A Linear Time Algorithm for Seeds Computation

TL;DR

This paper introduces a linear-time algorithm to compute a compact, linear-size representation of all seeds in a word, solving an open problem and improving efficiency over previous methods by leveraging combinatorial relations and simplifying the approach.

Contribution

It presents the first linear-time algorithm for seeds computation and a simpler representation that avoids using the reversed word, advancing the state of the art.

Findings

Linear-time algorithm for seeds computation

Compact representation of all seeds in a word

Improves upon previous O(n log n) algorithm

Abstract

A seed in a word is a relaxed version of a period in which the occurrences of the repeating subword may overlap. We show a linear-time algorithm computing a linear-size representation of all the seeds of a word (the number of seeds might be quadratic). In particular, one can easily derive the shortest seed and the number of seeds from our representation. Thus, we solve an open problem stated in the survey by Smyth (2000) and improve upon a previous O(n log n) algorithm by Iliopoulos, Moore, and Park (1996). Our approach is based on combinatorial relations between seeds and subword complexity (used here for the first time in context of seeds). In the previous papers, the compact representation of seeds consisted of two independent parts operating on the suffix tree of the word and the suffix tree of the reverse of the word, respectively. Our second contribution is a simpler representation of all seeds which avoids dealing with the reversed word. A preliminary version of this work, with a much more complex algorithm constructing the earlier representation of seeds, was presented at the 23rd Annual ACM-SIAM Symposium of Discrete Algorithms (SODA 2012).