缺失子序列数量的紧界
Tight Bounds for the Number of Absent Subsequences
AI总结：
本文针对具有相同泛在性指标的单词，建立了给定长度缺失子序列数量的紧上下界，并给出了最优及增量枚举算法。
摘要：
单词 $w$ 的一个子序列是指可以通过删除 $w$ 中的某些字母、同时保持剩余字母相对顺序而得到的单词 $u$，例如 $\mathtt{lala}$ 是 $\mathtt{alfalfa}$ 的一个子序列。在某个字母表 $Σ$ 上，如果一个单词包含 $Σ$ 上所有长度为 $ι$ 的可能单词作为子序列，则称其为 $ι$-泛在的；使这一性质成立的最大 $ι$ 称为 $w$ 的泛在性指标，记为 $ι(w)$。此外，不是 $w$ 子序列的单词称为 $w$ 的缺失子序列（AS），对其的研究始于（Kosche et al., 2022）。本文给出了在所有具有相同泛在性指标 $ι$ 的单词中，给定长度 $k$ 的缺失子序列数量的紧界。对于下界和上界，我们分别构造了在相应长度 $k$ 上具有最少和最多缺失子序列数量的单词；并且在下界情形中，我们以闭式形式给出了缺失子序列的精确数量。最后，我们提出了用于枚举一个单词给定长度子序列集合的高效枚举算法：我们给出了一种新颖的最优枚举算法，其输出线性延迟，预处理时间为 $O(|w|)$；并进一步改进为增量枚举算法，延迟为 $O(1)$，预处理时间为 $O(|w|)$。
英文摘要：
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 $O(|w|)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
