Palindromes are important objects in strings which have been extensively studied from combinatorial, algorithmic, and bioinformatics points of views. It is known that the length of the longest palindromic substrings (LPSs) of a given string T of length n can be computed in $O(n)$ time by Manacher's algorithm [12]. In this paper, we consider the problem of finding the LPS after the string is edited. We present an algorithm that uses $O(n)$ time and space for preprocessing, and answers the length of the LPSs in $O(\log (\min \{\sigma ,\log n\}))$ time after a single character substitution, insertion, or deletion, where σ denotes the number of distinct characters appearing in T. We also propose an algorithm that uses $O(n)$ time and space for preprocessing, and answers the length of the LPSs in $O(\ell +\log \log n)$ time, after an existing substring in T is replaced by a string of arbitrary length ℓ.

回文是字符串中的重要对象，已经从组合，算法和生物信息学的角度进行了广泛研究。已知可以计算长度为n的给定字符串T的最长回文子字符串（LPSs）的长度$Ø（ñ）$通过Manacher算法[12]。在本文中，我们考虑在字符串编辑后查找LPS的问题。我们提出一种使用$Ø（ñ）$ 预处理的时间和空间，并回答LPS的长度 $Ø（\mathrm{日志}（\mathrm{分}\{\sigma ，\mathrm{日志}ñ\}））$单个字符替换，插入或删除之后的时间，其中σ表示T中出现的不同字符的数量。我们还提出了一种使用$Ø（ñ）$ 预处理的时间和空间，并回答LPS的长度 $Ø（\ell +\mathrm{日志}\mathrm{日志}ñ）$时间，在现有的子串后Ť由任意长度的字符串替换ℓ。