Analyzing and Leveraging the $k$-Sensitivity of LZ77

TL;DR

This paper investigates how small edits to a string affect its LZ77 compression, providing tight bounds on sensitivity, a novel analysis contrasting with LZ78, and an algorithm for optimizing compression through pre-editing.

Contribution

It introduces tight bounds on the $k$-sensitivity of LZ77 compression and classifies the impact based on string compressibility, along with an approximation algorithm for pre-editing to improve compression.

Findings

Tight upper bounds for $k$-edits on LZ77 compression.

A trichotomy of compression sensitivity based on string compressibility.

An $ ext{epsilon}$-approximation algorithm for pre-editing to reduce compressed size.

Abstract

We study the sensitivity of the Lempel-Ziv 77 compression algorithm to edits, showing how modifying a string $w$ can deteriorate or improve its compression. Our first result is a tight upper bound for $k$ edits: $\forall w' \in B(w,k)$, we have $C_{\mathrm{LZ77}}(w') \leq 3 \cdot C_{\mathrm{LZ77}}(w) + 4k$. This result contrasts with Lempel-Ziv 78, where a single edit can significantly deteriorate compressibility, a phenomenon known as a *one-bit catastrophe*. We further refine this bound, focusing on the coefficient $3$ in front of $C_{\mathrm{LZ77}}(w)$, and establish a surprising trichotomy based on the compressibility of $w$. More precisely we prove the following bounds: - if $C_{\mathrm{LZ77}}(w) \lesssim k^{3/2}\sqrt{n}$, the compression may increase by up to a factor of $\approx 3$, - if $k^{3/2}\sqrt{n} \lesssim C_{\mathrm{LZ77}}(w) \lesssim k^{1/3}n^{2/3}$, this factor is at most $\approx 2$, - if $C_{\mathrm{LZ77}}(w) \gtrsim k^{1/3}n^{2/3}$, the factor is at most $\approx 1$. Finally, we present an $\varepsilon$-approximation algorithm to pre-edit a word $w$ with a budget of $k$ modifications to improve its compression. In favorable scenarios, this approach yields a total compressed size reduction by up to a factor of~$3$, accounting for both the LZ77 compression of the modified word and the cost of storing the edits, $C_{\mathrm{LZ77}}(w') + k \log |w|$.