Tight Additive Sensitivity on LZ-style Compressors and String Attractors

TL;DR

This paper establishes tight bounds on the additive sensitivity of string repetitiveness measures, including string attractors and Lempel-Ziv variants, revealing their stability under single-character edits.

Contribution

It provides matching upper and lower bounds for the additive sensitivity of key string compression measures, advancing understanding of their robustness.

Findings

Bounds of $O(\sqrt{n})$ for attractor and bidirectional scheme sizes

Bounds of $ heta(n^{2/3})$ for LZSS and LZ-End

Bounds of $ heta(n)$ for LZ78

Abstract

The worst-case additive sensitivity of a string repetitiveness measure $c$ is defined to be the largest difference between $c(w)$ and $c(w')$, where $w$ is a string of length $n$ and $w'$ is a string that can be obtained by performing a single-character edit operation on $w$. We present $O(\sqrt{n})$ upper bounds for the worst-case additive sensitivity of the smallest string attractor size $\gamma$ and the smallest bidirectional scheme size $b$, which match the known lower bounds $\Omega(\sqrt{n})$ for $\gamma$ and $b$ [Akagi et al. 2023]. Further, we present matching upper and lower bounds for the worst-case additive sensitivity of the Lempel-Ziv family - $\Theta(n^{\frac{2}{3}})$ for LZSS and LZ-End, and $\Theta(n)$ for LZ78.