Smallest Suffixient Sets: Effectiveness, Resilience, and Calculation

TL;DR

This paper introduces the suffixient set as a new measure of string repetitiveness, analyzes its properties under various string operations, and compares it with existing measures, providing algorithms and bounds.

Contribution

It defines the smallest suffixient set, studies its sensitivity to string modifications, and relates it to known measures like runs and lexicographic parsing.

Findings

   the size  of the smallest suffixient set cannot increase by more than 2 after appending or prepending a character.

 the ratio  of suffixient sets of a string and its reverse is at most 2.

 bounds for  the increase of   suffixient set size after edit operations and rotations.

Abstract

A suffixient set is a novel combinatorial object that captures the essential information of repetitive strings in a way that, provided with a random access mechanism, supports various forms of pattern matching. In this paper, we study the size $\chi$ of the smallest suffixient set as a repetitiveness measure. First, we study its sensitivity to various string operations. We show that $\chi$ cannot increase by more than 2 after appending or prepending a character to the string. As a consequence, we are able to give simple linear-time online algorithms to compute smallest suffixient sets. We also show that, although reversing the string can increase $\chi$ by an arbitrary $O(n)$ value, it always holds $\chi(T)/\chi(T^R)\le 2$. We also prove lower and upper bounds for the additive or multiplicative increase of $\chi$ after applying arbitrary edit operations, or rotating the text. In particular, we show that the additive increase can be as large as $\Omega(\sqrt{n})$ for all those operations. Secondly, we place $\chi$ in between known repetitiveness measures. In particular, we show $\chi = O(r)$ (where $r$ is the number of runs in the Burrows-Wheeler Transform of the string), that there are string families where $\chi=o(v)$ (where $v$ is the size of the smallext lexicographic parse of the string), and that $\chi$ is uncomparable to almost all reachable measures based on copy-paste mechanisms. In passing, we give precise bounds for $\chi$ for some relevant string families, for example $\chi \le \sigma+2$ on episturmian words over alphabets of size $\sigma$ (e.g., $\chi \le 4$ on Fibonacci strings, for which we precisely characterize the only two smallest suffixient sets).