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

摘要

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 θ, i.e., a function 8 such that θ(xy) = θ(y)θ(x) for any x, y ∈ ∑~*, and θ~2 is the identity. The WK-complementarity being thus modelled, any word which is a repetition of u and θ(u) such as uu, uθ(u)u, and uθ(u)θ(u)θ(u) can be regarded repetitive in this sense, and hence, called a θ-power of u. Taking the notion of θ-power into account, the Fine and Wilf's theorem was extended as "given an antimorphic involution θ and words u, v, if a θ-power of u and a θ-power of v have a common prefix of length at least b(|u|, |v|) = 2|u| + |u| - gcd(|u|, |v|), then u and v are θ-powers of a same word." In this paper, we obtain an improved bound b'(|u|, |v|) = 6(|u|, |v|) - [gcd(|u|, |w|)/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 0-powers of a same word, but one can construct a θ-power of u and a θ-power of v whose maximal common prefix is of length equal to σ'(|w|, |v|) - 1. Furthermore, we characterize such words in terms of Sturmian words.