Repetitions in Sturmian strings

Franěk, František; Karaman, A.; Smyth, W. Franklin

doi:10.1016/s0304-3975(00)00063-3

Cited by 12 publications

(8 citation statements)

References 9 publications

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“…In [1] we showed how to recognize prefixes of infinite Sturmian strings in time proportional to their length, a result extended in [2] to complete two-pattern strings. In [3] we described an algorithm to compute all the repetitions and near repetitions in linear time for complete two-pattern strings, again extending the same result on prefixes of infinite Sturmian strings [1].…”

Section: Introductionmentioning

confidence: 99%

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Two-pattern strings II—frequency of occurrence and substring complexity

Franěk

Jiang

Smyth

2007

Journal of Discrete Algorithms

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The previous paper in this series introduced a class of infinite binary strings, called two-pattern strings, that constitute a significant generalization of, and include, the much-studied Sturmian strings. The class of two-pattern strings is a union of a sequence of increasing (with respect to inclusion) subclasses T λ of two-pattern strings of scope λ, λ = 1, 2, . . . . Prefixes of two-pattern strings are interesting from the algorithmic point of view (their recognition, generation, and computation of repetitions and nearrepetitions) and since they include prefixes of the Fibonacci and the Sturmian strings, they merit investigation of how many finite two-pattern strings of a given size there are among all binary strings of the same length. In this paper we first consider the frequency f λ (n) of occurrence of two-pattern strings of length n and scope λ among all strings of length n on {a, b}: we show that lim n→∞ f λ (n) = 0, but that for strings of lengths n 2λ, two-pattern strings of scope λ constitute more than one-quarter of all strings. Since the class of Sturmian strings is a subset of two-pattern strings of scope 1, it was natural to focus the study of the substring complexity of two-pattern strings to those of scope 1. Though preserving the aperiodicity of the Sturmian strings, the generalization to two-pattern strings greatly relaxes the constrained substring complexity (the number of distinct substrings of the same length) of the Sturmian strings. We derive upper and lower bounds on C 1 (k) (the number of distinct substring of length k) of two-pattern strings of scope 1, and we show that it can be considerably greater than that of a Sturmian string. In fact, we describe circumstances in which lim k→∞ (C 1 (k) − k) = ∞.

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Section: Introductionmentioning

confidence: 99%

“…In [3] we described an algorithm to compute all the repetitions and near repetitions in linear time for complete two-pattern strings, again extending the same result on prefixes of infinite Sturmian strings [1].…”

Section: Introductionmentioning

confidence: 99%

Two-pattern strings II—frequency of occurrence and substring complexity

Franěk

Jiang

Smyth

2007

Journal of Discrete Algorithms

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“…A run is a maximal repetition in a word so that it is not a proper factor of another repetition in the same word. [FKS00] studied the computation of repetitions in Sturmian words. Prior to that, [IMS97] presented a linear-time algorithm to compute the runs in Fibonacci words.…”

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confidence: 99%

Weak repetitions in Sturmian strings

Karaman

2003

Theoretical Computer Science

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“…We continue the work of [8], where it was shown how to compute the number of runs for block-complete Sturmian words (not all standard Sturmian words have this property) in time linear with respect to the size of the whole word (while our algorithm is linear with respect to the size of compressed representation). A similar approach as in [8] is used in this paper -a kind of a reduction sequence, however our reductions are different than those in [8] and correspond closely to the structure of the recurrences (directive sequences). Also our aim is different -derivation of a simple formula for ρ(w) and asymptotic behavior of ρ(w).…”

Section: Introductionmentioning

confidence: 82%

“…Essentially we use an idea of a reduction sequence introduced in [8]. The computation of runs in S(γ 0 , γ 1 , .…”

Section: Morphic Representations and The Numbers N γ (K)mentioning

confidence: 99%

The Number of Runs in Sturmian Words

Baturo

Piatkowski

Rytter

Implementation and Applications of Automata

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Abstract. Denote by S the class of standard Sturmian words. 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 w ∈ S, with respect to their recurrences (directive sequences). We show thatfor each w ∈ S, and there is an infinite sequence of strictly growing words w k ∈ S such that lim k→∞. The complete understanding of the function ρ for a large class S 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).

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Repetitions in Sturmian strings

Cited by 12 publications

References 9 publications

Two-pattern strings II—frequency of occurrence and substring complexity

Two-pattern strings II—frequency of occurrence and substring complexity

Weak repetitions in Sturmian strings

The Number of Runs in Sturmian Words

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