Online Computation of Palindromes and Suffix Trees on Tries

2026-01-23

AI Summary

This work addresses the problem of efficiently enumerating maximal and distinct palindromic substrings online in a dynamic trie that supports leaf insertions and deletions. We propose the first subquadratic-time dynamic algorithm for this task, which jointly maintains a suffix tree and an EERTREE (palindromic tree) of the trie to enable efficient updates and queries of palindromic structures. Our main contributions include the first subquadratic algorithm for palindrome enumeration on a dynamic trie, the design of multiple online frameworks supporting dynamic updates with both theoretical efficiency and practical utility, and novel online construction methods for the trie’s suffix tree and EERTREE. The algorithm achieves $O(N \cdot \min(\log h, \sigma))$ time and $O(N)$ space for maximal palindrome enumeration, while distinct palindrome enumeration also enjoys rigorous efficiency guarantees.

Technology Category

• Search and Optimization: Distributed Search
• Machine Learning: Online Learning & Bandits
• Data Mining & Knowledge Management: Intelligent Query Processing

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• palindromes
• trie
• online computation
• dynamic data structures
• suffix tree

Innovation

• dynamic trie
• online palindrome enumeration
• EERTREE
• suffix tree of trie
• sub-quadratic algorithm