Sensitivity of Repetitiveness Measures to String Reversal

Hideo Bannai; Yuto Fujie; Peaker Guo; Shunsuke Inenaga; Yuto Nakashima; Simon J. Puglisi; Cristian Urbina

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arXiv · 2602.14385

Sensitivity of Repetitiveness Measures to String Reversal

Hideo Bannai·Yuto Fujie·Peaker Guo·Shunsuke Inenaga·Yuto Nakashima·Simon J. Puglisi·Cristian Urbina

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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.

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Hideo Bannai, Yuto Fujie, Peaker Guo, Shunsuke Inenaga, Yuto Nakashima, Simon J. Puglisi, Cristian Urbina. 2026-02-16. Sensitivity of Repetitiveness Measures to String Reversal. https://arxiv.org/abs/2602.14385

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