A string $w$ is called a minimal absent word (MAW) for another string $T$ if $w$ does not occur in $T$ but the proper substrings of $w$ occur in $T$. For example, let $\Sigma = \{\mathtt{a, b, c}\}$ be the alphabet. Then, the set of MAWs for string $w = \mathtt{abaab}$ is $\{\mathtt{aaa, aaba, bab, bb, c}\}$. In this paper, we study combinatorial properties of MAWs in the sliding window model, namely, how the set of MAWs changes when a sliding window of fixed length $d$ is shifted over the input string $T$ of length $n$, where $1 \leq d < n$. We present \emph{tight} upper and lower bounds on the maximum number of changes in the set of MAWs for a sliding window over $T$, both in the cases of general alphabets and binary alphabets. Our bounds improve on the previously known best bounds [Crochemore et al., 2020].

中文翻译：

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滑动窗的最小缺席词组合

如果$ w $不在$ T $中出现，但是$ w $的适当子字符串在$ T $中出现，则字符串$ w $被称为另一个字符串$ T $的最小缺席单词（MAW）。例如，假设$ \ Sigma = \ {\ mathtt {a，b，c} \} $是字母。然后，字符串$ w = \ mathtt {abaab} $的MAW集为$ \ {\ mathtt {aaa，aaba，bab，bb，c} \} $。在本文中，我们研究了滑动窗口模型中MAW的组合特性，即，当固定长度$ d $的滑动窗口在长度为$ n $的输入字符串$ T $上移动时，MAW的集合如何变化，其中$ 1 \ leq d <n $。对于普通字母和二进制字母，我们给出了$ T $上滑动窗口的MAW集合最大变化数的\ emph {tight}上限和下限。我们的界限在以前已知的最佳界限上有所改善[Crochemore et al。，2020]。