  A palindromic substring $T[i.. j]$ of a string $T$ is said to be a shortest
unique palindromic substring (SUPS) in $T$ for an interval $[p, q]$ if $T[i..
j]$ is a shortest palindromic substring such that $T[i.. j]$ occurs only once
in $T$, and $[i, j]$ contains $[p, q]$. The SUPS problem is, given a string $T$
of length $n$, to construct a data structure that can compute all the SUPSs for
any given query interval. It is known that any SUPS query can be answered in
$O(\alpha)$ time after $O(n)$-time preprocessing, where $\alpha$ is the number
of SUPSs to output [Inoue et al., 2018]. In this paper, we first show that
$\alpha$ is at most $4$, and the upper bound is tight. We also show that the
total sum of lengths of minimal unique palindromic substrings of string $T$,
which is strongly related to SUPSs, is $O(n)$. Then, we present the first
$O(n)$-bits data structures that can answer any SUPS query in constant time.
Also, we present an algorithm to solve the SUPS problem for a sliding window
that can answer any query in $O(\log\log W)$ time and update data structures in
amortized $O(\log\sigma + \log\log W)$ time, where $W$ is the size of the
window, and $\sigma$ is the alphabet size. Furthermore, we consider the SUPS
problem in the after-edit model and present an efficient algorithm. Namely, we
present an algorithm that uses $O(n)$ time for preprocessing and answers any
$k$ SUPS queries in $O(\log n\log\log n + k\log\log n)$ time after single
character substitution. Finally, as a by-product, we propose a fully-dynamic
data structure for range minimum queries (RmQs) with a constraint where the
width of each query range is limited to poly-logarithmic. The constrained RmQ
data structure can answer such a query in constant time and support a
single-element edit operation in amortized constant time.
