Further Algebraic Algorithms in the Congested Clique Model and Applications to Graph-Theoretic Problems

TL;DR

This paper advances algebraic algorithms in the congested clique model, enabling efficient solutions for matrix operations and graph problems, with improved algorithms for shortest paths, diameter, and graph decompositions.

Contribution

It introduces new deterministic and randomized algebraic algorithms in the congested clique model for matrix computations and applies them to various graph-theoretic problems, improving existing results.

Findings

Efficient algorithms for multiple matrix products, determinant, rank, and inverse in the congested clique.

Improved algorithms for all-pairs shortest paths and diameter in weighted graphs.

Algorithms for maximum matching, Gallai-Edmonds decomposition, and minimum vertex cover.

Abstract

Censor-Hillel et al. [PODC'15] recently showed how to efficiently implement centralized algebraic algorithms for matrix multiplication in the congested clique model, a model of distributed computing that has received increasing attention in the past few years. This paper develops further algebraic techniques for designing algorithms in this model. We present deterministic and randomized algorithms, in the congested clique model, for efficiently computing multiple independent instances of matrix products, computing the determinant, the rank and the inverse of a matrix, and solving systems of linear equations. As applications of these techniques, we obtain more efficient algorithms for the computation, again in the congested clique model, of the all-pairs shortest paths and the diameter in directed and undirected graphs with small weights, improving over Censor-Hillel et al.'s work. We also obtain algorithms for several other graph-theoretic problems such as computing the number of edges in a maximum matching and the Gallai-Edmonds decomposition of a simple graph, and computing a minimum vertex cover of a bipartite graph.