Block Palindromes: A New Generalization of Palindromes

TL;DR

This paper introduces block palindromes, a new generalization of palindromes, and presents properties, a compact representation, and an optimal algorithm for enumerating all maximal block palindromes in a string.

Contribution

It defines block palindromes, explores their properties, and provides an efficient algorithm for enumerating all maximal block palindromes in a string.

Findings

Introduces the concept of block palindromes and maximal block palindromes.

Provides a compact representation of all block palindromes in a string.

Develops an optimal algorithm with linear time complexity relative to output size.

Abstract

We study a new generalization of palindromes and gapped palindromes called block palindromes. A block palindrome is a string that becomes a palindrome when identical substrings are replaced with a distinct character. We investigate several properties of block palindromes and in particular, study substrings of a string which are block palindromes. In so doing, we introduce the notion of a \emph{maximal block palindrome}, which leads to a compact representation of all block palindromes that occur in a string. We also propose an algorithm which enumerates all maximal block palindromes that appear in a given string $T$ in $O(|T| + \|\mathit{MBP}(T)\|)$ time, where $\|\mathit{MBP}(T)\|$ is the output size, which is optimal unless all the maximal block palindromes can be represented in a more compact way.