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_diamond=A cup {diamond}$where {diamond} stands for a hole and matches (or is compatible with every letter in A. The subword complexity of a partial word w, denoted by p_w(n), is the number of distinct full words (those without holes) over the alphabet that are compatible with factors of length n of w. A function f: N -> N is (k,h)-feasible if for each integer N geq 1, there exists a k-ary partial word w with h holes such that p_w(n) = f(n) for all n, 1 = 3$holes. Finally, we give upper bounds on the lengths of minimal partial words with respect to f(n)=2n$ which are tight for h=0, 1 or 2

Counting Minimal Semi-Sturmian Words

A finite Sturmian word w is a balanced word over the binary alphabet {a,b}, that is, for all subwords u andv of w of equal length, ||u|a-|v|a|=1, where |u|a and |v|a denote the number of occurrences of the lettera in u and v, respectively. There are several other characterizations, some leading to efficient algorithms for testing whether a finite word is Sturmian. These algorithms find important applications in areas such as pattern recognition, image processing, and computer graphics. Recently, Blanchet-Sadri and Lensmire considered finite semi-Sturmian words of minimal length and provided an algorithm for generating all of them using techniques from graph theory. In this paper, we exploit their approach in order to count the number of minimal semi-Sturmian words. We also present some other results that come from applying this graph theoretical framework to subword complexity

De bruijn partial words

In a kn-complex word over an alphabet Σ of size k each of the kn words of length n appear as a subword at least once. Such a word is said to have maximum subword complexity. De Bruijn sequences of order n over Σ are the shortest words of maximum subword complexity and are well known to have length kn+n-1. They are efficiently constructed by finding Eulerian cycles in so-called de Bruijn graphs. In this thesis, we investigate partial words, or sequences with wildcard symbols or hole symbols, 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 h, k and n, a de Bruijn partial word of order n with h holes over an alphabet Σ of size k is a partial word w with h holes over Σ of minimal length with the property that pw(n)=kn. In some cases, they are efficiently constructed by finding Eulerian paths in modified de Bruijn graphs. 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