On extended special factors of a word

Abstract

An extended special factor of a word x is a factor of x whose longest infix can be extended by at least two distinct letters to the left or to the right and still occur in x. It is called extended bispecial if it can be extended in both directions and still occur in x. Let ρ(n) be the maximum number of extended bispecial factors over all words of length n. Almirantis et al. have shown that 2n - 6≤ ρ(n)≤ 3n-4 [WABI 2017]. In this article, we show that there is no constant c <3 such that ρ(n)≤cn. We then exploit the connection between extended special factors and minimal absent words to construct a data structure for computing minimal absent words of a specific length in optimal time for integer alphabets generalising a result by Fujishige et al. [MFCS 2016]. As an application of our data structure, we show how to compare two words over an integer alphabet in optimal time improving on another result by Charalampopoulos et al. [Inf. Comput. 2018].