On the Right-Seed Array of a String

摘要

We consider the problem of finding the repetitive structure of a given fixed string y. A factor u of y is a cover of y, if every letter of y falls within some occurrence of u in y. A factor v of y is a seed of y, if it is a cover of a superstring of y. There exist linear-time algorithms for solving the minimal cover problem. The minimal seed problem is of much higher algorithmic difficulty, and no linear-time algorithm is known. In this article, we solve one of its variants - computing the minimal and maximal right-seed array of a given string. A right seed of y is the shortest suffix of y that it is a cover of a superstring of y. An integer array RS is the minimal right-seed (resp. maximal right-seed) array of y, if RS[i] is the minimal (resp. maximal) length of right seeds of y[0 .. I]. We present an O(nlogn) time algorithm that computes the minimal right-seed array of a given string, and a linear-time solution to compute the maximal right-seed array by detecting border-free prefixes of the given string.