Smallest Suffixient Sets: Effectiveness, Resilience, and Calculation

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).