Algebraic Methods in the Congested Clique

TL;DR

This paper introduces algebraic techniques for faster graph algorithms in the congested clique model, achieving significant improvements in triangle counting, shortest paths approximation, and girth computation.

Contribution

It adapts parallel matrix multiplication to the congested clique, enabling faster algorithms for key graph problems with new algebraic methods and a constant-round 4-cycle detection algorithm.

Findings

Triangle counting in O(n^{0.158}) rounds

Approximate all-pairs shortest paths in O(n^{0.158}) rounds

Girth computation in O(n^{0.158}) rounds

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: -- triangle and 4-cycle counting in $O(n^{0.158})$ rounds, improving upon the $O(n^{1/3})$ triangle detection 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.