Better size estimation for sparse matrix products

TL;DR

This paper introduces a new, faster method for estimating the number of non-zero entries in sparse boolean matrix products, improving efficiency over previous algorithms and demonstrating practical accuracy through experiments.

Contribution

It presents a novel size estimation algorithm combining hash functions and sampling, achieving expected linear time and matching space lower bounds, with practical improvements over prior methods.

Findings

Expected time O(n) for approximation with small error probability

Significantly better accuracy in experiments on real data

Matching space lower bounds for size estimation

Abstract

We consider the problem of doing fast and reliable estimation of the number of non-zero entries in a sparse boolean matrix product. This problem has applications in databases and computer algebra. Let n denote the total number of non-zero entries in the input matrices. We show how to compute a 1 +- epsilon approximation (with small probability of error) in expected time O(n) for any epsilon > 4/\sqrt[4]{z}. The previously best estimation algorithm, due to Cohen (JCSS 1997), uses time O(n/epsilon^2). We also present a variant using O(sort(n)) I/Os in expectation in the cache-oblivious model. In contrast to these results, the currently best algorithms for computing a sparse boolean matrix product use time omega(n^{4/3}) (resp. omega(n^{4/3}/B) I/Os), even if the result matrix has only z=O(n) nonzero entries. Our algorithm combines the size estimation technique of Bar-Yossef et al. (RANDOM 2002) with a particular class of pairwise independent hash functions that allows the sketch of a set of the form A x C to be computed in expected time O(|A|+|C|) and O(sort(|A|+|C|)) I/Os. We then describe how sampling can be used to maintain (independent) sketches of matrices that allow estimation to be performed in time o(n) if z is sufficiently large. This gives a simpler alternative to the sketching technique of Ganguly et al. (PODS 2005), and matches a space lower bound shown in that paper. Finally, we present experiments on real-world data sets that show the accuracy of both our methods to be significantly better than the worst-case analysis predicts.