Abelian periods, partial words, and an extension of a theorem
          of Fine and Wilf

Blanchet-Sadri, Francine; Simmons, Sean; Tebbe, Amelia; Veprauskas, Amy

doi:10.1051/ita/2013034

Cited by 10 publications

(14 citation statements)

References 21 publications

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“…Other known extensions of the periodicity lemma include a variant with three [8] and an arbitrary number of specified periods [13,24], the so-called new periodicity lemma [1,10], a periodicity lemma for repetitions with morphisms [17], extensions into abelian [9] and k-abelian [14] periodicity, into abelian periodicity for partial words [7], into bidimensional words [18], and other variations [12,19].…”

Section: Previous Resultsmentioning

confidence: 99%

“…• each partial word of length at least 12 with one hole and periods 5, 7 has also period 1 = gcd (5,7),…”

Section: Introductionmentioning

confidence: 99%

“…• H(11, 5, 7) ≥ 1: due to the classic periodicity lemma, every solid word of length 11 with periods 5 and 7 has period 1 = gcd (5,7), and • H(11, 5, 7) ≤ 1: ababaababa♦ is non-unary, has one hole and periods 5, 7. We have H(12, 5, 7) ≤ H(11, 5, 7) + 1 = 2 since appending ♦ preserves periods.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Periodicity Lemma for Partial Words

Kociumaka

Radoszewski

Rytter

et al. 2018

Lecture Notes in Computer Science

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We investigate the function L(h, p, q), called here the threshold function, related to periodicity of partial words (words with holes). The value L(h, p, q) is defined as the minimum length threshold which guarantees that a natural extension of the periodicity lemma is valid for partial words with h holes and (strong) periods p, q. We show how to evaluate the threshold function in O(log p + log q) time, which is an improvement upon the best previously known O(p + q)-time algorithm. In a series of papers, the formulae for the threshold function, in terms of p and q, were provided for each fixed h ≤ 7. We demystify the generic structure of such formulae, and for each value h we express the threshold function in terms of a piecewise-linear function with O(h) pieces.

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Section: Previous Resultsmentioning

confidence: 99%

“…• each partial word of length at least 12 with one hole and periods 5, 7 has also period 1 = gcd (5,7),…”

Section: Introductionmentioning

confidence: 99%

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On Periodicity Lemma for Partial Words

Kociumaka

Radoszewski

Rytter

et al. 2018

Lecture Notes in Computer Science

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show abstract

“…3,b is connected, so all strings having this graph are strictly monotone. On the other hand, some strings with the graph G(17,8,5,5,2) inFig. 3,c have no monotonicity properties.…”

mentioning

confidence: 99%

“…The graph G(17,8,1,5,3) inFig. 3,b is connected, so all strings having this graph are strictly monotone.…”

mentioning

confidence: 99%

String periods in the order-preserving model

Gourdel

Kociumaka

Radoszewski

et al. 2020

Information and Computation

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The order-preserving model (op-model, in short) was introduced quite recently but has already attracted significant attention because of its applications in data analysis. We introduce several types of periods in this setting (op-periods). Then we give algorithms to compute these periods in time O(n), O(n log log n), O(n log 2 log n/ log log log n), O(n log n) depending on the type of periodicity. In the most general variant the number of different periods can be as big as Ω(n 2 ), and a compact representation is needed. Our algorithms require novel combinatorial insight into the properties of such periods.

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How Much Different Are Two Words with Different Shortest Periods

Alzamel¹,

Crochemore²,

Iliopoulos³

et al. 2018

IFIP Advances in Information and Communication Technology

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Sometimes the difference between two distinct words of the same length cannot be smaller than a certain minimal amount. In particular if two distinct words of the same length are both periodic or quasiperiodic, then their Hamming distance is at least 2. We study here how the minimum Hamming distance dist(x, y) between two words x, y of the same length n depends on their periods. Similar problems were considered in [1] in the context of quasiperiodicities. We say that a period p of a word x is primitive if x does not have any smaller period p which divides p. For integers p, n (p ≤ n) we define Pp(n) as the set of words of length n with primitive period p. We show several results related to the following functions introduced in this paper for p = q and n ≥ max(p, q).Lemma 1 (Periodicity Lemma [14]). If a word x has periods p and q and |x| ≥ p + q − GCD(p, q), then x also has a period GCD(p, q).Other known extensions of this lemma include a variant with three [10] and an arbitrary number of specified periods [11,16,17,23], the so-called new periodicity lemma [13,3], a periodicity lemma for repetitions that involve morphisms [19], and extensions into periodicity of partial words [4

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Abelian periods, partial words, and an extension of a theorem of Fine and Wilf

Cited by 10 publications

References 21 publications

On Periodicity Lemma for Partial Words

On Periodicity Lemma for Partial Words

String periods in the order-preserving model

How Much Different Are Two Words with Different Shortest Periods

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