On Maximal Repetitions in Words
FCT '99 Proceedings of the 12th International Symposium on Fundamentals of Computation Theory
European Journal of Combinatorics
Finding Maximal Repetitions in a Word in Linear Time
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Automatic Sequences: Theory, Applications, Generalizations
Automatic Sequences: Theory, Applications, Generalizations
The structure of subword graphs and suffix trees of Fibonacci words
Theoretical Computer Science - Implementation and application of automata
Towards a Solution to the "Runs" Conjecture
CPM '08 Proceedings of the 19th annual symposium on Combinatorial Pattern Matching
The Number of Runs in Sturmian Words
CIAA '08 Proceedings of the 13th international conference on Implementation and Applications of Automata
Compressed string-matching in standard Sturmian words
Theoretical Computer Science
Suffix automata and standard sturmian words
DLT'07 Proceedings of the 11th international conference on Developments in language theory
Sturmian and episturmian words: a survey of some recent results
CAI'07 Proceedings of the 2nd international conference on Algebraic informatics
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
Analysis of maximal repetitions in strings
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
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The standard Sturmian words (standard words in short) are extensively studied in combinatorics of words. They are complicated enough to have many interesting properties, and, at the same time, due to their recurrent structure, they are highly compressible. In this paper, we present compact formulas for the number of cubic runs in any standard word w (denoted by @r^(^3^)(w)). We show also that lim sup|w|-~@r^(^3^)(w)|w|=3@F+29@F+4~0.36924841, where @F=5+12 is the golden ratio, and present a sequence of strictly growing standard words achieving this limit. The exact asymptotic ratio here is irrational, contrary to the situation of squares and runs in the same class of words. Furthermore, we design an efficient algorithm for computing the number of cubic runs in standard words in linear time with respect to the size of a directive sequence, i.e., the compressed representation describing the word (recurrences). The explicit size of a word can be exponential with respect to this representation, and hence this is yet another example of a very fast computation on highly compressible texts.