Sensitivity of Repetitiveness Measures to String Reversal

2026-02-16

AI Summary

This study investigates the instability of mainstream repetitiveness measures—such as the number of runs in the Burrows–Wheeler Transform (\(r\)), the size of the LZ77 parsing (\(z\)), and the LZ-end size (\(v\))—under string reversal. By constructing specific families of strings and employing combinatorial analysis, asymptotic methods, and information-theoretic tools, the authors establish the first tight additive sensitivity bound of \(\Theta(n)\) for \(r\) under reversal, substantially improving the previous \(\Omega(\log n)\) lower bound. They also demonstrate that the ratio \(z(w^R)/z(w)\) can approach 3 and that the multiplicative sensitivity of \(v\) is \(\Theta(\log n)\). All derived bounds are asymptotically tight, revealing the pronounced vulnerability of these measures to such a simple transformation.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Data Compression
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

We study the impact that string reversal can have on several repetitiveness measures. First, we exhibit an infinite family of strings where the number, $r$, of runs in the run-length encoding of the Burrows--Wheeler transform (BWT) can increase additively by $\Theta(n)$ when reversing the string. This substantially improves the known $\Omega(\log n)$ lower-bound for the additive sensitivity of $r$ and it is asymptotically tight. We generalize our result to other variants of the BWT, including the variant with an appended end-of-string symbol and the bijective BWT. We show that an analogous result holds for the size $z$ of the Lempel--Ziv 77 (LZ) parsing of the text, and also for some of its variants, including the non-overlapping LZ parsing, and the LZ-end parsing. Moreover, we describe a family of strings for which the ratio $z(w^R)/z(w)$ approaches $3$ from below as $|w|\rightarrow \infty$. We also show an asymptotically tight lower-bound of $\Theta(n)$ for the additive sensitivity of the size $v$ of the smallest lexicographic parsing to string reversal. Finally, we show that the multiplicative sensitivity of $v$ to reversing the string is $\Theta(\log n)$, and this lower-bound is also tight. Overall, our results expose the limitations of repetitiveness measures that are widely used in practice, against string reversal -- a simple and natural data transformation.

Problem

• repetitiveness measures
• string reversal
• Burrows-Wheeler transform
• Lempel-Ziv parsing
• sensitivity

Innovation

• repetitiveness measures
• string reversal
• Burrows-Wheeler Transform
• LZ77 parsing
• sensitivity analysis