Distributed algorithms for the Lovász local lemma and graph coloring

Chung, Kai-Min; Pettie, Seth; Su, Hsin-Hao

doi:10.1145/2611462.2611465

Cited by 38 publications

(82 citation statements)

References 30 publications

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“…(C6) We get a Weak-MIS algorithm with complexity O(log ∆), which thus improves the round complexity of the distributed algorithmic version of the Lovász Local Lemma presented by Chung, Pettie, and Su [CPS14] from O(log 1…”

Section: (C3) We Get An O(mentioning

confidence: 82%

“…Using ∆ to denote the maximum degree, one can obtain answers such as O(log 2 ∆ + log 1/ε) rounds for Luby's algorithm, or O(log ∆ log log ∆ + log ∆ log 1/ε) rounds for the variant of Luby's used by Barenboim, Elkin, Pettie, and Schneider [BEPSv3] and Chung, Pettie, and Su [CPS14]. However, both of these bounds seem to be off from the right answer; e.g., one cannot recover from these the standard O(log n) high probability global complexity bound.…”

Section: Local Complexitymentioning

confidence: 99%

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An Improved Distributed Algorithm for Maximal Independent Set

Ghaffari¹

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

149

325

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The Maximal Independent Set (MIS) problem is one of the basics in the study of locality in distributed graph algorithms. This paper presents an extremely simple randomized algorithm providing a near-optimal local complexity for this problem, which incidentally, when combined with some known techniques, also leads to a near-optimal global complexity.Classical MIS algorithms of Luby [STOC'85] and Alon, Babai and Itai [JALG'86] provide the global complexity guarantee that, with high probability 1 , all nodes terminate after O(log n) rounds. In contrast, our initial focus is on the local complexity, and our main contribution is to provide a very simple algorithm guaranteeing that each particular node v terminates after O(log deg(v) + log 1/ε) rounds, with probability at least 1 − ε. The guarantee holds even if the randomness outside 2-hops neighborhood of v is determined adversarially. This degree-dependency is optimal, due to a lower bound of Kuhn, Moscibroda, and Wattenhofer [PODC'04].Interestingly, this local complexity smoothly transitions to a global complexity: by adding techniques of Barenboim, Elkin, Pettie, and Schneider [FOCS'12; arXiv: 1202], we 2 get a randomized MIS algorithm with a high probability global complexity of O(log ∆) + 2 O( √ log log n) , where ∆ denotes the maximum degree. This improves over the O(log 2 ∆) + 2 O( √ log log n) result of Barenboim et al., and gets close to the Ω(min{log ∆, √ log n}) lower bound of Kuhn et al.Corollaries include improved algorithms for MIS in graphs of upper-bounded arboricity, or lower-bounded girth, for Ruling Sets, for MIS in the Local Computation Algorithms (LCA) model, and a faster distributed algorithm for the Lovász Local Lemma.1 As standard, we use the phrase with high probability to indicate that an event has probability at least 1 − 1/n. 2 quasi nanos, gigantium humeris insidentes

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Section: (C3) We Get An O(mentioning

confidence: 82%

Section: Local Complexitymentioning

confidence: 99%

An Improved Distributed Algorithm for Maximal Independent Set

Ghaffari¹

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

149

325

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“…Their algorithm makes extensive use of the distributed Lovász local lemma [12] and runs in Ω(log n) time. Pettie and Su sketched a proof that ∆-coloring trees takes O(log ∆ log n + log * n) time, at least for sufficiently large ∆.…”

Section: Theorem 9 ([3]mentioning

confidence: 99%

“…This second phenomenon is no coincidence! It is a direct result of the graph shattering approach to symmetry breaking used in [5] and further in [12,14,16,18,7,21,29]. The idea is to apply some randomized procedure that fixes some fragment of the output (e.g., part of the MIS is fixed, part of the coloring is fixed, etc.…”

Section: Introductionmentioning

confidence: 99%

An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model

Chang

Kopelowitz

Pettie

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

Self Cite

184

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Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish numerous connections between the deterministic and randomized complexities in the LOCAL model. Each of our results has a very compelling take-away message:Fast ∆-coloring of trees requires random bits. Building on the recent randomized lower bounds of Brandt et al.[11], we prove that the randomized complexity of ∆-coloring a tree with maximum degree ∆ is Θ(log ∆ log n), for any ∆ ≥ 55, whereas its deterministic complexity is Θ(log ∆ n) for any ∆ ≥ 3. 1 This also establishes a large separation between the deterministic complexity of ∆-coloring and (∆ + 1)-coloring trees.Randomized lower bounds imply deterministic lower bounds. We prove that any deterministic algorithm for a natural class of problems that runs in O(1) + o(log ∆ n) rounds can be transformed to run in O(log * n − log * ∆ + 1) rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires Ω(log ∆ n) time deterministically. (This gives an alternate proof that deterministically ∆-coloring a tree with small ∆ takes Ω(log ∆ n) rounds.)Deterministic lower bounds imply randomized lower bounds. We prove that the randomized complexity of any natural problem on instances of size n is at least its deterministic complexity on instances of size √ log n. This shows that a deterministic Ω(log ∆ n) lower bound for any problem (∆-coloring a tree, for example) implies a randomized Ω(log ∆ log n) lower bound. It also illustrates that the graph shattering technique employed in recent randomized symmetry breaking algorithms is absolutely essential to the LOCAL model. For example, it is provably impossible to improve the 2 O( √ log log n) terms in the complexities of the best MIS and (∆ + 1)-coloring algorithms without also improving the 2 O( √ log n) -round Panconesi-Srinivasan algorithms.

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“…This natural idea has been considered in [28], also in a much broader context such as parallel job scheduling [11] or distributed Lovász local lemma [44,10]. For sampling from locally defined joint distributions, it is especially suitable because of the conditional independence property of MRFs.…”

Section: Our Resultsmentioning

confidence: 99%

What Can be Sampled Locally?

Feng

Sun

Yin

2017

Proceedings of the ACM Symposium on Principles of Distributed Computing

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The local computation of Linial [FOCS'87] and Naor and Stockmeyer [STOC'93] concerns with the question of whether a locally definable distributed computing problem can be solved locally: more specifically, for a given local CSP whether a CSP solution can be constructed by a distributed algorithm using local information. In this paper, we consider the problem of sampling a uniform CSP solution by distributed algorithms, and ask whether a locally definable joint distribution can be sampled from locally. More broadly, we consider sampling from Gibbs distributions induced by weighted local CSPs, especially the Markov random fields (MRFs), in the LOCAL model.We give two Markov chain based distributed algorithms which we believe to represent two fundamental approaches for sampling from Gibbs distributions via distributed algorithms. The first algorithm generically parallelizes the single-site sequential Markov chain by iteratively updating a random independent set of variables in parallel, and achieves an O(∆ log n) time upper bound in the LOCAL model, where ∆ is the maximum degree, when the Dobrushin's condition for the Gibbs distribution is satisfied. The second algorithm is a novel parallel Markov chain which proposes to update all variables simultaneously yet still guarantees to converge correctly with no bias. It surprisingly parallelizes an intrinsically sequential process: stabilizing to a joint distribution with massive local dependencies, and may achieve an optimal O(log n) time upper bound independent of the maximum degree ∆ under a stronger mixing condition.We also show a strong Ω(diam) lower bound for sampling: in particular for sampling independent set in graphs with maximum degree ∆ ≥ 6. Independent sets are trivial to construct locally and the sampling lower bound holds even when every node is aware of the entire graph. This gives a strong separation between sampling and constructing locally checkable labelings.

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Distributed algorithms for the Lovász local lemma and graph coloring

Cited by 38 publications

References 30 publications

An Improved Distributed Algorithm for Maximal Independent Set

An Improved Distributed Algorithm for Maximal Independent Set

An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model

What Can be Sampled Locally?

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