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∪{♢}A♢=A∪{♢}, where ♢♢ stands for a hole and matches (or is compatible   with) every letter in AA. The subword complexity   of a partial word ww, denoted by pw(n)pw(n), is the number of distinct full words (those without holes) over the alphabet that are compatible with factors of length nn of ww. A function f:N→Nf:N→N is (k,h)(k,h)-feasible   if for each integer N≥1N≥1, there exists a kk-ary partial word ww with hh holes such that pw(n)=f(n)pw(n)=f(n) for all nn such that 1≤n≤N1≤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+1f(n)=n+1 and f(n)=2nf(n)=2n. It turns out that both are (2,h)(2,h)-feasible for all non-negative integers hh. We classify all minimal partial words with hh holes of order NN with respect to f(n)=n+1f(n)=n+1, called Sturmian, computing their lengths as well as their numbers, except when h=0h=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 ai♢ajbalai♢ajbal, where i,j,li,j,l are integers satisfying some restrictions, that all minimal Sturmian partial words with two holes are one-periodic, and that up to complement, ♢(aN−1♢)h−1♢(aN−1♢)h−1 is the only minimal Sturmian partial word with h≥3h≥3 holes. Finally, we give upper bounds on the lengths of minimal partial words with respect to f(n)=2nf(n)=2n, showing them tight for h=0,1h=0,1 or 22.