Local periods and binary partial words: an algorithm

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 are 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 ∗This material is based upon work supported by the National Science Foundation under Grants CCR9700228 and CCR–0207673. A Research Assignment from the University of North Carolina at Greensboro is gratefully acknowledged. I thank Phuongchi Thi Le for very valuable comments and suggestions. She received a research assistantship from the University of North Carolina at Greensboro to work with me on this project.