An Improved Bound for an Extension of Fine and Wilf’s Theorem and Its Optimality

  • Authors:

  • Lila Kari;Shinnosuke Seki

  • Affiliations:

  • (Correspd.) Department of Computer Science, University of Western Ontario, London, Ontario, N6A 5B7, Canada. {lila, sseki} @csd.uwo.ca;Department of Computer Science, University of Western Ontario, London, Ontario, N6A 5B7, Canada. {lila, sseki} @csd.uwo.ca

  • Venue:
  • Fundamenta Informaticae
  • Year:
  • 2010

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Abstract

Considering two DNA molecules which are Watson-Crick (WK) complementary to each other “equivalent” with respect to the information they encode enables us to extend the classical notions of repetition, period, and power. WK-complementarity has been modelled mathematically by an antimorphic involution &thgr;, i.e., a function &thgr; such that &thgr;(xy) = &thgr;(y)&thgr;(x) for any x, y ∞ &Sgr;*, and &thgr; 2 is the identity. The WK-complementarity being thus modelled, any word which is a repetition of u and &thgr;(u) such as uu, u&thgr;(u)u, and u&thgr;(u)&thgr;(u)&thgr;(u) can be regarded repetitive in this sense, and hence, called a &thgr;-power of u. Taking the notion of &thgr;-power into account, the Fine and Wilf’s theorem was extended as “given an antimorphic involution &thgr; and words u, v, if a &thgr;-power of u and a &thgr;-power of v have a common prefix of length at least b(|u|, |v|) = 2|u| + |v| - gcd(|u|, |v|), then u and v are &thgr;-powers of a same word.” In this paper, we obtain an improved bound b′(|u|, |v|) = b(|u|, |v|) - [gcd(|u|, |v|)/2]. Then we show all the cases when this bound is optimal by providing all the pairs of words (u, v) such that they are not &thgr;-powers of a same word, but one can construct a &thgr;-power of u and a &thgr;-power of v whose maximal common prefix is of length equal to b′(|u|, |v|) − 1. Furthermore, we characterize such words in terms of Sturmian words.