Constructing partial words with subword complexities not achievable by full words

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, pw(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 pw(n) = Θ(n ) for any real number α > 1. In addition, these partial words have the property that there exist infinitely many non-negative integers m satisfying pw(m + 1) − pw(m) ≥ m. 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. ∗This material is based upon work supported by the National Science Foundation under Grant No. DMS–0754154. The Department of Defense is also gratefully acknowledged. We thank Emily Allen, as well as the referees of preliminary versions of this paper for their very valuable comments and suggestions. Department of Computer Science, University of North Carolina, P.O. Box 26170, Greensboro, NC 27402–6170, USA, [email protected] Department of Computer Science, University of Colorado at Boulder, 430 UCB, Boulder, CO 80309–0430, USA Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544–1000, USA Department of Computer Science, Princeton University, 35 Olden Street, Princeton, NJ 08540–5233, USA