“Tri, Tri Again”: Finding Triangles and Small Subgraphs in a Distributed Setting

(Extended Abstract)
  • Danny Dolev
  • Christoph Lenzen
  • Shir Peled
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7611)

Abstract

Let G = (V,E) be an n-vertex graph and M d a d-vertex graph, for some constant d. Is M d a subgraph of G? We consider this problem in a model where all n processes are connected to all other processes, and each message contains up to \(\mathcal{O}(\log n)\) bits. A simple deterministic algorithm that requires \(\mathcal{O}(n^{(d-2)/d}/\log n)\) communication rounds is presented. For the special case that M d is a triangle, we present a probabilistic algorithm that requires an expected \(\mathcal{O}(n^{1/3}/(t^ {2/3}+1))\) rounds of communication, where t is the number of triangles in the graph, and \(\mathcal{O}(\min\{n^{1/3}\log^{2/3}n/(t^ {2/3}+1),n^{1/3}\})\) with high probability.

We also present deterministic algorithms that are specially suited for sparse graphs. In graphs of maximum degree Δ, we can test for arbitrary subgraphs of diameter D in \(\mathcal{O}(\Delta^{D+1}/n)\) rounds. For triangles, we devise an algorithm featuring a round complexity of \(\mathcal{O}((A^2\log_{2+n/A^2} n)/n)\), where A denotes the arboricity of G.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Danny Dolev
    • 1
  • Christoph Lenzen
    • 2
  • Shir Peled
    • 1
  1. 1.School of Engineering and Computer ScienceHebrew University of JerusalemIsrael
  2. 2.Department for Computer Science and Applied MathematicsWeizmann Institute of ScienceIsrael

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