On generalized Lyndon words

Dolce, Francesco; Restivo, Antonio; Reutenauer, Christophe

doi:10.1016/j.tcs.2018.12.015

Cited by 20 publications

(31 citation statements)

References 16 publications

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“…Such a characterization also holds for Galois words, as shown in the following proposition. A different proof of this result, involving infinite words, is given in [10]. Proposition 7.4.…”

Section: Galois Words and Abwt Computation For Arbitrary Rotationsmentioning

confidence: 99%

“…We consider generalized lexicographic orders introduced in [43] (cf. also [10]) that, for the purposes of this paper, can be formalized as follows.…”

Section: Generalized Bwts: a Synopsismentioning

confidence: 99%

“…Although, in general, Galois and Lyndon words are distinct within a conjugacy class, some properties that hold for Lyndon words are preserved. Some characterizations of Galois words by using infinite words and some properties of words that are obtained as a nonincreasing factorization in Galois words, are studied in [10].…”

Section: Galois Words and Abwt Computation For Arbitrary Rotationsmentioning

confidence: 99%

“…The existence of the ABW T shows that the classical lexicographic order is not the only order relation that one can use to obtain a reversible transformation. Indeed, lexicographic and alternating lexicographic order are two particular cases of a more general class of order relations considered in [10,43]. In a preliminary version of this paper [20] we introduce therefore a class of reversible word transformations based on the above order relations that includes both the original BW T and the Alternating BW T .…”

Section: Introductionmentioning

confidence: 99%

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The Alternating BWT: An algorithmic perspective

Giancarlo

Manzini

Restivo

et al. 2020

Theoretical Computer Science

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The Burrows-Wheeler Transform (BWT) is a word transformation introduced in 1994 for Data Compression. It has become a fundamental tool for designing self-indexing data structures, with important applications in several area in science and engineering. The Alternating Burrows-Wheeler Transform (ABWT) is another transformation recently introduced in [Gessel et al. 2012] and studied in the field of Combinatorics on Words. It is analogous to the BWT, except that it uses an alternating lexicographical order instead of the usual one. Building on results in [Giancarlo et al. 2018], where we have shown that BWT and ABWT are part of a larger class of reversible transformations, here we provide a combinatorial and algorithmic study of the novel transform ABWT. We establish a deep analogy between BWT and ABWT by proving they are the only ones in the above mentioned class to be rank-invertible, a novel notion guaranteeing efficient invertibility. In addition, we show that the backward-search procedure can be efficiently generalized to the ABWT; this result implies that also the ABWT can be used as a basis for efficient compressed full text indices. Finally, we prove that the ABWT can be efficiently computed by using a combination of the Difference Cover suffix sorting algorithm [Kärkkäinen et al., 2006] with a linear time algorithm for finding the minimal cyclic rotation of a word with respect to the alternating lexicographical order.

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“…Such a characterization also holds for Galois words, as shown in the following proposition. A different proof of this result, involving infinite words, is given in [10]. Proposition 7.4.…”

Section: Galois Words and Abwt Computation For Arbitrary Rotationsmentioning

confidence: 99%

“…We consider generalized lexicographic orders introduced in [43] (cf. also [10]) that, for the purposes of this paper, can be formalized as follows.…”

Section: Generalized Bwts: a Synopsismentioning

confidence: 99%

Section: Galois Words and Abwt Computation For Arbitrary Rotationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Alternating BWT: An algorithmic perspective

Giancarlo

Manzini

Restivo

et al. 2020

Theoretical Computer Science

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“…Lemma 1. ( [3], Lemma 31) Let u, v be nonempty finite words. Then the following four conditions are equivalent:…”

Section: Existence and Uniqueness Of Finite Factorizationsmentioning

confidence: 99%

Generalized Lyndon factorizations of infinite words

Burcroff

Winsor

2020

Theoretical Computer Science

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A generalized lexicographic order on words is a lexicographic order where the total order of the alphabet depends on the position of the comparison. A generalized Lyndon word is a finite word which is strictly smallest among its class of rotations with respect to a generalized lexicographic order. This notion can be extended to infinite words: an infinite generalized Lyndon word is an infinite word which is strictly smallest among its class of suffixes. We prove a conjecture of Dolce, Restivo, and Reutenauer: every infinite word has a unique nonincreasing factorization into finite and infinite generalized Lyndon words. When this factorization has finitely many terms, we characterize the last term of the factorization. Our methods also show that the infinite generalized Lyndon words are precisely the words with infinitely many generalized Lyndon prefixes.Keywords: generalized lexicographic order · infinite generalized Lyndon word · unique nonincreasing Lyndon factorization arXiv:1905.04746v2 [math.CO]

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Generalized Lyndon Factorizations of Infinite Words

Burcroff

Winsor

2019

Lecture Notes in Computer Science

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On generalized Lyndon words

Cited by 20 publications

References 16 publications

The Alternating BWT: An algorithmic perspective

The Alternating BWT: An algorithmic perspective

Generalized Lyndon factorizations of infinite words

Generalized Lyndon Factorizations of Infinite Words

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