Graph Expansion and Communication Costs of Fast Matrix Multiplication

TL;DR

This paper establishes fundamental lower bounds on the communication costs of fast matrix multiplication algorithms, linking these costs to the expansion properties of their computation graphs, and demonstrates these bounds are achievable.

Contribution

It introduces a novel graph-theoretic approach to derive tight lower bounds on communication costs for fast matrix multiplication algorithms, including Strassen's and others.

Findings

Derived lower bounds on communication costs for sequential algorithms.

Extended bounds to parallel algorithms with multiple processors.

Showed bounds are attainable by existing fast algorithms in linear algebra.

Abstract

The communication cost of algorithms (also known as I/O-complexity) is shown to be closely related to the expansion properties of the corresponding computation graphs. We demonstrate this on Strassen's and other fast matrix multiplication algorithms, and obtain first lower bounds on their communication costs. In the sequential case, where the processor has a fast memory of size $M$, too small to store three $n$-by-$n$ matrices, the lower bound on the number of words moved between fast and slow memory is, for many of the matrix multiplication algorithms, $\Omega((\frac{n}{\sqrt M})^{\omega_0}\cdot M)$, where $\omega_0$ is the exponent in the arithmetic count (e.g., $\omega_0 = \lg 7$ for Strassen, and $\omega_0 = 3$ for conventional matrix multiplication). With $p$ parallel processors, each with fast memory of size $M$, the lower bound is $p$ times smaller. These bounds are attainable both for sequential and for parallel algorithms and hence optimal. These bounds can also be attained by many fast algorithms in linear algebra (e.g., algorithms for LU, QR, and solving the Sylvester equation).