Faster Online Matrix-Vector Multiplication

We consider the Online Boolean Matrix-Vector Multiplication (OMV) problem studied by Henzinger et al. [STOC’15]: given an n× n Boolean matrix M, we receive n Boolean vectors v1, . . . ,vn one at a time, and are required to output Mvi (over the Boolean semiring) before seeing the vector vi+1, for all i. Previous known algorithms for this problem are combinatorial, running in O(n3/ log2 n) time. Henzinger et al. conjecture there is no O(n3−ε) time algorithm for OMV, for all ε > 0; their OMV conjecture is shown to imply strong hardness results for many basic dynamic problems. We give a substantially faster method for computing OMV, running in n3/2Ω( √ logn) randomized time. In fact, after seeing 2ω( √ logn) vectors, we already achieve n2/2Ω( √ logn) amortized time for matrix-vector multiplication. Our approach gives a way to reduce matrix-vector multiplication to solving a version of the Orthogonal Vectors problem, which in turn reduces to “small” algebraic matrix-matrix multiplication. Applications include faster independent set detection, partial match retrieval, and 2-CNF evaluation. We also show how a modification of our method gives a cell probe data structure for OMV with worst case O(n7/4/ √ w) time per query vector, where w is the word size. This result rules out an unconditional proof of the OMV conjecture using purely information-theoretic arguments. ∗Department of Computer Science, Aarhus University, [email protected]. Supported by Center for Massive Data Algorithmics, a Center of the Danish National Research Foundation, grant DNRF84, a Villum Young Investigator Grant and an AUFF Starting Grant. †Computer Science Department, Stanford University, [email protected]. Supported in part by NSF CCF-1212372 and CCF-1552651 (CAREER). Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.