Almost succinct representation of maximal palindromes

TL;DR

This paper introduces a space-efficient data structure for representing all maximal palindromes in a string, enabling constant-time length retrieval and improving space efficiency for palindrome-related problems.

Contribution

It proposes a novel compact $(3(1+\epsilon)n + o(n))$-bit representation of maximal palindromes with $O(n)$ construction time, enhancing space efficiency and query speed.

Findings

Enables $O(1)$-time retrieval of palindrome lengths at any position.

Constructed in $O(n)$ time from the input string.

Supports a data structure to find the longest palindrome in any substring in $O(\log n)$ time.

Abstract

Palindromes are strings that read the same forward and backward. The computation of palindromic structures within strings is a fundamental problem in string algorithms, being motivated by potential applications in formal language theory and bioinformatics. Although the number of palindromic factors in a string of length $n$ can be quadratic, they can be implicitly represented in $O(n \log n)$ bits of space by storing the lengths of all maximal palindromes in an integer array, which can be computed in $O(n)$ time [Manacher, 1975]. In this paper, for any positive constant $\epsilon < 1$, we propose a novel $(3(1+\epsilon)n + o(n))$-bit representation of all maximal palindromes in a string, which enables $O(1)$-time retrieval of the length of the maximal palindrome centered at any given position. The data structure can be constructed in $O(n)$ time from the input string of length $n$. Since Manacher's algorithm and the notion of maximal palindromes are widely utilized for solving numerous problems involving palindromic structures, our compact representation will accelerate the development of more space-efficient solutions to such problems. Indeed, as the first application of our compact representation of maximal palindromes, we present a data structure of size $O(n)$ bits that can compute the longest palindrome appearing in any given factor of a string of length $n$ in $O(\log n)$ time.