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Repetitions in Sturmian strings

Franěk, F., Karaman, A. and Smyth, W.F. (1998) Repetitions in Sturmian strings. In: 9th Australasian Workshop on Combinatorial Algorithms (AWOCA '98), 27 - 30 July 1998, Perth, Western Australia



In this paper we apply a simple representation of Sturmian strings, which we call a "reduction sequence", to three algorithms. The first algorithm accepts as input a given finite string x and determines in time O(|x|) whether or not x is Sturmian. The second algorithm is a modification of the first that, in the case that x is Sturmian, outputs a reduction sequence for a superstring u of x that is a prefix of an infinite Sturmian string. The third algorithm uses the reduction sequence of u to compute all the repetitions in u in time Θ(|u|), thus extending a recent result for Fibonacci strings. The third algorithm is also based on a characterization of the repetitions in a Sturmian string that describes them compactly in terms of "runs". Finally, for every integer r ≥ 4, we show how to construct an infinite Sturmian string that contains maximal repetitions of exponents 2, 3, ..., r-1, but none of exponent r.

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