  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.
