Usefulness of Directed Acyclic Subword Graphs in Problems Related to Standard Sturmian Words

Baturo, Paweł; Piatkowski, Marcin; Rytter, Wojciech

doi:10.1142/s0129054109007017

Cited by 6 publications

(11 citation statements)

References 7 publications

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“…Finally, we have computed the asymptotic limit of the ratio of the number of cubic runs to the length of the word. In contrast to the maximal repetitions ratio (0.8; see [2]) and the distinct squares ratio (0.9; see [3]), in this case the limit is an irrational number:…”

Section: Final Remarksmentioning

confidence: 90%

“…The following lemma is a consequence of the very special structure of subword graphs (especially their compacted versions) of those words. See [3,25] for more information. To prove the above lemma, it is sufficient to show that no factor of the word Sw(γ 0 , .…”

Section: Formulas For the Number Of Cubic Runsmentioning

confidence: 99%

“…Denote by ρ (3) S (w), ρ (3) M (w), and ρ (3) L (w) the number of short, medium, and large cubic runs in the word w, respectively. We will consider each type separately.…”

Section: Formulas For the Number Of Cubic Runsmentioning

confidence: 99%

“…The number of medium cubic runs in the word w corresponds to the value of γ 2 . It is easy to see that γ 2 must be at most 2, otherwise the length of the word increases significantly and the value ρ (3) M (w) does not change. For γ 2 = 1 we have, by Eqs.…”

Section: Asymptotic Behavior Of the Maximal Number Of Cubic Runsmentioning

confidence: 99%

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Computing the number of cubic runs in standard Sturmian words

Pitkowski

Rytter

2014

Discrete Applied Mathematics

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a b s t r a c tThe 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 ρ (3) (w)). We show also thatis 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.

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Section: Final Remarksmentioning

confidence: 90%

Section: Formulas For the Number Of Cubic Runsmentioning

confidence: 99%

“…Denote by ρ (3) S (w), ρ (3) M (w), and ρ (3) L (w) the number of short, medium, and large cubic runs in the word w, respectively. We will consider each type separately.…”

Section: Formulas For the Number Of Cubic Runsmentioning

confidence: 99%

Section: Asymptotic Behavior Of the Maximal Number Of Cubic Runsmentioning

confidence: 99%

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Computing the number of cubic runs in standard Sturmian words

Pitkowski

Rytter

2014

Discrete Applied Mathematics

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“…For some lexicographic properties and structure of repetitions of standard Sturmian words see [5,3,1,4].…”

Section: Examplementioning

confidence: 99%

Asymptotic Behaviour of the Maximal Number of Squares in Standard Sturmian Words

Piatkowski

Rytter

2012

Int. J. Found. Comput. Sci.

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Denote by sq(w) the number of distinct squares in a string w and let S be the class of standard Sturmian words. They are generalizations of Fibonacci words and are important in combinatorics on words. For Fibonacci words the asymptotic behaviour of the number of runs and the number of squares is the same. We show that for Sturmian words the situation is quite different. The tight bound 8 10 |w| for the number of runs was given in [3]. In this paper we show that the tight bound for the maximal number of squares is 9 10 |w|. We use the results of [11], where exact (but not closed) complicated formulas were given for sq(w) for w ∈ S. We show that for all w ∈ S we have sq(w) ≤ 9 10 |w| and there is an infinite sequence of words w k ∈ S such that lim k→∞ |w k | = ∞ and lim k→∞ sq(w k ) |w k | = 9 10 . Surprisingly the maximal number of squares is reached by the words with recurrences of length only 5. This contrasts with the situation of Fibonacci words, though standard Sturmian words are natural extension of Fibonacci words. If this length drops to 4, the asymptotic behaviour of the maximal number of squares falls down significantly below 9 10 |w|. The structure of Sturmian words rich in squares has been discovered by us experimentally and verified theoretically. The upper bound is much harder, its proof is not a matter of simple calculations. The summation formulas for the number of squares are complicated, no closed formula is known. Some nontrivial reductions were necessary.

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