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DOI: 10.4230/LIPIcs.DISC.2019.26
URN: urn:nbn:de:0030-drops-113337
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Konrad, Christian ; Pemmaraju, Sriram V. ; Riaz, Talal ; Robinson, Peter

The Complexity of Symmetry Breaking in Massive Graphs

LIPIcs-DISC-2019-26.pdf (0.6 MB)


The goal of this paper is to understand the complexity of symmetry breaking problems, specifically maximal independent set (MIS) and the closely related beta-ruling set problem, in two computational models suited for large-scale graph processing, namely the k-machine model and the graph streaming model. We present a number of results. For MIS in the k-machine model, we improve the O~(m/k^2 + Delta/k)-round upper bound of Klauck et al. (SODA 2015) by presenting an O~(m/k^2)-round algorithm. We also present an Omega~(n/k^2) round lower bound for MIS, the first lower bound for a symmetry breaking problem in the k-machine model. For beta-ruling sets, we use hierarchical sampling to obtain more efficient algorithms in the k-machine model and also in the graph streaming model. More specifically, we obtain a k-machine algorithm that runs in O~(beta n Delta^{1/beta}/k^2) rounds and, by using a similar hierarchical sampling technique, we obtain one-pass algorithms for both insertion-only and insertion-deletion streams that use O(beta * n^{1+1/2^{beta-1}}) space. The latter result establishes a clear separation between MIS, which is known to require Omega(n^2) space (Cormode et al., ICALP 2019), and beta-ruling sets, even for beta = 2. Finally, we present an even faster 2-ruling set algorithm in the k-machine model, one that runs in O~(n/k^{2-epsilon} + k^{1-epsilon}) rounds for any epsilon, 0 <=epsilon <=1. For a wide range of values of k this round complexity simplifies to O~(n/k^2) rounds, which we conjecture is optimal.
Our results use a variety of techniques. For our upper bounds, we prove and use simulation theorems for beeping algorithms, hierarchical sampling, and L_0-sampling, whereas for our lower bounds we use information-theoretic arguments and reductions to 2-party communication complexity problems.

BibTeX - Entry

  author =	{Christian Konrad and Sriram V. Pemmaraju and Talal Riaz and Peter Robinson},
  title =	{{The Complexity of Symmetry Breaking in Massive Graphs}},
  booktitle =	{33rd International Symposium on Distributed Computing (DISC 2019)},
  pages =	{26:1--26:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-126-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{146},
  editor =	{Jukka Suomela},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-113337},
  doi =		{10.4230/LIPIcs.DISC.2019.26},
  annote =	{Keywords: communication complexity, information theory, k-machine model, maximal independent set, ruling set, streaming algorithms}

Keywords: communication complexity, information theory, k-machine model, maximal independent set, ruling set, streaming algorithms
Collection: 33rd International Symposium on Distributed Computing (DISC 2019)
Issue Date: 2019
Date of publication: 08.10.2019

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