String covering with optimal covers

In this paper we introduce the notion of an optimal cover. Let M denote the maximum number of positions in w covered by any repeating substring of w . Then a longest (shortest) optimal cover u is a longest (shortest) repeating substring of w that covers M positions. The advantage of this notion is that it is not only applicable to all strings, but also that it does not share the deficiencies of the existing definitions of covers. We show that both the longest and the shortest optimal covers for a given string w of length n can be computed easily and efficiently in O ( n log ⁡ n ) time and O ( n ) space. We further show that the data structures used to compute optimal covers also compute the α-partial covers introduced in [10].