Faster Algorithms for Rectangular Matrix Multiplication

TL;DR: A new algorithm for multiplying an n × n^k matrix by an n–k × n matrix, which is better than all known algorithms for rectangular matrix multiplication and recovers exactly the complexity of the algorithm by Coppersmith and Winograd.

Abstract: Let $\alpha$ be the maximal value such that the product of an $n\times n^\alpha$ matrix by an $n^\alpha\times n$ 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 $n\times n^k$ matrix by an $n^k\times n$ 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 Wino grad (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, where $n$ denotes the number of vertices, and also improve the time complexity of sparse square matrix multiplication.