Faster Cycle Detection in the Congested Clique

TL;DR

This paper introduces a faster distributed algorithm for detecting h-cycles in the Congested Clique model, with improvements for odd cycles and directed graphs, and extends to quantum models for triangle detection.

Contribution

It presents a novel, faster distributed cycle detection algorithm that adapts to graph density and extends to quantum models, including a new parallel matrix product technique.

Findings

Improved cycle detection algorithms for odd cycles in undirected graphs.

Faster cycle detection in directed graphs for all cycle lengths.

Quantum triangle detection algorithm surpassing previous methods.

Abstract

We provide a fast distributed algorithm for detecting $h$-cycles in the \textsf{Congested Clique} model, whose running time decreases as the number of $h$-cycles in the graph increases. In undirected graphs, constant-round algorithms are known for cycles of even length. Our algorithm greatly improves upon the state of the art for odd values of $h$. Moreover, our running time applies also to directed graphs, in which case the improvement is for all values of $h$. Further, our techniques allow us to obtain a triangle detection algorithm in the quantum variant of this model, which is faster than prior work. A key technical contribution we develop to obtain our fast cycle detection algorithm is a new algorithm for computing the product of many pairs of small matrices in parallel, which may be of independent interest.