Almost optimal Boolean matrix multiplication [BMM]-by multi-encoding of rows and columns

TL;DR

This paper presents a near-optimal algorithm for Boolean matrix multiplication that reduces complexity from cubic to nearly quadratic, enabling faster graph analysis and string parsing.

Contribution

It introduces a multi-encoding approach for rows and columns, achieving near-optimal Boolean matrix multiplication complexity with practical algorithms.

Findings

Reduces Boolean matrix multiplication complexity to O(m^{2+e}) for small e

Enables faster algorithms for graph property testing like triangle detection

Improves string parsing efficiency for context-free grammars

Abstract

The Boolean product $R = P \cdot Q$ of two $\{ 0, 1\} \; m \times m \; $ matrices is $$R(j,k) = 1 \; \mathrm{\ IF\ for\ some\ } \; t \; \,P(j, t) = Q(t, k) = 1\; \; \mathrm{ELSE\ } \, R(j, k) = 0. $$ The near-optimal design reduces the complexity of computing $R$ from the standard $m^3$ to $O(m^{(2+e)})$, for arbitrary small $e > 0$, by a practical algorithm. This renders reduced complexity to several graph-property tests: Finding triangles and higher-size cliques; finding all-pairs shortest paths, and more. Also, parsing a string $w$ by a context-free grammar is reduced to near quadratic in $w$-size. The design uses several distinct 2-digit encodings: $j$ by $(j_1, j_2), \; k \, $ by $\, (k_1, k_2)$. Each one gives rise to bunches of short digraphs from sources $j$'s to sinks $k$'s via switching nodes, and walks between them. The combined information, using the Chinese remainder theorem, leads to the correct values of $R(j, k)$.