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Occurrence and Lexicographic Properties of Standard Sturmian Words

Standard Sturmian words form an interesting class of structurally highly compressible strings. represented by simple recurrences. We show that simplicity of representation implies new algorithmic results: the compressed string matching for such words can be solved by very special algorithms and the output is simply characterized. Each standard Sturmian word w has Occurrence Shift Property: the set of occurrences of any subword v in w is an arithmetic progression (modulo the length of the word). For texts given by grammar-based compressed representation of size n the best known algorithms for searching a pattern of size m work in superquadratic time, see [4, 10], while in case of standard Sturmian texts our algorithm works in linear time. We consider also an unexpected relation of lexicography of cyclic shifts of standard Sturmian words to numeration systems called here standard systems (corresponding to q-systems in [5]). If we have two cyclic shifts by i and j of a same standard Sturmian word w, then we describe how the lexicographic order between these two shifts is immediately visible from the representations of i, j in our numeration system. This shows new relations between lexicographic and number-theoretic properties of standard Sturmian words and generalizes similar results related to Fibonacci words and Fibonacci number system, see [1].