Partial words and the critical factorization theorem revisited

In this paper, we consider one of the most fundamental results on periodicity of words, namely the critical factorization theorem. Given a word w and nonempty words u, v satisfying w = uv, the minimal local period associated to the factorization (u, v) is the length of the shortest square at position |u| − 1. The critical factorization theorem shows that for any word, there is always a factorization whose minimal local period is equal to the minimal period (or global period) of the word. Crochemore and Perrin presented a linear time algorithm (in the length of the word) that finds a critical factorization from the computation of the maximal suffixes of the word with respect to two total orderings on words: the lexicographic ordering related to a fixed total ordering on the alphabet, and the lexicographic ordering obtained by reversing the order of letters in the alphabet. Here, by refining Crochemore and Perrin’s algorithm, we give a version of the critical factorization theorem for partial words (such sequences may contain “do not know” symbols or “holes”). Our proof provides an efficient algorithm which computes a critical factorization when one exists. Our results extend those of Blanchet-Sadri and Duncan for partial words with one hole. A World Wide Web server interface at http://www.uncg.edu/mat/research/cft2/ has been established for automated use of the program.