Efficient Seeds Computation Revisited

TL;DR

This paper develops efficient algorithms for computing various seed-related measures in strings, including shortest and longest seed arrays, and improves understanding of the computational complexity of seed problems.

Contribution

It introduces linear time algorithms for key seed computations and corrects previous inaccuracies in shortest seed algorithms.

Findings

Linear time algorithms for shortest and longest seed arrays

An $O(n^2)$ algorithm for seed array computation of all prefixes

An $O(n ext{log}(n/m))$ algorithm for seed length verification

Abstract

The notion of the cover is a generalization of a period of a string, and there are linear time algorithms for finding the shortest cover. The seed is a more complicated generalization of periodicity, it is a cover of a superstring of a given string, and the shortest seed problem is of much higher algorithmic difficulty. The problem is not well understood, no linear time algorithm is known. In the paper we give linear time algorithms for some of its versions --- computing shortest left-seed array, longest left-seed array and checking for seeds of a given length. The algorithm for the last problem is used to compute the seed array of a string (i.e., the shortest seeds for all the prefixes of the string) in $O(n^2)$ time. We describe also a simpler alternative algorithm computing efficiently the shortest seeds. As a by-product we obtain an $O(n\log{(n/m)})$ time algorithm checking if the shortest seed has length at least $m$ and finding the corresponding seed. We also correct some important details missing in the previously known shortest-seed algorithm (Iliopoulos et al., 1996).