The input/output complexity of sparse matrix multiplication

Abstract

We consider the problem of multiplying sparse matrices (over a semiring) where the number of non-zero entries is larger than main memory. In the classical paper of Hong and Kung (STOC '81) it was shown that to compute a product of dense U ×U matrices θ (U3/B√M)), I/Os are necessary and sufficient in the I/O model with internal memory size M and memory block size B. In this paper we generalize the upper and lower bounds of Hong and Kung to the sparse case. Our bounds depend of the number N = nnz(A)+nnz(C) of nonzero entries in A and C, as well as the number Z =nnz(AC) of nonzero entries in AC. We show that using Õ(N/B min (√Z/M, N/M)) I/Os, AC can be computed with high probability. This is tight (up to polylogarithmic factors) when only semiring operations are allowed, even for dense rectangular matrices: We show a lower bound of ω (N/B min (√Z/M, N/M)) I/Os. While our lower bound uses fairly standard techniques, the upper bound makes use of "compressed matrix multiplication" sketches, which is new in the context of I/O-efficient algorithms, and a new matrix product size estimation technique that avoids the "no cancellation" assumption. © 2014 Springer-Verlag Berlin Heidelberg.