## Linear-time computation of local periods

Venue: | Theoret. Comput. Sci |

Citations: | 3 - 0 self |

### BibTeX

@ARTICLE{Duval_linear-timecomputation,

author = {Jean-pierre Duval and Roman Kolpakov and Gregory Kucherov and Thierry Lecroq and Arnaud Lefebvre},

title = {Linear-time computation of local periods},

journal = {Theoret. Comput. Sci},

year = {},

pages = {2004}

}

### OpenURL

### Abstract

Abstract. We present a linear-time algorithm for computing all local periods of a given word. This subsumes (but is substantially more powerful than) the computation of the (global) period of the word and on the other hand, the computation of a critical factorization, implied by the Critical Factorization Theorem. 1

### Citations

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Algorithms on Strings, Tree and Sequences
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Citation Context ...ore of many fundamental results [18,2,19]. Besides, notions and techniques related to periodic structures in words find their applications in different areas: data compression [24], molecular biology =-=[12]-=-, as well as for designing more efficient string search algorithms [11,3,5]. In this paper, we concentrate, from the algorithmic perspective, on the important notion of local periods, that characteriz... |

664 |
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Citation Context ...es in words has been for a long time an active area of research. It is well-known that the (global) period of a word can be computed in linear time, using the KnuthMorris-Pratt string matching method =-=[16,4]-=-.On the other hand, in [3] it has been shown that a critical factorization can be constructed in linear time, by computing the smallest and largest suffixes under the lexicographical ordering. In the ... |

662 |
Algebraic Combinatorics On Words
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- 2002
(Show Context)
Citation Context ...factorization, implied by the Critical Factorization Theorem. 1 Introduction Periodicities in words have been classically studied in word combinatorics and are at the core of many fundamental results =-=[18,2,19]-=-. Besides, notions and techniques related to periodic structures in words find their applications in different areas: data compression [24], molecular biology [12], as well as for designing more effic... |

611 |
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Citation Context ...use the s-factorization without copy overlap: Definition 3. The s-factorization of w without copy overlap is the factorization w = f1f2 ...fm, where fi’s are defined inductively as follows: (i) f1 = w=-=[1]-=-, (ii) assume we have computed f1f2 ...fi−1 (i ≥ 2), and let w[bi] be the letter immediately following f1f2 ...fi−1 (i.e. bi = |f1f2 ...fi−1| +1). If w[bi] does not occur in f1f2 ...fi−1, then fi = w[... |

341 |
Text Algorithms
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(Show Context)
Citation Context ...es in words has been for a long time an active area of research. It is well-known that the (global) period of a word can be computed in linear time, using the KnuthMorris-Pratt string matching method =-=[16,4]-=-.On the other hand, in [3] it has been shown that a critical factorization can be constructed in linear time, by computing the smallest and largest suffixes under the lexicographical ordering. In the ... |

161 |
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Citation Context ...atorics and are at the core of many fundamental results [18,2,19]. Besides, notions and techniques related to periodic structures in words find their applications in different areas: data compression =-=[24]-=-, molecular biology [12], as well as for designing more efficient string search algorithms [11,3,5]. In this paper, we concentrate, from the algorithmic perspective, on the important notion of local p... |

118 | Combinatorics of words
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(Show Context)
Citation Context ...factorization, implied by the Critical Factorization Theorem. 1 Introduction Periodicities in words have been classically studied in word combinatorics and are at the core of many fundamental results =-=[18,2,19]-=-. Besides, notions and techniques related to periodic structures in words find their applications in different areas: data compression [24], molecular biology [12], as well as for designing more effic... |

86 |
An optimal algorithm for computing the repetitions in a word
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Citation Context ...he smallest one. A direct application of this approach, however, cannot result in a linear-time algorithm, for the reason that the overall number of squares in a word can be as big as Θ(n log n) (see =-=[6]-=-). Therefore, manipulating the set of all squares explicitly is prohibitive for our purpose. In [13], maximal repetitions have been studied, which are maximally extended runs of consecutive squares. I... |

82 |
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Citation Context ...factorization, implied by the Critical Factorization Theorem. 1 Introduction Periodicities in words have been classically studied in word combinatorics and are at the core of many fundamental results =-=[18,2,19]-=-. Besides, notions and techniques related to periodic structures in words find their applications in different areas: data compression [24], molecular biology [12], as well as for designing more effic... |

81 | Periods in strings
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Citation Context ...of existing methods. An interesting research direction would be to study the combinatorics of possible sets of local periods, in a similar way as it was done for the structure of all (global) periods =-=[10,23]-=-. The results presented in this paper might provide an initial insight for a such study. Acknowledgments GK, TL and AL have been supported by the french Action Spécifique “Algorithmes et Séquences” of... |

72 | An o(n log n) algorithm for finding all repetitions in a string - Main, Lorentz - 1984 |

64 |
Linear algorithm for data compression via string matching
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Citation Context ...he longest factor previously occurred). We refer to [12] for a discussion on these two variants of factorization. A salient property of both factorizations is that they can be computed in linear time =-=[22]-=- in the case of constant alphabet. In their original definition, both of these factorizations allow an overlap between a factor and its left copy. However, we can restrict this and require the copy to... |

54 | Finding maximal repetitions in a word in linear time
- Kolpakov, Kucherov
- 1999
(Show Context)
Citation Context ...imum of all local periods, and each such maximal value corresponds to a distinct critical factorization. Note that a great deal of work has been done on finding periodicities occurring in a word (see =-=[13]-=- for a survey). However, none of them allows to compute all local periods in linear time. The reason is that most of those algorithms are intrinsically super-linear, which can be explained by the fact... |

51 | Two-way string matching
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(Show Context)
Citation Context ...ues related to periodic structures in words find their applications in different areas: data compression [24], molecular biology [12], as well as for designing more efficient string search algorithms =-=[11,3,5]-=-. In this paper, we concentrate, from the algorithmic perspective, on the important notion of local periods, that characterize a local periodic structure at each location of the word [9,8]. In informa... |

27 | Finding approximate repetitions under Hamming distance
- Kolpakov, Kucherov
- 2003
(Show Context)
Citation Context ...n several different string matching techniques. Some of those techniques (sfactorization, Main-Lorentz extension functions) have already been successfully used for several repetition finding problems =-=[7,21,20,13,14,15]-=-. In particular, in [13], it has been shown that all maximal repetitions can be found in linear time, providing an exhaustive information about the periodic structure of the word. However, here again,... |

24 |
Detecting leftmost maximal periodicities
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(Show Context)
Citation Context ...n several different string matching techniques. Some of those techniques (sfactorization, Main-Lorentz extension functions) have already been successfully used for several repetition finding problems =-=[7,21,20,13,14,15]-=-. In particular, in [13], it has been shown that all maximal repetitions can be found in linear time, providing an exhaustive information about the periodic structure of the word. However, here again,... |

23 |
Squares, cubes, and time-space efficient string searching
- Crochemore, Rytter
(Show Context)
Citation Context ...ues related to periodic structures in words find their applications in different areas: data compression [24], molecular biology [12], as well as for designing more efficient string search algorithms =-=[11,3,5]-=-. In this paper, we concentrate, from the algorithmic perspective, on the important notion of local periods, that characterize a local periodic structure at each location of the word [9,8]. In informa... |

20 |
Time-space optimal string matching
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(Show Context)
Citation Context ...ues related to periodic structures in words find their applications in different areas: data compression [24], molecular biology [12], as well as for designing more efficient string search algorithms =-=[11,3,5]-=-. In this paper, we concentrate, from the algorithmic perspective, on the important notion of local periods, that characterize a local periodic structure at each location of the word [9,8]. In informa... |

16 |
Computation of squares in a string
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(Show Context)
Citation Context ...er-linear, which can be explained by the fact that they tend, explicitly or implicitly, to enumerate all squares in the word, the number of which can be super-linear. The closest result is the one of =-=[17]-=- which claims a linear-time algorithm for finding, for each position i of the string, the smallest square starting at i. The approach is based on a sophisticated analysis of the suffix tree. The absen... |

15 | Combinatorics of periods in strings
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- 2003
(Show Context)
Citation Context ...of existing methods. An interesting research direction would be to study the combinatorics of possible sets of local periods, in a similar way as it was done for the structure of all (global) periods =-=[10,23]-=-. The results presented in this paper might provide an initial insight for a such study. Acknowledgments GK, TL and AL have been supported by the french Action Spécifique “Algorithmes et Séquences” of... |

14 |
Recherche linéaire d’un carré dans un mot. Comptes Rendus des Séances de l’Académie des Sciences
- Crochemore
- 1983
(Show Context)
Citation Context ...n several different string matching techniques. Some of those techniques (sfactorization, Main-Lorentz extension functions) have already been successfully used for several repetition finding problems =-=[7,21,20,13,14,15]-=-. In particular, in [13], it has been shown that all maximal repetitions can be found in linear time, providing an exhaustive information about the periodic structure of the word. However, here again,... |

8 | Finding repeats with fixed gap
- Kolpakov, Kucherov
- 2000
(Show Context)
Citation Context ...can then be processed in time O(n). The reader is referred to [20,13] for full details. This general approach, initiated in [7,20] has been applied successfully to various repetition finding problems =-=[13,14,15]-=-. In this work we show that it can be also applied to obtain a linear-time algorithm for computing internal local periods. This gives yet another illustration of the power of the approach. 3.2 Finding... |

4 |
Périodes locales et propagation de périodes dans un mot, Theoretical Computer Science 204
- Duval
- 1998
(Show Context)
Citation Context ...gorithms [11,3,5]. In this paper, we concentrate, from the algorithmic perspective, on the important notion of local periods, that characterize a local periodic structure at each location of the word =-=[9,8]-=-. In informal terms, the local period at a given position is the size of the smallest square centered at this position. An importance of local periods is evidenced by the fundamental Critical Factoriz... |

2 |
Recurrence and periodicity in infinite words from local periods, Theoretical Computer Science 262
- Duval, Mignosi, et al.
- 2001
(Show Context)
Citation Context ...gorithms [11,3,5]. In this paper, we concentrate, from the algorithmic perspective, on the important notion of local periods, that characterize a local periodic structure at each location of the word =-=[9,8]-=-. In informal terms, the local period at a given position is the size of the smallest square centered at this position. An importance of local periods is evidenced by the fundamental Critical Factoriz... |