Longest Gapped Repeats and Palindromes

TL;DR

This paper presents efficient algorithms to compute the longest gapped repeats and palindromes at each position in a word, considering various restrictions on the gap length, which is useful for pattern analysis in strings.

Contribution

It introduces methods to efficiently find the longest gapped repeats and palindromes at every position with restricted gap lengths, advancing string pattern detection techniques.

Findings

Algorithms run in linear or near-linear time

Applicable to various gap length restrictions

Enhances pattern detection in string analysis

Abstract

A gapped repeat (respectively, palindrome) occurring in a word $w$ is a factor $uvu$ (respectively, $u^Rvu$) of $w$. In such a repeat (palindrome) $u$ is called the arm of the repeat (respectively, palindrome), while $v$ is called the gap. We show how to compute efficiently, for every position $i$ of the word $w$, the longest gapped repeat and palindrome occurring at that position, provided that the length of the gap is subject to various types of restrictions. That is, that for each position $i$ we compute the longest prefix $u$ of $w[i..n]$ such that $uv$ (respectively, $u^Rv$) is a suffix of $w[1..i-1]$ (defining thus a gapped repeat $uvu$ -- respectively, palindrome $u^Rvu$), and the length of $v$ is subject to the aforementioned restrictions.