Symmetry breaking in the congest model: Timeand message-efficient algorithms for ruling sets

Shreyas Pai, Gopal Pandurangan, Sriram V. Pemmaraju, Talal Riaz, Peter Robinson

Research output: Chapter in Book/Report/Conference proceedingConference contribution

12 Scopus citations

Abstract

We study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. The time (round) complexity of MIS (and ruling sets) have attracted much attention in the Local model. Indeed, recent results (Barenboim et al., FOCS 2012, Ghaffari SODA 2016) for the MIS problem have tried to break the long-standing O(log n)-round "barrier" achieved by Luby's algorithm, but these yield o(log n)-round complexity only when the maximum degree Δ is somewhat small relative to n. More importantly, these results apply only in the Local model. In fact, the best known time bound in the Congest model is still O(log n) (via Luby's algorithm) even for moderately small Δ (i.e., for Δ =Ω(log n) and Δ = o(n)). Furthermore, message complexity has been largely ignored in the context of local symmetry breaking. Luby's algorithm takes O(m) messages on m-edge graphs and this is the best known bound with respect to messages. Our work is motivated by the following central question: can we break the Θ(log n) time complexity barrier and the Θ(m) message complexity barrier in the Congest model for MIS or closelyrelated symmetry breaking problems? This paper presents progress towards this question for the distributed ruling set problem in the Congest model. A β-ruling set is an independent set such that every node in the graph is at most β hops from a node in the independent set. We present the following results: Time Complexity: We show that we can break the O(log n) "barrier" for 2- and 3-ruling sets. We compute 3-ruling sets in O (log n/loglogn) rounds with high probability (whp). More generally we show that 2-ruling sets can be computed in O (logΔ(log n)1/2+ϵ + log n/loglog n) rounds for any ϵ > 0, which is o(log n) for a wide range of Δ values (e.g., Δ = 2(log n)1/2-ϵ). These are the first 2- and 3-ruling set algorithms to improve over the O(log n)-round complexity of Luby's algorithm in the Congest model. Message Complexity: We show an Ω(n2) lower bound on the message complexity of computing an MIS (i.e., 1-ruling set) which holds also for randomized algorithms and present a contrast to this by showing a randomized algorithm for 2-ruling sets that, whp, uses only O(n log2 n) messages and runs in O(Delta; log n) rounds. This is the first message-efficient algorithm known for ruling sets, which has message complexity nearly linear in n (which is optimal up to a polylogarithmic factor).

Original languageEnglish (US)
Title of host publication31st International Symposium on Distributed Computing, DISC 2017
EditorsAndrea W. Richa
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770538
DOIs
StatePublished - Oct 1 2017
Externally publishedYes
Event31st International Symposium on Distributed Computing, DISC 2017 - Vienna, Austria
Duration: Oct 16 2017Oct 20 2017

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume91
ISSN (Print)1868-8969

Conference

Conference31st International Symposium on Distributed Computing, DISC 2017
Country/TerritoryAustria
CityVienna
Period10/16/1710/20/17

Keywords

  • Congest model
  • Local model
  • Maximal independent set
  • Message complexity
  • Round complexity
  • Ruling sets
  • Symmetry breaking

ASJC Scopus subject areas

  • Software

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