Partial words and the critical factorization theorem revisited

  • Authors:

  • F. Blanchet-Sadri;Nathan D. Wetzler

  • Affiliations:

  • Department of Computer Science, University of North Carolina, P.O. Box 26170, Greensboro, NC 27402-6170, USA;Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, USA

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2007

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Abstract

In this paper, we consider one of the most fundamental results on the 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 with 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.