Data structures for computing unique palindromes in static and non-static strings

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.