Double-Ended Palindromic Trees in Linear Time

TL;DR

This paper introduces a dynamic double-ended eertree data structure that efficiently manages palindromic substrings with support for online operations, leveraging a novel concept called surfaces for linear-time performance.

Contribution

It presents a new double-ended eertree supporting online operations and introduces the concept of surfaces to achieve linear-time complexity.

Findings

Supports online double-ended operations on palindromic substrings.

Achieves linear-time complexity for the data structure.

Introduces the concept of surfaces for analyzing palindromic substring occurrences.

Abstract

The palindromic tree (a.k.a. eertree) is a data structure that provides access to all palindromic substrings of a string. In this paper, we propose a dynamic version of eertree, called double-ended eertree, which supports online operations on the stored string, including double-ended queue operations, counting distinct palindromic substrings, and finding the longest palindromic prefix/suffix. At the heart of our construction, we identify a new class of substring occurrences, called surfaces, that are palindromic substring occurrences that are neither prefixes nor suffixes of any other palindromic substring occurrences, which is of independent interest. Surfaces characterize the link structure of all palindromic substrings in the eertree, thereby allowing a linear-time implementation of double-ended eertrees through a linear-time maintenance of surfaces.