A palindrome is a string that reads the same forward and backward. A palindromic substring $w$ of a string $T$ is called a minimal unique palindromic substring (MUPS) of $T$ if $w$ occurs only once in $T$ and any proper palindromic substring of $w$ occurs at least twice in $T$. MUPSs are utilized for answering the shortest unique palindromic substring problem, which is motivated by molecular biology [Inoue et al., 2018]. Given a string $T$ of length $n$, all MUPSs of $T$ can be computed in $O(n)$ time. In this paper, we study the problem of updating the set of MUPSs when a character in the input string $T$ is substituted by another character. We first analyze the number $d$ of changes of MUPSs when a character is substituted, and show that $d$ is in $O(\log n)$. Further, we present an algorithm that uses $O(n)$ time and space for preprocessing, and updates the set of MUPSs in $O(\log\sigma + (\log\log n)^2 + d)$ time where $\sigma$ is the alphabet size. We also propose a variant of the algorithm, which runs in optimal $O(1+d)$ time when the alphabet size is constant.
翻译:等离子线是一个字符串, 读出相同的前向和后向。 等离子线是一个字符串。 等离子线以一字符串为单位, 以美元为单位, 如果美元仅发生一次, 而任何适当的低离子字符串以美元为单位, 则至少发生两倍于美元。 MUPS 用于应对由分子生物学[ Inoue 和 Al., 2018] 驱动的最短独特的低离子字符串问题。 由于字符串以美元为单位, 所有T$的MUPS, 都被称为最低特异的低离子串( MUPS) 美元。 在本文中, 当输入字符串中的一个字符以美元换成美元为单位时, 我们研究如何更新 MUPS 的数据集, 当我们用美元为单位时, 我们用美元 的平差值为n= 的平差值 。