🤖 AI Summary
This work addresses the core challenge of optimizing additive complexity in fast matrix multiplication (FMM). We propose an efficient parallel randomized search framework. Methodologically, we introduce the Greedy-Intersections heuristic scoring mechanism—replacing costly exact evaluation—and integrate multi-strategy random pairing replacement, heterogeneous selection, and cooperative parallel common subexpression elimination (CSE) with shared partial solutions. We systematically evaluate our approach on 164 ternary-coefficient schemes: 103 achieve strictly better additive complexity than the state-of-the-art Greedy-Potential method, 59 match it, and only 2 are marginally worse; average search speed is significantly accelerated. All code and results are publicly released. Our framework establishes a scalable, high-efficiency paradigm for FMM structural search, advancing automated discovery of low-complexity matrix multiplication algorithms.
📝 Abstract
This paper presents a parallel random-search method for reducing additive complexity in fast matrix multiplication. 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 164 ternary-coefficient schemes, the method achieves lower addition counts than the state-of-the-art Greedy-Potential on 103 schemes, matches it on 59, and is outperformed on 2. For most schemes, it gives equal or better results while being much faster, making it practical for algorithm exploration. All software and results are open source.