A new approach to the periodicity lemma on strings with holes

  • Authors:

  • W. F. Smyth;Shu Wang

  • Affiliations:

  • Algorithms Research Group, Department of Computing & Software, McMaster University, Hamilton, Ontario, Canada, L8S 4K1 and Digital Ecosystems & Business Intelligence Institute, Curtin University o ...;Algorithms Research Group, Department of Computing & Software, McMaster University, Hamilton, Ontario, Canada, L8S 4K1

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2009

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Abstract

We first give an elementary proof of the periodicity lemma for strings containing one hole (variously called a ''wild card'', a ''don't-care'' or an ''indeterminate letter'' in the literature). The proof is modelled on Euclid's algorithm for the greatest common divisor and is simpler than the original proof given in [J. Berstel, L. Boasson, Partial words and a theorem of Fine and Wilf, Theoret. Comput. Sci. 218 (1999) 135-141]. We then study the two-hole case, where our result agrees with the one given in [F. Blanchet-Sadri, Robert A. Hegstrom, Partial words and a theorem of Fine and Wilf revisited, Theoret. Comput. Sci. 270 (1-2) (2002) 401-419] but is more easily proved and enables us to identify a maximum-length prefix or suffix of the string to which the periodicity lemma does apply. Finally, we extend our result to three or more holes using elementary methods, and state a version of the periodicity lemma that applies to all strings with or without holes. We describe an algorithm that, given the locations of the holes in a string, computes maximum-length substrings to which the periodicity lemma applies, in time proportional to the number of holes. Our approach is quite different from that used by Blanchet-Sadri and Hegstrom, and also simpler.