Improved Network Decompositions Using Small Messages with Applications on MIS, Neighborhood Covers, and Beyond

Ghaffari, Mohsen; Portmann, Julian

doi:10.4230/lipics.disc.2019.18

Cited by 3 publications

(1 citation statement)

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“…Note that this is faster than the previously mentioned algorithms if the maximum degree ∆ is sufficiently small. In addition, the problem of computing an MIS has also been studied in the more restrictive CONGEST model, where in each round, every node can only send an O(log n)-bit message to each neighbor [16,23,24,26,39]. Together, these papers imply that also in the CONGEST model, an MIS can be computed deterministically in time O(log 5 n) and the fastest known randomized algorithm has a time complexity of O(log ∆ • log log n + log 6 log n) and is therefore almost as fast as the fastest known randomized LOCAL model algorithm.…”

Section: Additional Related Workmentioning

confidence: 99%

Improved Distributed Lower Bounds for MIS and Bounded (Out-)Degree Dominating Sets in Trees

Balliu¹,

Brandt²,

Kühn³

et al. 2021

Preprint

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and Olivetti [FOCS '20] showed the first ω(log * n) lower bound for the maximal independent set (MIS) problem in trees. In this work we prove lower bounds for a much more relaxed family of distributed symmetry breaking problems. As a by-product, we obtain improved lower bounds for the distributed MIS problem in trees.For a parameter k and an orientation of the edges of a graph G, we say that a subset S of the nodes of G is a k-outdegree dominating set if S is a dominating set of G and if in the induced subgraph G[S], every node in S has outdegree at most k. Note that for k = 0, this definition coincides with the definition of an MIS. For a given k, we consider the problem of computing a k-outdegree dominating set. We show that, even in regular trees of degree at most ∆, in the standard LOCAL model, there exists a constant > 0 such that for k ≤ ∆ , for the problem of computing a k-outdegree dominating set, any randomized algorithm requires at least Ω min log ∆, √ log log n rounds and any deterministic algorithm requires at least Ω min log ∆, √ log n rounds. The proof of our lower bounds is based on the recently highly successful round elimination technique. We provide a novel way to do simplifications for round elimination, which we expect to be of independent interest. Our new proof is considerably simpler than the lower bound proof in [FOCS '20]. In particular, our round elimination proof uses a family of problems that can be described by only a constant number of labels. The existence of such a proof for the MIS problem was believed impossible by the authors of [FOCS '20].

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