Faster Algorithms for Rectangular Matrix Multiplication

TL;DR

This paper presents improved algorithms for rectangular matrix multiplication, achieving a higher exponent .30298, which enhances the efficiency of related algorithms like all-pairs shortest paths.

Contribution

The authors develop a new algorithm for multiplying rectangular matrices with better complexity bounds than previous methods, extending to all k.30298 and improving related computational problems.

Findings

Achieved .30298 for rectangular matrix multiplication.

Improved time complexity for all-pairs shortest paths to O(n^{2.5302}).

Enhanced sparse square matrix multiplication algorithms.

Abstract

Let {\alpha} be the maximal value such that the product of an n x n^{\alpha} matrix by an n^{\alpha} x n matrix can be computed with n^{2+o(1)} arithmetic operations. In this paper we show that \alpha>0.30298, which improves the previous record \alpha>0.29462 by Coppersmith (Journal of Complexity, 1997). More generally, we construct a new algorithm for multiplying an n x n^k matrix by an n^k x n matrix, for any value k\neq 1. The complexity of this algorithm is better than all known algorithms for rectangular matrix multiplication. In the case of square matrix multiplication (i.e., for k=1), we recover exactly the complexity of the algorithm by Coppersmith and Winograd (Journal of Symbolic Computation, 1990). These new upper bounds can be used to improve the time complexity of several known algorithms that rely on rectangular matrix multiplication. For example, we directly obtain a O(n^{2.5302})-time algorithm for the all-pairs shortest paths problem over directed graphs with small integer weights, improving over the O(n^{2.575})-time algorithm by Zwick (JACM 2002), and also improve the time complexity of sparse square matrix multiplication.