PODC 2015 · 34th Annual ACM Symposium on Principles of Distributed Computing, Donostia-San Sebastián, Spain, July 2015 · doi:10.1145/2767386.2767414

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:

- triangle and 4-cycle counting in $O(n^{0.158})$ rounds, improving upon the $O(n^{1/3})$ triangle counting algorithm of Dolev et al. [DISC 2012],
- 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 of Nanongkai [STOC 2014], and
- 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.

Chryssis Georgiou and Paul G. Spirakis (Eds.): *PODC’15, Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, July 21–23, 2015, Donostia–San Sebastián, Spain*, pages 143–152, ACM Press, New York, 2015

ISBN 978-1-4503-3617-8

- Keren Censor-Hillel, Petteri Kaski, Janne H. Korhonen, Christoph Lenzen, Ami Paz, and Jukka Suomela: Algebraic methods in the congested clique ·
*Distributed Computing*32:461–478, 2019