legal colorings in the distributed setting [8,9,10,48]. We will discuss this relationship in detail in Chapters 6 -7, and in Section 8.2.…”

Section: Defective Coloringmentioning

confidence: 99%

“…The algorithms in Section 8.1 are due to Panconesi and Rizzi [68]. The algorithms in Section 8.2 are from [10].…”

Section: Edge-coloring and Maximal Matchingmentioning

confidence: 99%

“…We then show that in graphs with bounded arboricity these objects can be computed in O( log n log log n ) time [7]. Finally, we describe our ∆ 1+o(1) -edge-coloring algorithm from [10] that requires O(log ∆) + log * n time. The latter algorithm is based on the notions of defective and arbdefective colorings, described in Chapters 6 and 7.…”

Section: Introductionmentioning

confidence: 99%

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Distributed Graph Coloring: Fundamentals and Recent Developments

Barenboim¹,

Elkin²

2013

Synthesis Lectures on Distributed Computing Theory

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170

255

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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 ∆ the maximum degree of G. While coloring G with ∆ + 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 (see [60,61,39,40,18,55,31,5]) laid firm foundations for the area of distributed symmetry breaking already in the eighties. In particualr, they showed that all these problems can be solved in randomized logarithmic time. Also, Linial [55] showed that an O(∆ 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 (∆ + 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 less than ∆ 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 (∆ + 1)-coloring algorithms were achieved in [8,48,78,11]. Deterministic ∆ 1+o(1) -coloring algorithms with polylogarithmic running time were devised in [9]. Improved (and often sublogarithmic-time) randomized algorithms were devised in [47,78,11]. Drastically improved lower bounds were given in [50,52]. Wide families of graphs in which these problems are solvable much faster than on general graphs were identified in [49,51,30,77,7,9].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.

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Section: Defective Coloringmentioning

confidence: 99%

“…The algorithms in Section 8.1 are due to Panconesi and Rizzi [68]. The algorithms in Section 8.2 are from [10].…”

Section: Edge-coloring and Maximal Matchingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Distributed Graph Coloring: Fundamentals and Recent Developments

Barenboim¹,

Elkin²

2013

Synthesis Lectures on Distributed Computing Theory

Self Cite

170

255

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show abstract

“…Moreover, for more restricted scenarios and some related problems there are lower bounds of Ω(∆) [13,14,20,27]. The question of the possibility to devise algorithms that overcome this challenging barrier is one of the most fundamental questions in distributed symmetry breaking [4,6,13,14,19,24]. In this paper we settle this question for (∆+1)-vertex-coloring and (2∆−1)-edge-coloring by devising deterministic algorithms that require O(∆ 3/4 log ∆+log * n) time in the static, dynamic and faulty settings.…”

mentioning

confidence: 99%

Deterministic (Δ + 1)-Coloring in Sublinear (in Δ) Time in Static, Dynamic and Faulty Networks

Barenboim

2015

Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing

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In the distributed message passing model a communication network is represented by an n-vertex graph G = (V, E) of maximum degree ∆. Computation proceeds in discrete synchronous rounds consisting of sending and receiving messages and performing local computations. The running time of an algorithm is the number of rounds it requires. In the static setting the network remains unchanged throughout the entire execution. In the dynamic setting the topology of the network changes, and a new solution has to be computed after each change. In the faulty setting the network is static, but some vertices or edges may lose the computed solution as a result of faults. The goal of an algorithm in this setting is fixing the solution.The problems of (∆ + 1)-vertex-coloring and (2∆ − 1)-edge-coloring are among the most important and intensively studied problems in distributed computing. Despite a very intensive research in the last 30 years, no deterministic algorithms for these problems with sublinear (in ∆) time have been known so far. Moreover, for more restricted scenarios and some related problems there are lower bounds of Ω(∆) [13,14,20,27]. The question of the possibility to devise algorithms that overcome this challenging barrier is one of the most fundamental questions in distributed symmetry breaking [4,6,13,14,19,24]. In this paper we settle this question for (∆+1)-vertex-coloring and (2∆−1)-edge-coloring by devising deterministic algorithms that require O(∆ 3/4 log ∆+log * n) time in the static, dynamic and faulty settings. (The term log * n is unavoidable in view of the lower bound of Linial [21].) Moreover, for (1 + o(1))∆-vertex-coloring and (2 + o(1))∆-edge-coloring we devise algorithms withÕ( √ ∆ + log * n) deterministic time. This is roughly a quadratic improvement comparing to the state- * of-the-art that requires O(∆ + log * n) time [4,19,24]. Our results are actually more general than that since they apply also to a variant of the list-coloring problem that generalizes ordinary coloring.Our results are obtained using a novel technique for coloring partially-colored graphs (also known as fixing). We partition the uncolored parts into a small number of subgraphs with certain helpful properties. Then we color these subgraphs gradually using a technique that employs constructions of polynomials in a novel way. Our construction is inspired by the algorithm of Linial [21] for ordinary O(∆ 2 )-coloring. However, it is a more sophisticated construction that differs from [21] in several important respects. These new insights in using systems of polynomials allow us to significantly speed up the O(∆)-coloring algorithms. Moreover, they allow us to devise algorithms with the same running time also in the more complicated settings of dynamic and faulty networks.

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“…Panconesi and collaborators have produced a number of papers tying edge coloring to channel assignment and presenting novel edge coloring algorithms with communication complexity of as low as O(loglogn) [5], [11], [10], [9]. [1]. A limitation of this algorithm is that the constant factor for quality increases as decreases.…”

Section: B Prior Workmentioning

confidence: 99%

Two Edge Coloring Algorithms Using a Simple Matching Discovery Automata

Daigle

Prasad

2012

2012 IEEE 26th International Parallel and Distributed Processing Symposium Workshops &Amp; PhD Forum

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We here present two probabilistic edge coloring algorithms for a message passing model of distributed computing. The algorithms use a simple automata for finding a matching on a graph to produce the colorings. Our first algorithm for edge coloring finds an edge coloring of a graph which is guaranteed to use no more than 2Δ − 1 colors and completes in O(Δ) communication rounds using only one hop information, where Δ is the greatest degree of the graph. Our second algorithm finds a strong edge coloring of a symmetric digraph in O(Δ) communication rounds, using only one hop information.

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Distributed deterministic edge coloring using bounded neighborhood independence

Cited by 30 publications

References 29 publications

Distributed Graph Coloring: Fundamentals and Recent Developments

Distributed Graph Coloring: Fundamentals and Recent Developments

Deterministic (Δ + 1)-Coloring in Sublinear (in Δ) Time in Static, Dynamic and Faulty Networks

Two Edge Coloring Algorithms Using a Simple Matching Discovery Automata

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