Graph Expansion Analysis for Communication Costs of Fast Rectangular Matrix Multiplication

TL;DR

This paper extends graph expansion analysis to rectangular matrix multiplication algorithms, deriving new communication cost lower bounds and identifying some bounds as optimal, thereby broadening the understanding of computational efficiency.

Contribution

It introduces an extension of expansion analysis to rectangular matrix multiplication algorithms, providing new lower bounds and demonstrating their optimality.

Findings

Derived new communication lower bounds for rectangular matrix multiplication algorithms.

Extended graph expansion analysis to a broader class of algorithms.

Proved some bounds to be optimal.

Abstract

Graph expansion analysis of computational DAGs is useful for obtaining communication cost lower bounds where previous methods, such as geometric embedding, are not applicable. This has recently been demonstrated for Strassen's and Strassen-like fast square matrix multiplication algorithms. Here we extend the expansion analysis approach to fast algorithms for rectangular matrix multiplication, obtaining a new class of communication cost lower bounds. These apply, for example to the algorithms of Bini et al. (1979) and the algorithms of Hopcroft and Kerr (1971). Some of our bounds are proved to be optimal.