Longest Gapped Repeats and Palindromes

A gapped repeat (respectively, palindrome) occurring in a word $w$ is afactor $uvu$ (respectively, $u^Rvu$) of $w$. In such a repeat (palindrome) $u$is called the arm of the repeat (respectively, palindrome), while $v$ is calledthe 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 ofrestrictions. 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.Comment: This is an extension of the conference papers "Longest $\alpha$-Gapped Repeat and Palindrome", presented by the second and third authors at FCT 2015, and "Longest Gapped Repeats and Palindromes", presented by the first and third authors at MFCS 201