A Characterization of the Squares in a Fibonacci String

A ((nite) Fibonacci string F n is deened as follows: F 0 = b, F 1 = a; for every integer n 2, F n = F n?1 F n?2. For n 1, the length of F n is denoted by f n = jF n j. The innnite Fibonacci string F is the string which contains every F n , n 1, as a preex. Apart from their general theoretical importance, Fibonacci strings are often cited as worst case examples for algorithms which compute all the repetitions or all the \Abelian squares" in a given string. In this paper we provide a characterization of all the squares in F, hence in every preex F n ; this characterization naturally gives rise to a (f n) algorithm which speciies all the squares of F n in an appropriate encoding. This encoding is made possible by the fact that the squares of F n occur consecutively, in \runs", the number of which is (f n). By contrast, the known general algorithms for the computation of the repetitions in an arbitrary string require (f n log f n) time (and produce (f n log f n) outputs) when applied to a Fibonacci string F n .