Border Correlations of Partial Words

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 = A ∪ { }, A∗ denotes the set of all partial words over A. In this paper, we investigate the border correlation function β : A∗ → {a, b}∗ that specifies which conjugates (cyclic shifts) of a given partial word w of length n are bordered, that is, β(w) = c0c1 . . . cn−1 where ci = a or ci = b according to whether the ith cyclic shift σi (w) of w is unbordered or bordered. A partial word w is bordered if a proper prefix x1 of w is compatible with a proper suffix x2 of w, in which case any partial word x containing both x1 and x2 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 m0m1 . . . mn−1 where mi is the length of a shortest border of σi (w). Our results extend those of Harju and Nowotka.

• Type: Journal
• Year: 2010
• Where: MST
• Authors: Francine Blanchet-Sadri, E. Clader, O. Simpson
• Keywords: Border Correlation Function, Ith Cyclic Shift, MST 2010, Partial Word