The I/O complexity of Strassen's matrix multiplication with recomputation

TL;DR

This paper establishes a tight lower bound on the I/O complexity of Strassen's matrix multiplication algorithm with recomputation, using novel proof techniques involving Grigoriev's flow and CDAG properties, and extends previous bounds to include recomputation scenarios.

Contribution

It introduces a new proof method for I/O lower bounds that accounts for recomputation in Strassen's algorithm, expanding the understanding of its computational complexity.

Findings

Derived a tight lower bound on I/O complexity with recomputation.

Extended previous bounds to include recomputation scenarios.

Developed a new proof technique involving Grigoriev's flow and CDAG properties.

Abstract

A tight $\Omega((n/\sqrt{M})^{\log_2 7}M)$ lower bound is derived on the \io complexity of Strassen's algorithm to multiply two $n \times n$ matrices, in a two-level storage hierarchy with $M$ words of fast memory. A proof technique is introduced, which exploits the Grigoriev's flow of the matrix multiplication function as well as some combinatorial properties of the Strassen computational directed acyclic graph (CDAG). Applications to parallel computation are also developed. The result generalizes a similar bound previously obtained under the constraint of no-recomputation, that is, that intermediate results cannot be computed more than once. For this restricted case, another lower bound technique is presented, which leads to a simpler analysis of the \io complexity of Strassen's algorithm and can be readily extended to other "Strassen-like" algorithms.