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 $1\le j-i+1\le d$. The first version works in $O(n\log {\sigma }^{\prime })$ time with $O(d)$ space where ${\sigma }^{\prime }\le d$ 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.

References in corpus (1)

• A Subquadratic Algorithm for Minimum Palindromic Factorization