Towards efficiently searching triple product property triples: Deterministic and randomized algorithms

Abstract

The key objects in the group-theoretic approach to matrix multiplication are subsets of a group satisfying the so-called triple product property (TPP). In this paper, we focus on the problem of efficiently finding the triple product property triples. We deduce and present some new characteristics of the tripleproduct property. Using these new characteristics, we firstly propose an efficient deterministic algorithm in which a screening process based on historical information is designed to reduce the search space. In contrast to some of the recent heuristic search methods, the proposed deterministic algorithm can search all kinds of TPP triples in a highly efficient way with the help of a novel representation for subsets and a Moving 1 principle. In addition, we also propose an efficient randomized algorithm for finding TPP triples, which adopts a greedy randomized strategy to randomly generate possible TPP candidates. Experimental results demonstrate that our proposed deterministic algorithm can achieve a huge speed-up in terms of running time compared with the existing deterministic algorithm, and the proposed randomized algorithm outperforms other existing approaches for finding TPP triples.