Computing Longest Single-arm-gapped Palindromes in a String

Abstract

We introduce new types of approximate palindromes called single-arm-gapped palindromes (SAGPs). A SAGP contains a gap in either its left or right arm, which is in the form of either \(wguc u^R w^R\) or \(wuc u^Rgw^R\), where w and u are non-empty strings, \(w^R\) and \(u^R\) are their reversed strings respectively, g is a gap, and c is either a single character or the empty string. We classify SAGPs into two groups: those which have \(ucu^R\) as a maximal palindrome (type-1), and the others (type-2). We propose several algorithms to compute all type-1 SAGPs with longest arms occurring in a given string using suffix arrays, and them a linear-time algorithm based on suffix trees. We also show how to compute type-2 SAGPs with longest arms in linear time. We perform some preliminary experiments to evaluate practical performances of the proposed methods.