On minimal Sturmian partial words

Partial words, which are sequences that may have some undefined positions called holes, can be viewed as sequences over an extended alphabet A = A ∪ { }, where stands for a hole and matches (or is compatible with) every letter in A. The subword complexity of a partial word w, denoted by pw(n), is the number of distinct full words (those without holes) over the alphabet that are compatible with factors of length n of w. A function f : N → N is (k, h)-feasible if for each integer N ≥ 1, there exists a k-ary partial word w with h holes such that pw(n) = f(n) for all n, 1 ≤ n ≤ N. We show that when dealing with feasibility in the context of finite binary partial words, the only linear functions that need investigation are f(n) = n + 1 and f(n) = 2n. It turns out that both are (2, h)-feasible for all non-negative integers h. We classify all minimal partial words with h holes of order N with respect to f(n) = n + 1, called Sturmian, computing their lengths as well as their numbe...

• Type: Journal
• Year: 2011
• Where: DAM
• Authors: Francine Blanchet-Sadri, John Lensmire
• Keywords: DAM 2011, Minimal Partial Words, Partial Words, Sturmian Partial Word, Theoretical Computer Science