Logo image
Back
A linear algorithm for computing all the squares of aFibonacci string
Conference paper   Open access

A linear algorithm for computing all the squares of aFibonacci string

C.S. Iliopoulos, D. Moore and W.F. Smyth
Computing: The Australasian Theory Symposium (CATS '96) (Melbourne, Australia, 29/01/1996–30/01/1996)
1996
pdf
cats-1.pdfDownloadView
Open Access

Abstract

A (finite) Fibonacci string $F_n$ is defined as follows: $F_0 = b$, $F_1 = a$; for every integer $n \ge 2$, $F_n = F_{n-1}F_{n- 2}$. For $n \ge 1$, the length of $F_n$ is denoted by $f_n = |F_n|$, while it is convenient to define $f_0 \equiv 0$. The infinite Fibonacci string $F$ is the string which contains every $F_n$, $n \ge 1$, as a prefix. 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 prefix $F_n$; this characterization naturally gives rise to a $\Theta(f_n)$ algorithm which specifies 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 $\Theta(f_n)$. By contrast, the known general algorithms for the computation of the repetitions in an arbitrary string require $\Theta(f_n\log f_n)$ time (and produce $\Theta(f_n\log f_n)$ outputs) when applied to a Fibonacci string $F_n$.

Details

Metrics

81 File views/ downloads
75 Record Views
Logo image