A. Apostolico, F. Preparata. Optimal off--line detection of repetitions in a string. Theoretical Computer Science, 22, 1983, pp. 297--315.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....matching, string periodicity. 1 Introduction A suffix tree of a string s is a compacted trie of all suffixes of s. It is a powerful data structure which finds applications in many string processing algorithms. Some examples are string matching, finding all squares or repetitions in a string [AP83], computing substring statistics [AP85] approximate string matching [LV86] text compression [RPE81] analyzing genetic sequences [CHM86] etc. A very powerful feature of the suffix tree is that after construction and suitable preprocessing, the longest common prefix of any two substrings of s ....

A. Apostolico, F. Preparata. Optimal off--line detection of repetitions in a string. Theoretical Computer Science, 22, 1983, pp. 297--315.

....is simple, while, surprisingly, detecting whether it is cube free is still open. For D0L languages, in turn, even deciding whether the order of repetition of words in the language is unbounded is a nontrivial problem [12, 14] There are many algorithms for detecting squares in a string, see [5, 8, 20] for sequential algorithms and [2, 3, 11] for parallel ones. The sequential algorithms work in O(n log n) time which is proved to be optimal [20] The parallel ones are work optimal, i.e. the total number of operations in them is O(n log n) and they work in polylogarithmic time. The algorithm in ....

Apostolico A., Preparata F., Optimal off-line detection of repetitions in a string, Theor. Comput. Sci. 22, 297-315, 1983. 7

....for a systematic change of parameters. Suffix trees find a wide variety of applications in many different areas related to string processing, such as: string matching [6, 29, 101] approximate string matching [23, 42, 71, 72, 79, 98] finding longest repeated substrings [101] finding squares [10, 68] and repetitions in a string [10] computing statistics for the non overlapping occurrences [11] finding the longest match between all ordered suffix prefix pairs of a given set of strings [55] finding the longest substring that appears in h out of k strings, for any h 2 [58] computing ....

.... Suffix trees find a wide variety of applications in many different areas related to string processing, such as: string matching [6, 29, 101] approximate string matching [23, 42, 71, 72, 79, 98] finding longest repeated substrings [101] finding squares [10, 68] and repetitions in a string [10]; computing statistics for the non overlapping occurrences [11] finding the longest match between all ordered suffix prefix pairs of a given set of strings [55] finding the longest substring that appears in h out of k strings, for any h 2 [58] computing characteristic strings [59] matching a ....

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Apostolico, A., and Preparata, F. P., Optimal off-line detection of repetitions in a string, Theoret. Comp. Sci. 22, 297--315, (1983).

.... and most notably in molecular biology (cf. 40] In computer science and molecular biology many algorithms depend on a solution to the following problem: given a word X and a set of arbitrary b 1 suffixes S 1 , S b 1 of X , what is the longest common prefix of these suffixes (cf. 2] [3], 9] 12] 42] In coding theory (e.g. prefix codes) one asks for the shortest prefix of a suffix S i which is not a prefix of any other suffixes S j , 1 j n of a given sequence X (cf. 34] In data compression schemes, the following problem is of prime interest: for a given data base ....

....coined for this structure, and among these we mention here position trees, subword trees, directed acyclic graphs, etc. cf. 1] Suffix trees find a wide variety of applications in algorithms on words including: the longest repeated substring (cf. 41] squares or repetitions in strings (cf. [3]) string statistics (cf. 3] string matching (cf. 9] 42] approximate string matching (cf. 12] 9] 42] string comparison, compression schemes (cf. 25] implementation of the Lempel Ziv algorithm, genetic sequences, biologically significant motif patterns in DNA (cf. 9] 40] ....

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A. Apostolico and F.P. Preparata, Optimal Off-Line Detection of Repetitions in a String, Theoretical Computer Science, 22, 297-315 (1983).

.... integer powers in a word (i.e. those primitivelyrooted integer powers u k which are not followed or preceded by another occurrence of u) This is an asymptotically optimal bound, as the number of such powers can be n log n) Using a suffix tree technique, Apostolico and Preparata [1] described an O(n log n) algorithm for finding all right maximal repetitions, which are repetitions that cannot be extended to the right without increasing the period. Main and Lorentz [15] proposed another algorithm which actually finds all maximal repetitions in O(n log n) time. They also point ....

A. Apostolico and F. Preparata. Optimal off-line detection of repetitions in a string. Theoretical Computer Science, 22(3):297--315, 1983.

....for finding occurrences of contiguously repeated substrings. An occurrence of a substring of the form is called an occurrence of a square or a tandem repeat. Several methods have been presented that in time O(n log n z) find all z occurrences of tandem repeats in a string of length n, e.g. [2, 5, 17, 20, 26]. Methods that in time O(n) decide if a string of length n contains an occurrence of a tandem repeat have also been presented, e.g. 6, 21] Extending on the ideas presented in [6] two methods [12, 16] have been presented that find a compact representation of all tandem repeats in a string of ....

.... occurrence of a tandem repeat is just a right maximal Finding Maximal Pairs with Bounded Gap 3 pair with gap zero, the methods presented in this paper can be used to find all tandem repeats in time O(n log n z) This matches the time bounds of previous published methods for this problem, e.g. [2, 5, 17, 20, 26]. 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 define pairs and suffix trees and describe how in general to find pairs using the suffix tree. In the first part, Section 3, ....

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A. Apostolico and F. P. Preparata. Optimal off-line detection of repetitions in a string. Theoretical Computer Science, 22:297--315, 1983.

....tandem repeats which do not contain shorter repeats. These are called primitive tandem repeats. It is known that there can be at most O(n log n) occurrences of primitive tandem repeats in a string of length n, and several algorithms are known that identify those occurrences in O(n log n) time [Cro81, AP83, SG98]. In a very impressive, though highly technical, extended abstract, Kosaraju [Kos94] addresses the question of finding for each position i of S the shortest tandem repeat starting at position i, and sketches an O(n) time algorithm for that problem. He also mentions the problem of finding all ....

A. Apostolico and F. P. Preparata. Optimal off-line detection of repetitions in a string. Theor. Comput. Sci., 22:297--315, 1983.

....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 finding all the repetitions in a string [10, 5, 24]. 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 finding all the repetitions. A natural extension of the repetition problems is to allow errors. Approximate repetitions are common in applications such as ....

A. Apostolico, F.P. Preparata, Optimal off-line detection of repetitions in a string, Theoretical Computer Science 22, (1983), 297-315.

....also showed that, in his encoding, a Fibonacci string of length n contains Omega Gamma n log n) repetitions, so that, at least with respect to his encoding, his algorithm was optimal . Somewhat later, two other, quite different, algorithms for computing all the strong repetitions were published [AP83, ML84], both also requiring Theta(n log n) time, but now over an arbitrary alphabet. This paper discusses, apparently for the first time, the computation of all the weak repetitions in x. This problem generalizes and includes the corresponding strong repetitions problem, since every strong repetition ....

A. Apostolico & F. P. Preparata, Optimal off-line detection of repetitions in a string, Theoretical Comp. Sci. 22 (1983) 297-315.

....Strings (ab) 2 , ac) 2 , and a 2 . The study of repetitions in strings is motivated by the equivalent problem, encountered by molecular biologists, of automatically detecting repeated regions (with errors) in DNA and protein sequences (see [17] There exist three well known algorithms [2,4,13] for finding all the repetitions in a given string x = x[1: jxj] Each of these algorithms is asymptotically optimal, executing in time Theta(jxj log jxj) which is also the time required [13] merely to recognize whether or not x contains a repetition. Indeed, the execution time achieved by the ....

A. Apostolico and F. P. Preparata. Optimal off-line detection of repetitions in a string. Theoretical Computer Science, Volume 22, pages 297-315, (1983).

....must be separated from another, because each stage deals with equivalence relation E l . Thus the algorithm in [1] cannot be applied directly to our partitioning. Crochemore [C81] used this partitioning for computing all repetitions in a string. Although the approach of Apostolico and Preparata [AP83] computes the same equivalence classes with their elements sorted, the partitioning method in Fig. 2 is simpler and more elegant ( AP83] uses quite complicated data structures) 4. Data Structures for Gap Monitoring Let C = fi 1 ; i 2 ; i k g be a start set of a string x, with i j 2 1: jxj; j ....

....to our partitioning. Crochemore [C81] used this partitioning for computing all repetitions in a string. Although the approach of Apostolico and Preparata [AP83] computes the same equivalence classes with their elements sorted, the partitioning method in Fig. 2 is simpler and more elegant ([AP83] uses quite complicated data structures) 4. Data Structures for Gap Monitoring Let C = fi 1 ; i 2 ; i k g be a start set of a string x, with i j 2 1: jxj; j 2 f1: kg. We define the lists left gap(im ) i m Gamma i m Gamma1 for all m 2 f1: kg right gap(im ) i m 1 Gamma i m for all m 2 ....

A. Apostolico and F. P. Preparata, Optimal off-line detection of repetitions in a string, Theoret. Comput. Sci. 22 (1983), 297--315.

....were first studied by the mathematician Axel Thue [14] who showed how to construct strings of infinite length on three letters that contain no repetitions. In recent years, computer scientists have become interested in the algorithmic problem of computing all the repetitions in a given string [8,11,4], a task that requires Theta(n log n) time. It is interesting to note that Thue also showed in [14] how to construct infinite strings without overlaps that is, with no coverable substrings that are not repetitions. The corresponding algorithmic problem, the computation of the maximal ....

Alberto Apostolico & F. P. Preparata, Optimal off-line detection of repetitions in a string, TCS 22 (1983) 297-315.

....model. We obtain bounds of Theta(log log n) time on n log n log log n processors or Theta(d n log n p e log log d1 p=ne 2p) time on p processors. 4.1 Historical Overview There exist few sequential algorithms that test if a string is square free. Algorithms by Apostolico and Preparata [12], by Crochemore [35] Rabin [77] and by Main and Lorentz [72] find all the squares in a string of length n in O(n log n) time. Main and Lorentz [72] also show that O(n log n) comparisons are necessary even to decide if a string is square free. In another paper, Main and Lorentz [73] show that the ....

A. Apostolico and F. P. Preparata. Optimal off-line detection of repetitions in a string. Theoret. Comput. Sci., 22:297--315, 1983.

.... common to two strings [Cr87] 2) to construct a minimal transducer that outputs the position of first occurrence for each substring of a string [Cr86] and (3) to find squares in a string, or to determine whether as string is square free, where a square is an occurrence of xx for a substring x [AP,Cr86,Gu,ML,Rab,SeiG]. Problem (3) has also been called the problem of finding repetition in a string. In the [RPE] algorithm for Lempel Ziv data compression [ZL] the algorithm finds some maximal matches in the course of building a suffix tree, but does not find all maximal matches. The goal of reporting all pairs of ....

A. Apostolico and F. Preparata, Optimal off-line detection of repetitions in a string, Theoretical Comput. Sci. 22 (83), pp. 297-315.

....at the turn of the century, but it is only since about 1980 that computer scientists have begun to consider algorithms to find all the repetitions in a given string x what we shall call here the R Problem. Three such algorithms using very different methods have since been designed [C81,AP83,ML84], each executing in time Theta(n log n) where n = jxj. These algorithms encode repetitions as triples (i; p; k) k 1, signifying that (a) x[i: i p Gamma 1] is not a repetition; b) x[i: i pk Gamma 1] x[i: i p Gamma 1] k ; c) x[i: i p(k 1) Gamma 1] 6= x[i: i p Gamma 1] ....

A. Apostolico & F. P. Preparata, Optimal off-line detection of repetitions in a string, TCS 22 (1983) 297-315.

.... [Cr87] producing the minimal transducer that outputs the position of first occurrence of every distinct substring [Cr86] Lempel Ziv data compression [RPE,ZL] which finds some but not all maximal matches, and finding occurrences of specific patterns such as xx, where x is any substring, as in [AP,Cr86,ML,Ra]. Parameterized matching is reminiscent of unification [GeN] where the goal is to determine whether two expressions can be made equivalent via substitutions for variables, but unification differs in three ways from our problem: the domain is expressions (terms) rather than strings, terms (rather ....

A. Apostolico and F. Preparata, Optimal off-line detection of repetitions in a string, Theoretical Comput. Sci. 22 (83), pp. 297-315.

....squares (ab) 2 , ac) 2 , and a 2 . The study of repetitions in strings is motivated by the equivalent problem, encountered by molecular biologists, of automatically detecting repeated regions (with errors) in DNA and protein sequences (see [17] There exist three well known algorithms [2,4,13] for finding all the repetitions in a given string x = x[1: jxj] Each of these algorithms is asymptotically optimal, executing in time Theta(jxj log jxj) which is also the time required [13] merely to recognize whether or not x contains a repetition. Indeed, the execution time achieved by the ....

A. Apostolico and F. P. Preparata. Optimal off-line detection of repetitions in a string. Theoretical Computer Science, Volume 22, pages 297-315, (1983).

....[1: 6] aba) 2 , F 5 [3: 4] a 2 , F 5 [4: 7] ab) 2 , and F 5 [5: 8] ba) 2 . Note also that, according to this definition, x = a n contains only the single repetition a n . There are three well known algorithms which compute all the repetitions in a given string x of length n [AP83,C81,ML84]; each of these algorithms executes in time Theta(n log n) a bound that is known to be lowest possible [ML84] Thus Theta(n log n) is an upper bound on the number of repetitions which can possibly occur in any string x, and, as Crochemore has shown [C81] this bound is in fact achieved by the ....

Alberto Apostolico & F. P. Preparata, Optimal off-line detection of repetitions in a string, TCS 22 (1983) 297-315.

....a square. However, there exist infinite length strings on three letter alphabets that are square free as shown by Thue [36, 37] In the sequential setting, algorithms for testing if a string is square free and for finding all repetitions in a string were designed by Apostolico and Preparata [7], Crochemore [14, 15] Kosaraju [28] Main and Lorentz [31, 32] and Rabin [34] Main and Lorentz [31] proved that it is possible to find all repetition in a string in O(n log n) time using pairwise comparison of input symbols that test for equality. They have also shown that Omega Gamma n log n) ....

A. Apostolico and F.P. Preparata. Optimal off-line detection of repetitions in a string. Theoret. Comput. Sci., 22:297--315, 1983.

....full paper [2] can detect all squares in the same bounds. We prove also that this is the best time bound possible for an optimal parallel algorithm that solves this problem over a general alphabet. There exist few sequential algorithms to solve this problem. Algorithms by Apostolico and Preparata [3], by Crochemore [9] and by Main and Lorentz [17] find all the squares in a string of length n in O(n log n) time. Main and Lorentz [17] also show that O(n log n) comparisons are necessary even to decide if a string is square free. In another paper, Main and Lorentz [18] show that the latter ....

Apostolico, A. and Preparata, F. P. (1983), Optimal off-line detection of repetitions in a string, Theoretical Computer Science 22, 297-315.

....prove this is by giving an algorithm that enumerates all the squares. M. Crochemore showed in 1981 [57] that this number of squares is also tight: the Fibonacci strings, defined by F 0 = a; F 1 = b, and F i = F i Gamma1 F i Gamma2 , attain this bound. There are several efficient or optimal serial [101, 117, 57, 22, 84, 86] and parallel [67, 66, 17, 12] algorithms to test square freeness and detect all squares. We will discuss some simple criterion and algorithm later. 2.3 QUASIPERIODS AND COVERS In the Summer of 1990, A. Ehrenfeucht suggested that some repetitive structures defying the classical characterizations ....

A. Apostolico and F. P. Preparata. Optimal off-line detection of repetitions in a string. Theoret. Comput. Sci., 22:297--315, 1983.

....by order comparisons; and general alphabets where the only access an algorithm has to the input symbols is by equality comparisons. In the last decade, several sequential algorithms that find all squares in strings have been published. Algorithms that were discovered by Apostolico and Preparata [4] and by Crochemore [13, 15] find all squares in a string of length n over ordered alphabets in O(n log n) time. Rabin [27] gave a randomized algorithm that takes O(n log n) expected time over constant size alphabets. Any sequential algorithm that lists all squares in a string of length n must take ....

A. Apostolico and F.P. Preparata. Optimal off-line detection of repetitions in a string. Theoret. Comput. Sci., 22:297--315, 1983. 16

....input string of n symbols emitted by a symmetric source. We find that building the suffix tree for such a word, which takes linear time by clever methods [MC] takes O(n log n) time by the direct method; detecting all squares in that word, which takes optimal O(n log n) time by clever methods [AP, CR, ML], takes O(n log n) expected time by a simpler method; computing the full statistics without overlap of all substrings of that word, which takes O(n log 2 n) time by clever methods [AP1, AP2] takes O(n log n) expected time by a simpler method, etc. The same asymptotic bounds hold in the case of ....

....S 1 it is possible to spot all square prefixes of X as a byproduct of the construction of TX . The same straightforward strategy can be used for square suffixes. On the other hand, efficient algorithms for testing square freedom or detecting all squares in X require quite elaborate constructions [ML,CR,AP]. The number of distinct occurrences of squares in a word can be Theta(n log n) which sets a lower bound for all algorithms that find all squares [CR] For instance, infinitely many Fibonacci words, defined by: W 0 = b ; W 1 = a Wm 1 = Wm Wm Gamma1 for m 1 have Theta(n log n) distinct ....

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A. Apostolico and F.P. Preparata, Optimal Off-line Detection of Repetitions in String, Theoretical Computer Science, 22, 287--315 (1983).

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