Algebraic methods in the congested clique

Abstract

In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an \(O(n^{1-2/\omega })\) round matrix multiplication algorithm, where \(\omega < 2.3728639\) is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include:

  1. 1.

    triangle and 4-cycle counting in \(O(n^{0.158})\) rounds, improving upon the \(O(n^{1/3})\) algorithm of Dolev et al. [DISC 2012],

  2. 2.

    a \((1 + o(1))\)-approximation of all-pairs shortest paths in \(O(n^{0.158})\) rounds, improving upon the \(\tilde{O} (n^{1/2})\)-round \((2 + o(1))\)-approximation algorithm given by Nanongkai [STOC 2014], and

  3. 3.

    computing the girth in \(O(n^{0.158})\) rounds, which is the first non-trivial solution in this model.

In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.

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Acknowledgments

We thank Keijo Heljanko, Juho Hirvonen, Fabian Kuhn, Tuomo Lempiäinen, and Joel Rybicki for discussions, and the anonymous reviewers for comments.

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Correspondence to Keren Censor-Hillel.

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This work was supported by ISF Individual Research Grant 1696/14 (K. C.-H.) and the European Research Council under the European Union’s Seventh Framework programme (FP/2007–2013) / ERC Grant Agreement n. 338077 (P. K.). This work is an extended version of a preliminary conference report [18].

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Censor-Hillel, K., Kaski, P., Korhonen, J.H. et al. Algebraic methods in the congested clique. Distrib. Comput. 32, 461–478 (2019). https://doi.org/10.1007/s00446-016-0270-2

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Keywords

  • Distributed computing
  • Congested clique model
  • Lower bounds
  • Matrix multiplication
  • Subgraph detection
  • Distance computation