Superfast Coloring in CONGEST via Efficient Color Sampling

Halldórsson, Magnús M.; Nolin, Alexandre

doi:10.1007/978-3-030-79527-6_5

Cited by 4 publications

(1 citation statement)

References 21 publications

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“…For the ∆-coloring problem a very recent paper provided a genuinely different algorithm to solve the problem for ∆ ≥ poly log n, making a fix unnecessary [FMH22]. For the Lovász Local Lemma problem, one can use the algorithm by Chung, Pettie and Su to solve the problem in O(log n) rounds [CPS17b], which is faster than the O(∆ 2 ) + poly log log n rounds of [FG17] when ∆ ≥ poly log n. Last but not least, let us remark that the state-of-the-art randomized (∆+1)-coloring algorithms are not impacted by the flaw as they only rely on shattering when ∆ ≤ poly log n [CLP20,HN21]. For more details on the background of Theorem 1.4 and our solutions, see Section 3 and Section 7.…”

Section: Our Contributionsmentioning

confidence: 96%

Distributed Symmetry Breaking on Power Graphs via Sparsification

Maus¹,

Saku²,

Uitto³

2023

Preprint

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In this paper we present efficient distributed algorithms for classical symmetry breaking problems, maximal independent sets (MIS) and ruling sets, in power graphs. We work in the standard CONGEST model of distributed message passing, where the communication network is abstracted as a graph G. Typically, the problem instance in CONGEST is is identical to the communication network G, that is, we perform the symmetry breaking in G. In this work, we consider a setting where the problem instance corresponds to a power graph G k , where each node of the communication network G is connected to all of its k-hop neighbors.A β-ruling set is a set of non-adjacent nodes such that each node in G has a ruling neighbor within β hops; a natural generalization of an MIS. On top of being a natural family of problems, ruling sets (in power graphs) are well-motivated through their applications in the powerful shattering framework [BEPS JACM'16, Ghaffari SODA'19] (and others). We present randomized algorithms for computing maximal independent sets and ruling sets of G k in essentially the same time as they can be computed in G. Our main contribution is a deterministic polylogarithmic time algorithm for computing k-ruling sets of G k , which (for k > 1) improves exponentially on the current stateof-the-art runtimes. Our main technical ingredient for this result is a deterministic sparsification procedure which may be of independent interest.We also revisit the shattering algorithm for MIS [BEPS J'ACM'16] and present different approaches for the post-shattering phase. Our solutions are algorithmically and analytically simpler (also in the LOCAL model) than existing solutions and obtain the same runtime as [Ghaffari SODA '16].

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