Hiroto Fujimaru

Hiroto Fujimaru, Shunsuke Inenaga

The compact directed acyclic word graph (CDAWG) [Blumer et al. 1987] of a string is the minimal compact automaton that recognizes all the suffixes of the string. CDAWGs can be used for various string tasks including text pattern searching, data compression, and pattern discovery. The CDAWG-grammar [Belazzougui & Cunial 2017] is a grammar-based text compression based on the CDAWG, which allows for representing the CDAWG in $O(e)$ space without storing the string, where $e$ denotes the number of CDAWG edges. Let $g$ be the size of the CDAWG-grammar for the input string $T$. We show that the worst-case additive sensitivity of the CDAWG-grammar is lower bounded by $3g-21$ and is upper bounded by $8 g + 4$.

Hiroto Fujimaru, Gonzalo Navarro, Giuseppe Romana et al.

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 $χ$ of the smallest suffixient set as a repetitiveness measure. First, we study its sensitivity to various string operations. We show that $χ$ 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 $χ$ by an arbitrary $O(n)$ value, it always holds $χ(T)/χ(T^R)\le 2$. We also prove lower and upper bounds for the additive or multiplicative increase of $χ$ after applying arbitrary edit operations, or rotating the text. In particular, we show that the additive increase can be as large as $Ω(\sqrt{n})$ for all those operations. Secondly, we place $χ$ in between known repetitiveness measures. In particular, we show $χ= O(r)$ (where $r$ is the number of runs in the Burrows-Wheeler Transform of the string), that there are string families where $χ=o(v)$ (where $v$ is the size of the smallext lexicographic parse of the string), and that $χ$ is uncomparable to almost all reachable measures based on copy-paste mechanisms. In passing, we give precise bounds for $χ$ for some relevant string families, for example $χ\le σ+2$ on episturmian words over alphabets of size $σ$ (e.g., $χ\le 4$ on Fibonacci strings, for which we precisely characterize the only two smallest suffixient sets).