Tight Bounds for the Number of Absent Subsequences
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
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2024
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| _version_ | 1866912558075609088 |
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| author | Adamson, Duncan Fleischmann, Pamela Huch, Annika Manea, Florin Sarnighausen-Cahn, Paul Wiedenhöft, Max |
| author_facet | Adamson, Duncan Fleischmann, Pamela Huch, Annika Manea, Florin Sarnighausen-Cahn, Paul Wiedenhöft, Max |
| contents | A {\em subsequence} of a word $w$ is a word $u$ that can be obtained by deleting some letters from $w$ while maintaining the relative order of the remaining letters, e.g., $\mathtt{lala}$ is a subsequence of $\mathtt{alfalfa}$. A word, over some alphabet $Σ$, which has all possible words of length $ι$ over $Σ$ as subsequences is called $ι$-universal, and the largest $ι$ for which this holds is called the universality index of $w$, and denoted $ι(w)$. Moreover, words that are not subsequences of $w$ are called absent subsequences (AS) of $w$, and their investigation was started in (Kosche et al., 2022). In this paper, we present tight bounds on the number of AS of a given length $k$ among all words with the same universality index $ι$. For both the lower and upper bound, we construct words that have, respectively, a minimal and maximal number of absent subsequences of the respective length $k$, and, in the case of the lower bound, we provide the exact number of missing subsequences as a closed form. Finally, we present efficient enumeration algorithms for the set of subsequences of given length of a word: we give a novel, optimal enumeration algorithm with output linear delay of this set of subsequences, with preprocessing time $O(|w|)$, which is further improved to an incremental enumeration algorithm with $O(1)$ delay of this set of subsequences, with preprocessing time $O(|w|)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18599 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tight Bounds for the Number of Absent Subsequences Adamson, Duncan Fleischmann, Pamela Huch, Annika Manea, Florin Sarnighausen-Cahn, Paul Wiedenhöft, Max Formal Languages and Automata Theory Combinatorics 68R15 G.2.1; F.4.3 A {\em subsequence} of a word $w$ is a word $u$ that can be obtained by deleting some letters from $w$ while maintaining the relative order of the remaining letters, e.g., $\mathtt{lala}$ is a subsequence of $\mathtt{alfalfa}$. A word, over some alphabet $Σ$, which has all possible words of length $ι$ over $Σ$ as subsequences is called $ι$-universal, and the largest $ι$ for which this holds is called the universality index of $w$, and denoted $ι(w)$. Moreover, words that are not subsequences of $w$ are called absent subsequences (AS) of $w$, and their investigation was started in (Kosche et al., 2022). In this paper, we present tight bounds on the number of AS of a given length $k$ among all words with the same universality index $ι$. For both the lower and upper bound, we construct words that have, respectively, a minimal and maximal number of absent subsequences of the respective length $k$, and, in the case of the lower bound, we provide the exact number of missing subsequences as a closed form. Finally, we present efficient enumeration algorithms for the set of subsequences of given length of a word: we give a novel, optimal enumeration algorithm with output linear delay of this set of subsequences, with preprocessing time $O(|w|)$, which is further improved to an incremental enumeration algorithm with $O(1)$ delay of this set of subsequences, with preprocessing time $O(|w|)$. |
| title | Tight Bounds for the Number of Absent Subsequences |
| topic | Formal Languages and Automata Theory Combinatorics 68R15 G.2.1; F.4.3 |
| url | https://arxiv.org/abs/2407.18599 |