Sensitivity of Repetitiveness Measures to String Reversal

TL;DR

This paper investigates how string reversal affects various measures of repetitiveness, revealing significant sensitivities and limitations of these measures in practical applications.

Contribution

It provides tight bounds on the sensitivity of repetitiveness measures like BWT runs and LZ parsing to string reversal, extending understanding of their limitations.

Findings

Reversal can increase BWT run count by A(n)

The ratio of LZ parsing sizes approaches 3 for reversed strings

Sensitivity bounds for lexicographic parsing size are established

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.