Extracting powers and periods in a word from its runs structure

Crochemore, Maxime; Iliopoulos, Costas S.; Kubica, Marcin; Radoszewski, Jakub; Rytter, Wojciech; Waleń, Tomasz

doi:10.1016/j.tcs.2013.11.018

Cited by 51 publications

(52 citation statements)

References 9 publications

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“…The Lyndon root λ of a word U is the minimal cyclic shift of the shortest string period of U . The notion of a Lyndon root was introduced in the context of runs by Crochemore et al (2014).…”

Section: Computing Squares Induced By a Family Of Wordsmentioning

confidence: 99%

“…The algorithm of Crochemore et al (2014) actually computes the set ssquares(T ) together with all the data in assumption of Lemma 4.2. These are the short words in the construction.…”

Section: Corollary 45 For Partial Word T Of Length N the Set Ssquarmentioning

confidence: 99%

“…We compute Lyndon roots of all the sealed fragments in O(nk 3 ) time using Corollary 4.15. For each solid p-square we may compute its Lyndon root in O(k 2 ) time using Lemma 4.14; we can also use the Lyndon roots as computed in Crochemore et al (2014).…”

Section: Theorem 416 For a Partial Word T Of Length N With K Holes mentioning

confidence: 99%

“…W = a n ), however, it is known that such a word contains only O(n) distinct square factors (Fraenkel and Simpson 1998;Ilie 2005); currently the best known upper bound is 11 6 n (Deza et al 2015). Moreover, all distinct square factors of a word over an integer alphabet can be listed in O(n) time using the suffix tree (Gusfield and Stoye 2004;Bannai et al 2017) or the suffix array and the structure of runs (maximal repetitions) in the word (Crochemore et al 2014).…”

Section: Introductionmentioning

confidence: 99%

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Efficient enumeration of non-equivalent squares in partial words with few holes

et al. 2018

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A word of the form W W for some word W ∈ Σ * is called a square. A partial word is a word possibly containing holes (also called don't cares). The hole is a special symbol ♦ / ∈ Σ which matches any symbol from (n) and USQUARES k (n) be the maximum number of non-equivalent p-squares and non-equivalent unambiguous p-squares in T over all partial words T of length n with at most k holes. We show asymptotically tight bounds:We present an algorithm that reports all non-equivalent p-squares in O(nk 3 ) time for a partial word of length n with k holes, for an integer alphabet. In particular, it runs in linear time for k = O(1) and its time complexity near-matches the asymptotic bound for PSQUARES k (n). We also show an O(n)-time algorithm that reports all nonequivalent p-squares of a given length. The paper is a full and improved version of

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Section: Computing Squares Induced By a Family Of Wordsmentioning

confidence: 99%

“…The algorithm of Crochemore et al (2014) actually computes the set ssquares(T ) together with all the data in assumption of Lemma 4.2. These are the short words in the construction.…”

Section: Corollary 45 For Partial Word T Of Length N the Set Ssquarmentioning

confidence: 99%

Section: Theorem 416 For a Partial Word T Of Length N With K Holes mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Efficient enumeration of non-equivalent squares in partial words with few holes

et al. 2018

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“…Fraenkel and Simpson [20] showed in 1998 that the maximum number of distinct squares in such a word is asymptotically bounded from below by n − o(n), and bounded from above by 2n, since at each position of a word of length n at most two distinct squares have their rightmost, or last, occurrence begin. They conjectured that this maximum number is at most n. This work became the motivation for linear-time algorithms that find all repetitions in a string, encoded into maximal repetitions [1,9,12,23,30,37].…”

Section: Introductionmentioning

confidence: 99%

Constructing Words with High Distinct Square Densities

Blanchet-Sadri¹,

Osborne²

2017

Electron. Proc. Theor. Comput. Sci.

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Fraenkel and Simpson showed that the number of distinct squares in a word of length n is bounded from above by 2n, since at most two distinct squares have their rightmost, or last, occurrence begin at each position. Improvements by Ilie to $2n-\Theta(\log n)$ and by Deza et al. to 11n/6 rely on the study of combinatorics of FS-double-squares, when the maximum number of two last occurrences of squares begin. In this paper, we first study how to maximize runs of FS-double-squares in the prefix of a word. We show that for a given positive integer m, the minimum length of a word beginning with m FS-double-squares, whose lengths are equal, is 7m+3. We construct such a word and analyze its distinct-square-sequence as well as its distinct-square-density. We then generalize our construction. We also construct words with high distinct-square-densities that approach 5/6.Comment: In Proceedings AFL 2017, arXiv:1708.0622

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Shortest Covers of All Cyclic Shifts of a String

Crochemore

Iliopoulos

Radoszewski

et al. 2020

WALCOM: Algorithms and Computation

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A factor W of a string X is called a cover of X, if X can be constructed by concatenations and superpositions of W . Breslauer (IPL, 1992) proposed a well-known O(n)-time algorithm that computes the shortest cover of every prefix of a string of length n. We show an O(n log n)-time algorithm that computes the shortest cover of every cyclic shift of a string and an O(n)-time algorithm that computes the shortest among these covers. A related problem is the number of different lengths of shortest covers of cyclic shifts of the same string of length n. We show that this number is Ω(log n).

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Extracting powers and periods in a word from its runs structure

Cited by 51 publications

References 9 publications

Efficient enumeration of non-equivalent squares in partial words with few holes

Efficient enumeration of non-equivalent squares in partial words with few holes

Constructing Words with High Distinct Square Densities

Shortest Covers of All Cyclic Shifts of a String

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