Palindromic Trees for a Sliding Window and Its Applications

TL;DR

This paper develops algorithms for maintaining palindromic trees in sliding windows of a string, enabling efficient updates and applications to minimal unique palindromic substrings and absent palindromic words.

Contribution

It introduces two algorithms for maintaining palindromic trees in sliding windows with different time and space complexities, extending their applicability.

Findings

First algorithm operates in O(n log σ') time with O(d) space.

Second algorithm operates in O(n + dσ) time with (d+2)σ + O(d) space.

Algorithms enable efficient computation of MUPS and MAPW in sliding windows.

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.