Palindromic Trees for a Sliding Window and Its Applications

Abstract: The palindromic tree (a.k.a. eertree) for a string $S$ of length $n$ is a tree-like data structure that represents the set of all distinct palindromic substrings of $S$, using $O(n)$ space [Rubinchik and Shur, 2018]. It is known that, when $S$ is over an alphabet of size $\sigma$ and is given in an online manner, then the palindromic tree of $S$ can be constructed in $O(n\log\sigma)$ time with $O(n)$ space. In this paper, we consider the sliding window version of the problem: For a sliding window of length at most $d$, we present two versions of an algorithm which maintains the palindromic tree of size $O(d)$ for every sliding window $S[i..j]$ over $S$, where $1 \leq j-i+1 \leq d$. The first version works in $O(n\log\sigma')$ time with $O(d)$ space where $\sigma' \leq d$ is the maximum number of distinct characters in the windows, and the second one works in $O(n + d\sigma)$ time with $(d+2)\sigma + O(d)$ space. We also show how our algorithms can be applied to efficient computation of minimal unique palindromic substrings (MUPS) and minimal absent palindromic words (MAPW) for a sliding window.