Constructing partial words with subword complexities not achievable by full words

  • Authors:

  • F. Blanchet-Sadri;Aleksandar Chakarov;Lucas Manuelli;Jarett Schwartz;Slater Stich

  • Affiliations:

  • Department of Computer Science, University of North Carolina, P.O. Box 26170, Greensboro, NC 274026170, USA;Department of Computer Science, University of Colorado at Boulder, 430 UCB, Boulder, CO 803090430, USA;Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 085441000, USA;Department of Computer Science, Princeton University, 35 Olden Street, Princeton, NJ 085405233, USA;Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 085441000, USA

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2012

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Abstract

Partial words are sequences over a finite alphabet that may contain wildcard symbols, called holes, which match, or are compatible with, all letters in the alphabet ((full) words are just partial words without holes). The subword complexity function of a partial word w over a finite alphabet A assigns to each positive integer, n, the number, p"w(n), of distinct full words over A that are compatible with factors of length n of w. In this paper, with the help of our so-called hole functions, we construct infinite partial words w such that p"w(n)=@Q(n^@a) for any real number @a1. In addition, these partial words have the property that there exist infinitely many non-negative integers m satisfying p"w(m+1)-p"w(m)=m^@a. Combining these results with earlier ones on full words, we show that this represents a class of subword complexity functions not achievable by full words. We also construct infinite partial words with intermediate subword complexity, that is, between polynomial and exponential.