Suffix array and Lyndon factorization of a text

Mantaci, Sabrina; Restivo, Antonio; Rosone, Giovanna; Sciortino, Marinella

doi:10.1016/j.jda.2014.06.001

Cited by 22 publications

(55 citation statements)

References 14 publications

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“…We refer to [6] where a new method is presented for constructing the suffix array of a text by using its Lyndon factorization advantageously. Partitioning the text according to its Lyndon properties allows tackling the problem in local portions of the text, local suffixes, prior to extending the solution globally, to achieve the suffix array of the entire text -the local portions are determined by the Lyndon factors.…”

Section: Applicationsmentioning

confidence: 99%

Enhanced string factoring from alphabet orderings

Clare

Daykin

2019

Information Processing Letters

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Section: Applicationsmentioning

confidence: 99%

Enhanced string factoring from alphabet orderings

Clare

Daykin

2019

Information Processing Letters

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“…In [27,28], the authors found interesting relations between the sorting of the suffixes of a word w and that of its factors. Let w, x, u, y ∈ Σ * , and let u be a nonempty factor of w = xuy.…”

Section: Sorting Suffixes Of a Textmentioning

confidence: 99%

“…The suffix array (SA) of a word w is the lexicographically ordered list of the starting positions of the suffixes of w. The connection between Lyndon factorizations and suffix arrays has been pointed out in [17], where the authors show a method to construct the Lyndon factorization of a text from its SA. Conversely, the computation of the SA of a text from its Lyndon factorization has been proposed in [5] and then explored in [27,28].…”

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confidence: 99%

“…The algorithm proposed in [27,28] is based on the following interesting combinatorial result, proved in the same papers: if u is a concatenation of consecutive Lyndon factors of w = xuy, then the position of a nonempty suffix u i in the ordered list of suffixes of u (called local suffixes) is the same position of the nonempty suffix u i y in the ordered list of the suffixes of w (called global suffixes). In turn, this result suggests a divide and conquer strategy for the sorting of the suffixes of a word w = w 1 w 2 : we order the nonempty suffixes of w 1 and the nonempty suffixes of w 2 independently (or in parallel) and then we merge the resulting lists (see Section 2.5 for further details).…”

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Inverse Lyndon words and inverse Lyndon factorizations of words

Bonizzoni

Felice

Zaccagnino

et al. 2018

Advances in Applied Mathematics

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Motivated by applications to string processing, we introduce variants of the Lyndon factorization called inverse Lyndon factorizations. Their factors, named inverse Lyndon words, are in a class that strictly contains anti-Lyndon words, that is Lyndon words with respect to the inverse lexicographic order. The Lyndon factorization of a nonempty word w is unique but w may have several inverse Lyndon factorizations. We prove that any nonempty word w admits a canonical inverse Lyndon factorization, named ICFL(w), that maintains the main properties of the Lyndon factorization of w: it can be computed in linear time, it is uniquely determined, it preserves a compatibility property for sorting suffixes. In particular, the compatibility property of ICFL(w) is a consequence of another result: any factor in ICFL(w) is a concatenation of consecutive factors of the Lyndon factorization of w with respect to the inverse lexicographic order. IntroductionLyndon words were introduced in [25], as standard lexicographic sequences, and then used in the context of the free groups in [6]. A Lyndon word is a word which is strictly smaller than each of its proper cyclic shifts for the lexicographical ordering. A famous theorem concerning Lyndon words asserts that any nonempty word factorizes uniquely as a nonincreasing product of Lyndon words, called its Lyndon factorization. This theorem, that can be recovered from results in [6], provides an example of a factorization of a free monoid, as defined in [32] (see also [4,23]). Moreover, there are several results which give relations between Lyndon words, codes and combinatorics of words [3].The Lyndon factorization has recently revealed to be a useful tool also in string processing algorithms [2,29] with strong potentialities that have not been completely explored and understood. This is due also to the fact that it can be efficiently computed. Linear-time algorithms for computing this factorization can be found in [11,12] whereas an O(lg n)-time parallel algorithm has been proposed in [1,10]. A connection between the Lyndon factorization and the Lempel-Ziv (LZ) factorization has been given in [18], where it is shown that in general the size of the LZ factorization is larger than the size of the Lyndon factorization, and in any case the size of the Lyndon factorization cannot be larger than a factor of 2 with respect to the size of LZ.Relations between Lyndon words and the Burrows-Wheeler Transform (BWT) have been discovered first in [9,26] and, more recently, in [21]. Variants of BWT proposed in the previous papers are based on combinatorial results proved in [14] (see [30] for further details and [13] for more recent related results).Lyndon words are lexicographically smaller than all its proper nonempty suffixes. This explains why the Lyndon factorization has become of particular interest also in suffix sorting Lyndon words are also called prime words and their prefixes are also called preprime words in [19]. Interesting properties of Lyndon words are recalled below. Proposition 2.2 Eac...

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“…The Lyndon factorization has recently revealed to be a useful tool also in investigating queries related to sorting suffixes of a word, with strong potentialities for string comparison that have not been completely explored and understood [40,41]. Relations between Lyndon words and the Burrows-Wheeler Transform (BWT) have been discovered first in [13,38] and more recently in [32].…”

Section: Introductionmentioning

confidence: 99%

Lyndon Words versus Inverse Lyndon Words: Queries on Suffixes and Bordered Words

Bonizzoni

Felice

Zaccagnino

et al. 2020

Language and Automata Theory and Applications

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Lyndon words have been largely investigated and showned to be a useful tool to prove interesting combinatorial properties of words. In this paper we state new properties of both Lyndon and inverse Lyndon factorizations of a word w, with the aim of exploring their use in some classical queries on w.The main property we prove is related to a classical query on words. We prove that there are relations between the length of the longest common extension (or longest common prefix) lcp(x, y) of two different suffixes x, y of a word w and the maximum length M of two consecutive factors of the inverse Lyndon factorization of w. More precisely, M is an upper bound on the length of lcp(x, y). This result is in some sense stronger than the compatibility property, proved by Mantaci, Restivo, Rosone and Sciortino for the Lyndon factorization and here for the inverse Lyndon factorization. Roughly, the compatibility property allows us to extend the mutual order between local suffixes of (inverse) Lyndon factors to the suffixes of the whole word.A main tool used in the proof of the above results is a property that we state for factors m i with nonempty borders in an inverse Lyndon factorization: a nonempty border of m i cannot be a prefix of the next factor m i+1 . The last property we prove shows that if two words share a common overlap, then their Lyndon factorizations can be used to capture the common overlap of the two words.The above results open to the study of new applications of Lyndon words and inverse Lyndon words in the field of string comparison.

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Suffix array and Lyndon factorization of a text

Cited by 22 publications

References 14 publications

Enhanced string factoring from alphabet orderings

Enhanced string factoring from alphabet orderings

Inverse Lyndon words and inverse Lyndon factorizations of words

Lyndon Words versus Inverse Lyndon Words: Queries on Suffixes and Bordered Words

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