有界范围内的子序列：匹配与分析问题
Subsequences in Bounded Ranges: Matching and Analysis Problems
AI总结：
本文研究有界范围内的子序列匹配与分析问题，给出匹配问题的二次时间上下界，证明多数相关分析问题在该场景下难解，并给出循环单词匹配的应用示例。
摘要：
本文研究经典算法问题的一个变体：判断给定单词$v$是否为另一单词$w$的子序列。更具体地，我们研究如下判定问题：给定数值$p$（定义范围边界）以及两个单词$v$和$w$，是否存在$w$的一个因子$w[i:i+p-1]$（换言之，长度为$p$的区间）使得$v$是该因子的子序列（即$v$作为子序列出现在有界范围$w[i:i+p-1]$中）。我们给出了该问题时间复杂度的匹配二次上下界。此外，我们研究了该场景下的一系列算法问题：给定整数$k$、$p$和单词$w$，分析集合$p$-Subseq$_k(w)$，即所有长度为$k$、且作为子序列出现在$w$的某个长度为$p$的因子中的单词构成的集合。其中我们研究了$k$-通用性问题、$k$-等价性问题，以及与缺失子序列相关的问题。令人意外的是，不同于单词中子序列的经典模型下这类问题通常存在高效解的情况，我们证明在考虑有界范围内的子序列的新场景下，这些问题大多变得难解。最后，我们给出示例，说明部分结果可应用于循环单词的子序列匹配问题。
英文摘要：
In this paper, we consider a variant of the classical algorithmic problem of checking whether a given word $v$ is a subsequence of another word $w$. More precisely, we consider the problem of deciding, given a number $p$ (defining a range-bound) and two words $v$ and $w$, whether there exists a factor $w[i:i+p-1]$ (or, in other words, a range of length $p$) of $w$ having $v$ as subsequence (i.\,e., $v$ occurs as a subsequence in the bounded range $w[i:i+p-1]$). We give matching upper and lower quadratic bounds for the time complexity of this problem. Further, we consider a series of algorithmic problems in this setting, in which, for given integers $k$, $p$ and a word $w$, we analyse the set $p$-Subseq$_{k}(w)$ of all words of length $k$ which occur as subsequence of some factor of length $p$ of $w$. Among these, we consider the $k$-universality problem, the $k$-equivalence problem, as well as problems related to absent subsequences. Surprisingly, unlike the case of the classical model of subsequences in words where such problems have efficient solutions in general, we show that most of these problems become intractable in the new setting when subsequences in bounded ranges are considered. Finally, we provide an example of how some of our results can be applied to subsequence matching problems for circular words.
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