Faster Online Matrix-Vector Multiplication

TL;DR

This paper presents a significantly faster randomized algorithm for the Online Boolean Matrix-Vector Multiplication problem, reducing the known computational complexity and enabling improvements in related dynamic problems and data structures.

Contribution

The authors introduce a novel approach that reduces OMV complexity to near-quadratic time and connects it to algebraic matrix multiplication, surpassing previous combinatorial algorithms.

Findings

Achieves $n^3/2^{ ext{Omega}( oot{ ext{log} n}{})}$ randomized time for OMV

Reduces matrix-vector multiplication to small algebraic matrix-matrix multiplication

Provides a cell probe data structure with $O(n^{7/4}/ oot{w}{})$ worst-case query time

Abstract

We consider the Online Boolean Matrix-Vector Multiplication (OMV) problem studied by Henzinger et al. [STOC'15]: given an $n \times n$ Boolean matrix $M$, we receive $n$ Boolean vectors $v_1,\ldots,v_n$ one at a time, and are required to output $M v_i$ (over the Boolean semiring) before seeing the vector $v_{i+1}$, for all $i$. Previous known algorithms for this problem are combinatorial, running in $O(n^3/\log^2 n)$ time. Henzinger et al. conjecture there is no $O(n^{3-\varepsilon})$ time algorithm for OMV, for all $\varepsilon > 0$; 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 $n^3/2^{\Omega(\sqrt{\log n})}$ randomized time. In fact, after seeing $2^{\omega(\sqrt{\log n})}$ vectors, we already achieve $n^2/2^{\Omega(\sqrt{\log n})}$ 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 $O(n^{7/4}/\sqrt{w})$ 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.