M. G. Main and R. J. Lorentz, An O(n log n) algorithm for nding all repetitions in a string, Journal of Algorithms 5 (1984), 422{ 432.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... programming by keeping and computing only a relevant subset of important values, as demonstrated in [16] 17] 33] and [46] A similar unbalanced strategy has been successfully used for square detection in strings [11] to speed up the original algorithm based on a divide and conquer approach [36]. 2 Preliminaries 2.1 The Alignment Graph The dynamic programming solution to the string comparison computation problem can be represented in terms of a weighted alignment graph [24] See Figure 1) The weight of a given edge can be speci ed directly on the grid graph, or as is frequently the ....

Main, M. G., R. J. Lorentz, An O(n log n) algorithm for nding all repetitions in a string. J. Algorithms, 5, 422-432 (1984).

....techniques are presented in Appendices I and II. Exact repetitions have been studied extensively. The repetitions can be either concatenated with the original substring or they may overlap or they may not. Algorithms for nding non overlapping repetitions in a given string can be found in [1, 8, 15, 21, 18, 26] and algorithms for computing overlapping repetitions can be found in [3, 13, 14, 25] A natural extension of the repetitions problem is to allow the presence of errors; that is, the identi cation of substrings that are duplicated to within a certain tolerance k (usually edit distance or Hamming ....

....4 Text Size (k) d = 5 g = 30 Fig. 7. Timing curves for f; g approximate squares. 6 Conclusion and Open problems The running time of the computation of approximate squares can be reduced to O(n log n) A theoretical algorithm is presented in [16] that shadows the Main and Lorentz algorithm ([21]) The following two problems are still open: Problem 1. Given a string t = t 1 : t n and two integers m and , compute all positions j of t, that there exists a string t such that t[j: j m] t t[j m 1: j 2m] t : t[j m 1: j ( 1)m] t Problem 2 Given a ....

G. Main and R. Lorentz, An O(n log n) algorithm for nding all repetitions in a string, Journal of Algorithms 5 (1984), pp. 422-432.

....of the pattern in the text with at most k di erences; the LV algorithm requires O(n 2 (log m log j j) running time. The LV method uses a complicated data structure (the sux tree) that makes their algorithm unsuitable for practical use. Furthermore algorithms for exact repetitions are given in [1, 6, 14], approximate repeats are treated in [12] and quasiperiodicities in [8, 9] Here we present an O(n 2 m=w) algorithm for several variants of the problem of computing overlapping evolutionary chains with k di erences, where n is the length of the input string, m is the length of the motif and w ....

G. Main and R. Lorentz, An O(n log n) algorithm for nding all repetitions in a string, Journal of Algorithms 5, pp. 422-432 (1984).

....an O(n log n) time algorithm for computing all seeds of x. For the same problem, Berkman, Iliopoulos and Park [6] presented a parallel algorithm that requires O(log n) time and O(n log n) work. Repetitions: There are several O(n log n) time algorithms for nding all the repetitions in a string [10,5,26]. In parallel computation, Apostolico and Breslauer [1] gave an optimal O(log log n) time algorithm (i.e. total work is O(n log n) for nding all the repetitions. A natural extension of the repetition problems is to allow errors. Approximate repetitions are common in applications such as ....

M.G. Main and R.J. Lorentz, An algorithm for nding all repetitions in a string, Journal of Algorithms 5 (1984), 422-432.

....the pattern is placed upon itself at position (i; j) If all overlapping elements are the same, then we set Witness[i; j] See gure 1 for an example of a witness. The witness table is constructed in O(m 2 log ) time. The witness table construction uses the algorithm of Main and Lorentz [13] which nds the longest pre x of a pattern string occurring 3 a a b b b b a b a a a b a a a a a b a b a a a b a a a a c a b b b a a a (a) a a b b b b a b a a a b a a a a a b a b a a a b b b b a a a a a a a b b b b a a a a a b a b a a a b a a a a c a b b b a a a (b) 1 2 3 4 5 6 1 ....

M.G. Main and R.J. Lorentz. An O(n log n) algorithm for nding all repetitions in a string. J. of Algorithms, pages 422-432, 1984.

....(in secs. d = 5 g = 30 Fig. 7. Timing curves for f; g approximate squares. 6 Conclusion and Open problems The running time of the computation of approximate squares can be reduced to O(n log n) A theoretical algorithm is presented in [16] that shadows the Main and Lorentz algorithm ([22]) The following two problems are still open: Problem 1. Given a string t = t 1 : t n and an integer , compute all positions j of t, that there exists a string t such that t[j: j m] t t[j m 1: j 2m] t : t[j m 1: j ( 1)m] t Problem 2 Given a string t = t ....

G. Main and R. Lorentz, An O(n log n) algorithm for nding all repetitions in a string, Journal of Algorithms 5 (1984), pp. 422-432.

.... Several methods that determine if a string of length n contains a square in time O(n) have been presented, e.g. 12, 18, 23] Several methods that nd occurrences of squares in a string of length n in time O(n log n) plus the time it takes to output the detected squares have been presented, e.g. [6, 11, 17, 24]. Recently two methods [14, 16] have been presented that nd a compact representation of all squares in a string of length n in time O(n) Basic Research In Computer Science (BRICS) Center of the Danish National Research Foundation, Department of Computer Science, University of Aarhus, Ny ....

M. G. Main and R. J. Lorentz. An O(n log n) algorithm for nding all repetitions in a string. Journal of Algorithms, 5:422-432, 1984.

....Karp, Miller and Rosenberg [5] Their algorithm (henceforward called KMR) runs in O(n log n) on a string of length n but can not nd all repetitions. However, various solutions based on closely related ideas have been proposed by Crochemore [52] Apostolico and Preparata [53] and Main and Lorentz [54]. They all take O(n log n) time, and any algorithm that lists all occurrences of squares, or even maximal repetitions in a string, takes at least n log n) time because, for example, Fibonacci words contain that many occurrences of repetitions (see [52] A more speci c question arises when one ....

....speci c question arises when one considers the problem of detecting and locating the squares (words of the form uu, for a non empty string u) that possibly occur within a given string of length n. The lower bound for testing squarefreeness of a string is also n log n) on general alphabets (see [54]) However, on a xed alphabet the problem of testing an occurrence of a square can be done in O(n log j j) which implies linear time algorithms if the size of the alphabet is xed (see [11] Recently, Kolpakov and Kucherov [55] proposed a linear time algorithm to compute all the distinct ....

M. G. Main et R. J. Lorentz. An O(n log n) algorithm for nding all repetitions in a string. J. Algorithms, 5(3):422-432, 1984.

....1950s led naturally to an interest in strings and their applications, but the emphasis shifted gradually toward algorithms operating on nite strings. In the 1960s a fundamental theoretical result, the periodicity lemma , was published [FW65] Then in the early 1980s three papers were published [C81,AP83,ML84] describing three quite di erent algorithms for computing all the tandem repetitions in a given string x in time (n log n) In [ML84] it was shown in addition that this was best possible, in the sense that any deterministic method based on letter comparisons that recognized whether or not x was ....

....strings. In the 1960s a fundamental theoretical result, the periodicity lemma , was published [FW65] Then in the early 1980s three papers were published [C81,AP83,ML84] describing three quite di erent algorithms for computing all the tandem repetitions in a given string x in time (n log n) In [ML84] it was shown in addition that this was best possible, in the sense that any deterministic method based on letter comparisons that recognized whether or not x was free of tandem repetitions was shown to require n log n) time. Recently a new (n log n) algorithm for all tandem repetions has ....

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Michael G. Main & Richard J. Lorentz, An O(n log n) algorithm for nding all repetitions in a string, J. Algs. 5 (1984) 422-432.

.... to string processing, such as compression [Sto88] or biological sequence analysis [Gus97] A great deal of work, in word combinatorics and string matching, has been devoted to contiguous repetitions, when a fragment is repeated contiguously two or more times [Cro81, Sli83, Cro83, AP83, ML84, ML85, Mai89, Kos94, IMS97, SG98a, KK99b, SG98b, KK99a] A simplest form of contiguous repetition is a square (tandem repeat) which is a subword of the form uu. On the other hand, some applications bring up the problem of nding subwords repeated in a word in a possibly non contiguous way. As ....

....in [CR94] or Lempel Ziv factorization [Gus97] because of its use in the well known Lempel Ziv compression method [LZ76, ZL77] In this paper, for presentation purposes, we use the Lempel Ziv factorization. The second component of our method is longest common extension functions [ML84] To illustrate the idea, assume we are given two words u 1 ; u 2 , and want to compute, for each position i of u 2 , the length of the longest pre x of u 1 which occurs at position i in u 2 . A variation of the Knuth Morris Pratt algorithm (see [ML84, CR94] allows to compute all these lengths ....

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M.G. Main and R.J. Lorentz. An O(n log n) algorithm for nding all repetitions in a string. Journal of Algorithms, 5(3):422-432, 1984.

....an O(n log n) algorithm for nding all occurrences of primitively rooted maximal integer powers in a word. Using a sux tree technique, Apostolico and Preparata [AP83] described an O(n log n) algorithm for nding all positioned right maximal fractional repetitions. Finally, Main and Lorentz [ML84] proposed another algorithm which actually nds all maximal repetitions in O(n log n) time. In 1989, using Crochemore s s factorization, Main [Mai89] proposed a linear time algorithm which nds all leftmost occurrences of distinct maximal repetitions in a word. As far as other related works are ....

M.G. Main and R.J. Lorentz. An O(n log n) algorithm for nding all repetitions in a string. Journal of Algorithms, 5(3):422-432, 1984.

....of more than two characters that contain no squares. Since then a lot of work has been done to develop ecient methods to detect or count squares in strings. Several methods [12, 20, 25] have been presented that determine if a string of length n contains a square in time O(n) Several methods [6, 9, 11, 18, 19, 26] have been presented that nd occurrences of squares in a string of length n in time O(n log n) plus the time it takes to output the detected squares. Recently Basic Research In Computer Science (BRICS) Center of the Danish National Research Foundation, Department of Computer Science, ....

M. G. Main and R. J. Lorentz. An O(n log n) algorithm for nding all repetitions in a string. Journal of Algorithms, 5:422-432, 1984.

....strings over any alphabet of more than two characters that contain no squares. Since then a lot of work have focused on developing ecient methods to count or detect squares in strings. Several methods [12, 18, 23] can determine if a string of length n contains a square in time O(n) and methods [6, 11, 17, 24] can nd occurrences of squares in a string of length n in time O(n log n) plus the time it takes to output the detected squares. Recently two methods [14, 16] have been presented that nd a compact representation of all squares in a string of length n in time O(n) A way to describe the ....

M. G. Main and R. J. Lorentz. An O(n log n) algorithm for nding all repetitions in a string. Journal of Algorithms, 5:422-432, 1984.

....of a substring of the form is called an occurrence of a square or a tandem repeat. Most well known methods for nding the occurrences of all tandem repeats in a string require time O(n log n z) where n is the length of the string and z is the number of reported occurrences of tandem repeats [4, 2, 19, 16, 25]. Work has also been done on just detecting whether or not a string contains a tandem repeat [20, 5] Recently, extending on the idea presented in [5] two methods have been presented that nd a compact representation of all tandem repeats in a string in time O(n) 15, 11] Other papers consider ....

....the 2 number of tandem repeats. Since a branching occurrence of a tandem repeat is just a right maximal pair with gap zero, the methods presented in this paper can be used to nd all tandem repeats in time O(n log n z) This matches the time bounds of previous published methods for this problem [4, 2, 19, 16, 25]. The rest of this paper is organized in two parts which can be read independently. In Section 2 we present the preliminaries necessary to read either of the two parts; we de ne pairs and sux trees and describe how in general to nd pairs using the sux tree. In the rst part, Section 3, we present ....

M. G. Main and R. J. Lorentz. An O(n log n) algorithm for nding all repetitions in a string. Journal of Algorithms, 5:422-432, 1984. 30

....of a substring of the form is called an occurrence of a square or a tandem repeat. Most well known methods for nding the occurrences of all tandem repeats in a string require time O(n log n z) where n is the length of the string and z is the number of reported occurrences of tandem repeats [5, 2, 18, 15, 24]. Work has also been done on just detecting whether or not a string contains a tandem repeat [19, 6] Recently, extending on the idea presented in [6] two methods have been presented that nd a compact representation of all tandem repeats in a string in time O(n) 14, 10] Other papers consider ....

....to the number of tandem repeats. Since a branching occurrence of a tandem repeat is just a right maximal pair with gap zero, the methods presented in this paper can be used to nd all tandem repeats in time O(n log n z) This matches the time bounds of previous published methods for this problem [5, 2, 18, 15, 24]. The rest of this paper is organized as follows. In Sect. 2 we de ne pairs and su x trees and describe how in general to nd pairs using the su x tree. In Sect. 3 we present facts about e cient merging of search trees, and use them to formulate methods for nding all maximal pairs in a string ....

M. G. Main and R. J. Lorentz. An O(n log n) algorithm for nding all repetitions in a string. Journal of Algorithms, 5:422-432, 1984.

....word. It allows easily to extract all repetitions of other types, such as (primitively or non primitively rooted) squares, cubes, or integer powers. Thus, all these tasks can be done in time O(n T ) where T is the output size. This improves many known algorithms and answers some open questions [39, 34, 47, 46, 43, 51]. Besides, this allows to solve eciently some new string matching problems. For example, one can nd, in linear time, the number of repetitions of a given exponent starting at each position of the word (see [2] In [12] a new analysis of the Apostolico Giancarlo string matching algorithm ....

M.G. Main and R.J. Lorentz. An O(n log n) algorithm for nding all repetitions in a string. Journal of Algorithms, 5(3):422-432, 1984.

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M. G. Main and R. J. Lorentz, An O(n log n) algorithm for nding all repetitions in a string, Journal of Algorithms 5 (1984), 422{ 432.

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G. Main and R. Lorentz, An O(n log n) algorithm for nding all repetitions in a string, Journal of Algorithms 5 (1984), pp. 422-432.

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M.G. Main and R.J. Lorentz. An O(n log n) algorithm for nding all repetitions in a string. Journal of Algorithms, Vol. 5, 1984, pp. 422-432.

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G. Main and R. Lorentz, An O(n log n) algorithm for nding all repetitions in a string, Journal of Algorithms 5 (1984), pp. 422-432.

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