Handbook of formal languages, vol. 1
On the combinatorics of finite words
Theoretical Computer Science
Partial words and a theorem of Fine and Wilf
Theoretical Computer Science
Journal of Combinatorial Theory Series A
Partial words and a theorem of Fine and Wilf revisited
Theoretical Computer Science
Theoretical Computer Science
Partial words and the critical factorization theorem
Journal of Combinatorial Theory Series A
Codes, orderings, and partial words
Theoretical Computer Science
Discrete Applied Mathematics
Testing primitivity on partial words
Discrete Applied Mathematics
Partial words and the critical factorization theorem revisited
Theoretical Computer Science
Theoretical Computer Science
Periodicity properties on partial words
Information and Computation
Discrete Applied Mathematics
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
Combinatorics on partial word correlations
Journal of Combinatorial Theory Series A
Periodicity algorithms for partial words
MFCS'11 Proceedings of the 36th international conference on Mathematical foundations of computer science
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The study of the combinatorial properties of strings of symbols from a finite alphabet (also referred to as words) is profoundly connected to numerous fields such as biology, computer science, mathematics, and physics. Research in combinatorics on words goes back roughly a century. There is a renewed interest in combinatorics on words as a result of emerging new application areas such as molecular biology. Partial words were recently introduced in this context. The motivation behind the notion of a partial word is the comparison of genes (or proteins). Alignment of two genes (or two proteins) can be viewed as a construction of partial words that are said to be compatible. While a word can be described by a total function, a partial word can be described by a partial function. More precisely, a partial word of length n over a finite alphabet A is a partial function from {1,...,n} into A. Elements of {1,...,n} without an image are called holes. A word is just a partial word without holes. The notion of period of a word is central in combinatorics on words. In the case of partial words, there arc two notions: one is that of period, the other is that of local period. This paper extends to partial words with one hole the well known result of Guibas and Odlyzko which states that for every word u, there exists a word v of same length as u over the alphabet {0, 1} such that the set of all periods of u coincides with the set of all periods of v. Our result states that for every partial word u with one hole, there exists a partial word v of same length as u with at most one hole over the alphabet {0, 1} such that the set of all periods of u coincides with the set of all periods of v and the set of all local periods of u coincides with the set of all local periods of v. To prove our result, we use the technique of Halava, Harju and Ilie which they used to characterize constructively the set of periods of a given word. As a consequence of our constructive proof, we obtain a linear time algorithm which, given a partial word with one hole, computes a partial word with at most one hole over the alphabet {0, 1} with the same length and the same sets of periods and local periods. A World Wide Web server interface at http://www.uncg.edu/mat/AlgBin/ has been established for automated use of the program.