Better size estimation for sparse matrix products

Authors:
Rasmus Resen Amossen;Andrea Campagna;Rasmus Pagh
Affiliations:
IT University of Copenhagen, Copenhagen S, Denmark;IT University of Copenhagen, Copenhagen S, Denmark;IT University of Copenhagen, Copenhagen S, Denmark
Venue:
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Year:
2010

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Abstract

We consider the problem of doing fast and reliable estimation of the number of non-zero entries in a sparse boolean matrix product. Let n denote the total number of non-zero entries in the input matrices. We show how to compute a 1 ± ε approximation (with small probability of error) in expected time O(n) for any ε 4/√4n. The previously best estimation algorithm, due to Cohen (JCSS 1997), uses time O(n/ε2). We also present a variant using O(sort(n)) I/Os in expectation in the cache-oblivious model. We also 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.