Partial words and a theorem of Fine and Wilf
Theoretical Computer Science
Automatic Sequences: Theory, Applications, Generalizations
Automatic Sequences: Theory, Applications, Generalizations
Colored de Bruijn Graphs and the Genome Halving Problem
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
Algorithmic Combinatorics on Partial Words (Discrete Mathematics and Its Applications)
Algorithmic Combinatorics on Partial Words (Discrete Mathematics and Its Applications)
Binary de bruijn partial words with one hole
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
Hard counting problems for partial words
LATA'10 Proceedings of the 4th international conference on Language and Automata Theory and Applications
Deciding representability of sets of words of equal length
Theoretical Computer Science
Counting minimal semi-Sturmian words
Discrete Applied Mathematics
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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 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=1, there exists a k-ary partial word w with h holes such that p"w(n)=f(n) for all n such that 1@?n@?N. We show that when dealing with feasibility in the context of finite binary partial words, the only affine 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 numbers, except when h=0 in which case we describe an algorithm that generates all minimal Sturmian full words. We show that up to reversal and complement, any minimal Sturmian partial word with one hole is of the form a^i@?a^jba^l, where i,j,l are integers satisfying some restrictions, that all minimal Sturmian partial words with two holes are one-periodic, and that up to complement, @?(a^N^-^1@?)^h^-^1 is the only minimal Sturmian partial word with h=3 holes. Finally, we give upper bounds on the lengths of minimal partial words with respect to f(n)=2n, showing them tight for h=0,1 or 2.