Distributed Testing of Distance-k Colorings

Fraigniaud, Pierre; Halldórsson, Magnús M.; Nolin, Alexandre

doi:10.1007/978-3-030-54921-3_16

Cited by 4 publications

(3 citation statements)

References 29 publications

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“…Recent years have seen several results for problems in power graphs in the CONGEST model, reaching from verifying solutions efficiently (or showing that this is not possible) [FHN20], and answering several computational questions in the settings, e.g., for the already discussed problem of distance-2 coloring [HKM20, HKMN20] or optimization problems [BCM + 20].…”

Section: Further Related Workmentioning

confidence: 99%

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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Section: Further Related Workmentioning

confidence: 99%

Distributed Symmetry Breaking on Power Graphs via Sparsification

Maus¹,

Saku²,

Uitto³

2023

Preprint

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“…This is an exponential improvement over the previous best known bound O(log n) [HKMN20], as a function of n alone. Interestingly, for more general power graphs G k with k ≥ 3, it is provably hard to verify an arbitrary coloring [FHN20]. Thus, any poly log log n algorithm coloring G k when k ≥ 3 and ∆ ≫ poly log log n would need a different approach.…”

Section: Introductionmentioning

confidence: 99%

Coloring Fast with Broadcasts

Flin

Ghaffari

Halldórsson

et al. 2023

Proceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures

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We provide a O(log 6 log n)-round randomized algorithm for distance-2 coloring in CONGEST with ∆ 2 +1 colors. For ∆ ≫ poly log n, this improves exponentially on the O(log ∆+poly log log n) algorithm of [Halldórsson, Kuhn, Maus, Nolin, DISC'20].Our study is motivated by the ubiquity and hardness of local reductions in CONGEST. For instance, algorithms for the Local Lovász Lemma [Moser, Tardos, JACM'10; Fischer, Ghaffari, DISC'17; Davies, SODA'23] usually assume communication on the conflict graph, which can be simulated in LOCAL with only constant overhead, while this may be prohibitively expensive in CONGEST. We hope our techniques help tackle in CONGEST other coloring problems defined by local relations.

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“…We ask: are derandomized using the method of conditional expectation [21]. d2-coloring in CONGEST is further of special interest as it appears to lie at the edge of what is computable efficiently, i.e., in polylogarithmic time, while distance-3 coloring is even hard to verify [18].…”

Section: Introductionmentioning

confidence: 99%