Faster Algorithms for Rectangular Matrix Multiplication

Abstract

Let α be the maximal value such that the product of an n × n α matrix by an n α × n matrix can be computed with n 2+o(1) arithmetic operations. In this paper we show that α > 0.30298, which improves the previous record α > 0.29462 by Coppersmith (Journal of Complexity, 1997). More generally, we construct a new algorithm for multiplying an n × n k matrix by an n k × n matrix, for any value k = 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. Short description of the approach by Coppersmith and Winograd. The results [9,13,16,24,27] mentioned above are all obtained by extending the approach by Coppersmith and Winograd [10]. This approach is an illustration of a general methodology initiated in the 1970's based on the theory of bilinear and trilinear forms, through which most of the improvements for matrix multiplication have been obtained. Informally, the idea is to start with a basic construction (some small trilinear form), and then exploit general properties of matrix multiplication (in particular Schönhage's asymptotic sum inequality [23]) to derive an upper bound on the exponent ω from this construction. The main contributions of [10] consist of two parts: the discovery of new basic constructions and the introduction of strong techniques to analyze them. In their paper, Coppersmith and Winograd actually present three algorithms, based on three different basic constructions. The first basic construction (Section 6 in [10]) is the simplest of the three and leads to the upper bound ω < 2.40364. The second basic construction (Section 7 in [10]), that we will refer in this paper as F q (here q ∈ N is a parameter), leads to the upper bound

Fast approximation algorithms for the diameter and radius of sparse graphs

“…It is not known whether APSP in such graphs can be solved inÕ(n ω ) time -the best algorithm is by Zwick [36] running in O(n 2.54 ) time [25], and hence for directed unweighted graphs diameter and radius can be solved somewhat faster than APSP. For undirected unweighted graphs the best known algorithm for diameter and radius is Seidel'sÕ(n ω ) time APSP algorithm [32].…”

Klee's Measure Problem Made Easy

“…Setting b = ε min{log n/ log log n, log n/ log log log U } yields a time bound of [16,27] can in fact achieve near mb d/2 time when b = o(m ε )); unfortunately this bound is too big to yield o(n d/2 ) time at the end.…”

Counting Thin Subgraphs via Packings Faster than Meet-in-the-Middle Time

“…[23]) that the product of an N × M matrix and an M × N matrix can be computed using Remark. The bounds above are not the best possible [21]; however, to provide a clean exposition we will work with these somewhat sub-state-of-the-art bounds.…”

“…For now we will be content in simply defining the equations and providing an illustration in Fig. 2. (The eventual algorithmic serendipity of this construction will be revealed only later in (20) and (21).) Let i = 0, 1, .…”