Binary de bruijn partial words with one hole

Authors:
Francine Blanchet-Sadri;Jarett Schwartz;Slater Stich;Benjamin J. Wyatt
Affiliations:
Department of Computer Science, University of North Carolina, Greensboro, NC;Department of Computer Science, Princeton University, Princeton, NJ;Department of Mathematics, Princeton University, Princeton, NJ;Department of Computer Science, University of North Carolina, Greensboro, NC
Venue:
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
Year:
2010

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Abstract

In this paper, we investigate partial words, or finite sequences that may have some undefined positions called holes, of maximum subword complexity The subword complexity function of a partial word w over a given alphabet of size k assigns to each positive integer n, the number pw(n) of distinct full words over the alphabet that are compatible with factors of length n of w For positive integers n, h and k, we introduce the concept of a de Bruijn partial word of order n with h holes over an alphabet A of size k, as being a partial word w with h holes over A of minimal length with the property that $p_w(n)=k^n$ We are concerned with the following three questions: (1) What is the length of k-ary de Bruijn partial words of order n with h holes? (2) What is an efficient method for generating such partial words? (3) How many such partial words are there?