Compact representation for matrices of bounded twin-width

TL;DR

This paper introduces a space-efficient data structure for binary matrices with bounded twin-width, enabling fast entry queries and minimal storage, advancing the understanding of matrix compression for specific graph classes.

Contribution

It provides the first linear-space data structure for d-twin-ordered matrices with sub-logarithmic query time, optimizing storage and access.

Findings

Uses $O_d(n)$ bits of space for representation.

Supports matrix entry queries in $O_d(\log \log n)$ time.

Achieves minimal possible space for this class of matrices.

Abstract

For every fixed $d \in \mathbb{N}$, we design a data structure that represents a binary $n \times n$ matrix that is $d$-twin-ordered. The data structure occupies $O_d(n)$ bits, which is the least one could hope for, and can be queried for entries of the matrix in time $O_d(\log \log n)$ per query.