Online Computation of Palindromes and Suffix Trees on Tries

TL;DR

This paper introduces the first sub-quadratic online algorithms for enumerating palindromes in dynamic tries, supporting leaf insertions and deletions, and also presents online suffix tree and EERTREE construction methods.

Contribution

It develops novel online algorithms for palindrome enumeration in dynamic tries, improving efficiency and supporting dynamic updates, along with online suffix tree and EERTREE construction.

Findings

Algorithms run in $O(N ext{min}( ext{log} h, ext{σ}))$ time for maximal palindromes.

Supports leaf insertions and deletions with various time-space trade-offs.

Provides online construction algorithms for suffix trees and EERTREE of tries.

Abstract

We consider the problems of computing maximal palindromes and distinct palindromes in a trie. A trie is a natural generalization of a string, which can be seen as a single-path tree. There is a linear-time offline algorithm to compute maximal palindromes and distinct palindromes in a given (static) trie whose edge-labels are drawn from a linearly-sortable alphabet [Mieno et al., ISAAC 2022]. In this paper, we tackle problems of palindrome enumeration on dynamic tries which support leaf additions and leaf deletions. We propose the first sub-quadratic algorithms to enumerate palindromes in a dynamic trie. For maximal palindromes, we propose an algorithm that runs in $O(N \min(\log h, \sigma))$ time and uses $O(N)$ space, where $N$ is the maximum number of edges in the trie, $\sigma$ is the size of the alphabet, and $h$ is the height of the trie. For distinct palindromes, we develop several online algorithms based on different algorithmic frameworks, including approaches using the EERTREE (a.k.a. palindromic tree) and the suffix tree of a trie. These algorithms support leaf insertions and deletions in the trie and achieve different time and space trade-offs. Furthermore, as a by-product, we present online algorithms to construct the suffix tree and the EERTREE of the input trie, which is of independent interest.