Compressed string-matching in standard Sturmian words

We present a simple algorithm which for an explicitly given input string pat (a pattern) and a standard Sturmian word x described by the recurrences of size n computes, in time O(|pat|+n), the set of all occurrences of pat in x as a single arithmetic progression (modulo the length of x). The algorithm can be extended to the case when some letters of the pattern are replaced by a don’t care symbol. In this case the set of all occurrences does not need to be a single arithmetic progression and our algorithm produces linearly many (with respect to the size of pat) arithmetic progressions. It is an example of fast computations for the input given in a compressed form. In our special case the length of the standard Sturmian word x is usually exponential with respect to the size of the input.