Parallel Heuristic Exploration for Additive Complexity Reduction in Fast Matrix Multiplication

TL;DR

This paper introduces a parallel heuristic search method that significantly improves additive complexity reduction in fast matrix multiplication algorithms with ternary coefficients, outperforming previous methods in speed and results.

Contribution

A novel parallel random-search approach with heuristic scoring and new strategies for reducing additive complexity in matrix multiplication algorithms.

Findings

Achieved lower addition counts on 102 of 149 schemes.

Found 57 new best-known results for optimal-rank schemes.

Method is faster and often better than state-of-the-art algorithms.

Abstract

This paper presents a parallel random-search method for reducing additive complexity in fast matrix multiplication algorithms with ternary coefficients $\{-1,0,1\}$. The approach replaces expensive exact evaluation with fast heuristic scoring, including the new Greedy-Intersections strategy. The method runs many independent common subexpression elimination processes in parallel, exploring the search space through random pair substitutions and diverse selection strategies while sharing promising partial solutions. Tested on 149 ternary-coefficient schemes, the method achieves lower addition counts than the state-of-the-art Greedy-Potential on 102 schemes (including 57 new best-known results for optimal-rank schemes), matches it on 45, and is outperformed on only 2. For most schemes, it provides equal or better results while being significantly faster, making it practical for algorithm exploration. All software and results are open source.