Weighted ancestors in suffix trees

TL;DR

This paper presents a novel method to preprocess suffix trees in linear space, enabling constant-time weighted ancestor queries, significantly enhancing the efficiency of suffix tree applications.

Contribution

It introduces a new approach that achieves O(1) query time for weighted ancestors in suffix trees with linear space, solving a long-standing open problem.

Findings

Suffix trees can be preprocessed in O(n) space for constant-time queries.

The new method improves the efficiency of multiple suffix tree applications.

A periodicity-based insight is key to the new data structure design.

Abstract

The classical, ubiquitous, predecessor problem is to construct a data structure for a set of integers that supports fast predecessor queries. Its generalization to weighted trees, a.k.a. the weighted ancestor problem, has been extensively explored and successfully reduced to the predecessor problem. It is known that any solution for both problems with an input set from a polynomially bounded universe that preprocesses a weighted tree in O(n polylog(n)) space requires \Omega(loglogn) query time. Perhaps the most important and frequent application of the weighted ancestors problem is for suffix trees. It has been a long-standing open question whether the weighted ancestors problem has better bounds for suffix trees. We answer this question positively: we show that a suffix tree built for a text w[1..n] can be preprocessed using O(n) extra space, so that queries can be answered in O(1) time. Thus we improve the running times of several applications. Our improvement is based on a number of data structure tools and a periodicity-based insight into the combinatorial structure of a suffix tree.