Faster Online Matrix-Vector Multiplication

Kasper Green Larsen, R. Ryan Williams

Abstract: 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-∊) time algorithm for OMV, for all ∊ > 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 randomized time. In fact, after seeing vectors, we already achieve 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 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.