Edit distance in substitution systems

TL;DR

This paper investigates the edit distance among words generated by primitive substitutions, improving bounds on their diameter and analyzing specific cases like the Thue–Morse substitution.

Contribution

It provides an improved upper bound of the edit distance diameter for substitution words and analyzes the specific case of the Thue–Morse substitution.

Findings

Upper bound on edit distance diameter improved to O(n/√log n)

Diameter for Thue–Morse words is at least √(n/6) - 1

Handling non-uniform substitutions presents main challenge

Abstract

Let $\sigma$ be a primitive substitution on an alphabet $\mathcal{A}$, and let $\mathcal{W}_n$ be the set of words of length $n$ determined by $\sigma$ (i.e., $w \in \mathcal{W}_n$ if $w$ is a subword of $\sigma^k(a)$ for some $a \in \mathcal{A}$ and $k \geq 1$). It is known that the corresponding substitution dynamical system is loosely Kronecker (also known as zero-entropy loosely Bernoulli), so the diameter of $\mathcal{W}_n$ in the edit distance is $o(n)$. We improve this upper bound to $O(n/\sqrt{\log n})$. The main challenge is handling the case where $\sigma$ is non-uniform; a better bound is available for the uniform case. Finally, we show that for the Thue--Morse substitution, the diameter of $\mathcal{W}_n$ is at least $\sqrt {n/6} - 1$.