Repetitions in Sturmian strings
Theoretical Computer Science
On Maximal Repetitions in Words
FCT '99 Proceedings of the 12th International Symposium on Fundamentals of Computation Theory
Finding Maximal Repetitions in a Word in Linear Time
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
The structure of subword graphs and suffix trees of Fibonacci words
Theoretical Computer Science - Implementation and application of automata
The number of runs in a string
Information and Computation
Towards a Solution to the "Runs" Conjecture
CPM '08 Proceedings of the 19th annual symposium on Combinatorial Pattern Matching
Suffix automata and standard sturmian words
DLT'07 Proceedings of the 11th international conference on Developments in language theory
Analysis of maximal repetitions in strings
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
LATA '09 Proceedings of the 3rd International Conference on Language and Automata Theory and Applications
On the maximal number of cubic runs in a string
LATA'10 Proceedings of the 4th international conference on Language and Automata Theory and Applications
The maximal number of cubic runs in a word
Journal of Computer and System Sciences
On the maximum number of cubic subwords in a word
European Journal of Combinatorics
The total run length of a word
Theoretical Computer Science
Computing the number of cubic runs in standard Sturmian words
Discrete Applied Mathematics
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Denote by the class of standard Sturmianwords. It is a class of highly compressible words extensively studied in combinatorics of words, including the well known Fibonacci words. The suffix automata for these words have a very particular structure. This implies a simple characterization (described in the paper by the Structural Lemma) of the periods of runs (maximal repetitions) in Sturmian words. Using this characterization we derive an explicit formula for the number ρ(w) of runs in words , with respect to their recurrences(directive sequences). We show that $\frac{\rho(w)}{|w|}\le \frac{4}{5} \textrm{\ for each\ }\ w\in {\cal S},$ and there is an infinite sequence of strictly growing words $w_k\in {\cal S}$ such that $\lim_{k\rightarrow \infty}\ \frac{\rho(w_k)}{|w_k|}\ =\ \frac{4}{5}$. The complete understanding of the function ρfor a large class of complicated words is a step towards better understanding of the structure of runs in words. We also show how to compute the number of runs in a standard Sturmian word in linear time with respect to the size of its compressed representation (recurrences describing the word). This is an example of a very fast computation on texts given implicitly in terms of a special grammar-based compressed representation (usually of logarithmic size with respect to the explicit text).