  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
[J. ACM '75]. 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 \sigma 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 \ell.
