Border Correlations of Partial Words

Abstract

Partial words are finite sequences over a finite alphabet A that may contain a number of “do not know” symbols denoted by ⋄’s. Setting $A_{\diamond}=A\cup\lbrace{\diamond}\rbrace$ , A * denotes the set of all partial words over A. In this paper, we investigate the border correlation function $\beta:A_{\diamond}^{*}\rightarrow\lbrace a,b\rbrace^{*}$ that specifies which conjugates (cyclic shifts) of a given partial word w of length n are bordered, that is, β(w)=c 0 c 1…c n−1 where c i =a or c i =b according to whether the ith cyclic shift σ i (w) of w is unbordered or bordered. A partial word w is bordered if a proper prefix x 1 of w is compatible with a proper suffix x 2 of w, in which case any partial word x containing both x 1 and x 2 is called a border of w. In addition to β, we investigate an extension β′:A * →ℕ* that maps a partial word w of length n to m 0 m 1…m n−1 where m i is the length of a shortest border of σ i (w). Our results extend those of Harju and Nowotka.