Tight Additive Sensitivity on LZ-style Compressors and String Attractors

2025-06-28

AI Summary

This paper investigates the worst-case additive sensitivity of string repetitiveness measures under single-character edits. Specifically, it analyzes how γ (minimum attractor size), b (minimum bidirectional scheme size), and three Lempel–Ziv compression measures—LZSS, LZ-End, and LZ78—change when one character is inserted, deleted, or substituted. The work establishes the first tight bounds: Θ(√n) for γ and b; Θ(n²/³) for LZSS and LZ-End; and Θ(n) for LZ78. Methodologically, it combines combinatorial construction with compression-theoretic analysis, achieving matching upper and lower bounds via explicit adversarial examples and rigorous upper-bound proofs. The results fully characterize the edit sensitivity of these fundamental repetitiveness measures, resolving several long-standing open problems. Moreover, this work provides the first systematic theoretical framework for analyzing the robustness of string compression under small perturbations.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Natural Language Processing: Safety and Robustness

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 $γ$ and the smallest bidirectional scheme size $b$, which match the known lower bounds $Ω(sqrt{n})$ for $γ$ 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 - $Θ(n^{frac{2}{3}})$ for LZSS and LZ-End, and $Θ(n)$ for LZ78.

Problem

• Study worst-case additive sensitivity of string repetitiveness measures
• Establish upper bounds for sensitivity of string attractor size
• Derive bounds for Lempel-Ziv family sensitivity variations

Innovation

• O(√n) bounds for string attractor sensitivity
• Matching bounds for bidirectional scheme size
• Θ(n^(2/3)) and Θ(n) for LZSS and LZ78