Minimal unique palindromic substrings after single-character substitution

TL;DR

This paper investigates how minimal unique palindromic substrings (MUPSs) change after a single-character substitution in a string, providing bounds and efficient algorithms for updating MUPSs.

Contribution

It introduces bounds on the number of MUPS changes after a character substitution and presents efficient algorithms for updating MUPSs with logarithmic time complexity.

Findings

Number of MUPS changes is O(log n) after substitution.

Proposed algorithms update MUPSs in O(log σ + (log log n)^2 + d) time.

Optimal O(1+d) update time for constant alphabet size.

Abstract

A palindrome is a string that reads the same forward and backward. A palindromic substring $w$ of a string $T$ is called a minimal unique palindromic substring (MUPS) of $T$ if $w$ occurs only once in $T$ and any proper palindromic substring of $w$ occurs at least twice in $T$. MUPSs are utilized for answering the shortest unique palindromic substring problem, which is motivated by molecular biology [Inoue et al., 2018]. Given a string $T$ of length $n$, all MUPSs of $T$ can be computed in $O(n)$ time. In this paper, we study the problem of updating the set of MUPSs when a character in the input string $T$ is substituted by another character. We first analyze the number $d$ of changes of MUPSs when a character is substituted, and show that $d$ is in $O(\log n)$. Further, we present an algorithm that uses $O(n)$ time and space for preprocessing, and updates the set of MUPSs in $O(\log\sigma + (\log\log n)^2 + d)$ time where $\sigma$ is the alphabet size. We also propose a variant of the algorithm, which runs in optimal $O(1+d)$ time when the alphabet size is constant.