Smallest Suffixient Sets: Effectiveness, Resilience, and Calculation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916059759509504 |
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| author | Fujimaru, Hiroto Navarro, Gonzalo Romana, Giuseppe Urbina, Cristian |
| author_facet | Fujimaru, Hiroto Navarro, Gonzalo Romana, Giuseppe Urbina, Cristian |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_05638 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Smallest Suffixient Sets: Effectiveness, Resilience, and Calculation Fujimaru, Hiroto Navarro, Gonzalo Romana, Giuseppe Urbina, Cristian Formal Languages and Automata Theory Data Structures and Algorithms Combinatorics 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). |
| title | Smallest Suffixient Sets: Effectiveness, Resilience, and Calculation |
| topic | Formal Languages and Automata Theory Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2506.05638 |