Computing palindromes on a trie in linear time

TL;DR

This paper presents an efficient linear-time algorithm for computing all distinct and maximal palindromes in a trie, significantly improving previous methods and enabling fast palindrome queries in trie-structured data.

Contribution

The authors develop a novel linear-time algorithm for palindrome computation on tries, surpassing prior $O(n \\log h)$ solutions, and introduce a space-efficient data structure for rapid palindrome reporting.

Findings

All distinct and maximal palindromes can be computed in $O(n)$ time.

The new algorithms improve over previous $O(n \\log h)$-time methods.

The data structure allows output-optimal palindrome reporting in trie-based queries.

Abstract

A trie $\mathcal{T}$ is a rooted tree such that each edge is labeled by a single character from the alphabet, and the labels of out-going edges from the same node are mutually distinct. Given a trie $\mathcal{T}$ with $n$ edges, we show how to compute all distinct palindromes and all maximal palindromes on $\mathcal{T}$ in $O(n)$ time, in the case of integer alphabets of size polynomial in $n$. This improves the state-of-the-art $O(n \log h)$-time algorithms by Funakoshi et al. [PCS 2019], where $h$ is the height of $\mathcal{T}$. Using our new algorithms, the eertree with suffix links for a given trie $\mathcal{T}$ can readily be obtained in $O(n)$ time. Further, our trie-based $O(n)$-space data structure allows us to report all distinct palindromes and maximal palindromes in a query string represented in the trie $\mathcal{T}$, in output optimal time. This is an improvement over an existing (na\"ive) solution that precomputes and stores all distinct palindromes and maximal palindromes for each and every string in the trie $\mathcal{T}$ separately, using a total $O(n^2)$ preprocessing time and space, and reports them in output optimal time upon query.