# The I/O complexity of hybrid algorithms for square matrix multiplication

## TL;DR

This paper establishes tight lower bounds on the I/O complexity of hybrid matrix multiplication algorithms combining Strassen-like and standard methods, providing new insights into their efficiency limits in hierarchical memory systems.

## Contribution

It introduces the first I/O lower bounds for non-uniform, non-stationary hybrid matrix multiplication algorithms, extending the understanding of their computational complexity.

## Key findings

- Derived tight lower bounds for hybrid algorithms in hierarchical memory.
- Extended bounds to non-uniform, non-stationary algorithms with recursive variations.
- Discussed implications for parallel processing models.

## Abstract

Asymptotically tight lower bounds are derived for the I/O complexity of a general class of hybrid algorithms computing the product of $n \times n$ square matrices combining ``\emph{Strassen-like}'' fast matrix multiplication approach with computational complexity $\Theta{n^{\log_2 7}}$, and ``\emph{standard}'' matrix multiplication algorithms with computational complexity $\Omega\left(n^3\right)$. We present a novel and tight $\Omega\left(\left(\frac{n}{\max\{\sqrt{M},n_0\}}\right)^{\log_2 7}\left(\max\{1,\frac{n_0}{M}\}\right)^3M\right)$ lower bound for the I/O complexity a class of ``\emph{uniform, non-stationary}'' hybrid algorithms when executed in a two-level storage hierarchy with $M$ words of fast memory, where $n_0$ denotes the threshold size of sub-problems which are computed using standard algorithms with algebraic complexity $\Omega\left(n^3\right)$. The lower bound is actually derived for the more general class of ``\emph{non-uniform, non-stationary}'' hybrid algorithms which allow recursive calls to have a different structure, even when they refer to the multiplication of matrices of the same size and in the same recursive level, although the quantitative expressions become more involved. Our results are the first I/O lower bounds for these classes of hybrid algorithms. All presented lower bounds apply even if the recomputation of partial results is allowed and are asymptotically tight. The proof technique combines the analysis of the Grigoriev's flow of the matrix multiplication function, combinatorial properties of the encoding functions used by fast Strassen-like algorithms, and an application of the Loomis-Whitney geometric theorem for the analysis of standard matrix multiplication algorithms. Extensions of the lower bounds for a parallel model with $P$ processors are also discussed.