A Separation of $\gamma$ and $b$ via Thue--Morse Words

TL;DR

This paper demonstrates a fundamental separation between the sizes of the smallest string attractor and the smallest bidirectional scheme for Thue--Morse words, revealing new insights into string compression complexities.

Contribution

It establishes the first known separation between string attractor size and bidirectional scheme size for a specific family of words.

Findings

b(t_n) = n+2 for n ≥ 2

γ(t_n) = 4 for n ≥ 4

Shows γ = o(b) for Thue--Morse words

Abstract

We prove that for $n\geq 2$, the size $b(t_n)$ of the smallest bidirectional scheme for the $n$th Thue--Morse word $t_n$ is $n+2$. Since Kutsukake et al. [SPIRE 2020] show that the size $\gamma(t_n)$ of the smallest string attractor for $t_n$ is $4$ for $n \geq 4$, this shows for the first time that there is a separation between the size of the smallest string attractor $\gamma$ and the size of the smallest bidirectional scheme $b$, i.e., there exist string families such that $\gamma = o(b)$.