A. Apostolico, D. Breslauer and Z. Galil, Optimal parallel algorithms for periods, palindromes and squares. Proc. 19th Int. Colloq. Automata Languages and Programming, Lecture Notes in Computer Science, Vol. 623, 1992, pp. 296--307.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....de ne them. A signi cant amount of research has been done on each of these four notions: Periods: The preprocessing of the Knuth Morris Pratt algorithm [21] nds all periods of x in linear time in fact, all periods of every pre x of x. In parallel computation, Apostolico, Breslauer and Galil [2] gave an optimal O(log log n) time algorithm for nding all periods, where n is the length of x. Covers: Apostolico, Farach and Iliopoulos [4] introduced the notion of covers and described a linear time algorithm to test whether x is superprimitive or not (see also [7,8,18] Moore and Smyth ....

A. Apostolico, D. Breslauer and Z. Galil, Optimal parallel algorithms for periods, palindromes and squares, in: Proc. 19th Int. Colloq. Automata Languages and Programming, Lecture Notes in Computer Science, 623 (1992), 296-307.

....ff, which is a leaf in this case, contains the list of positions of all its occurrences. So, factors of size longer than h can be detected by (i) a search on the tree followed by (ii) a search on the input string. Algorithms to detect classical palindromes have been developed in particular in [ABG92] The palindromes in the DNA differ from classical palindromes, for they admit gaps. We didn t optimize the search of the palindromes in biocompress 2 yet. They are found using an automaton similar to the one presented above. At each step of the compression, the longest factor beginning at the ....

A. Apostolico, D. Breslauer, and Z. Galil. Optimal parallel algorithms for periods, palindromes and squares. In unpublished, 1992.

....25000 characters. Note also that DNA sequences are always read in the same direction, 3 such as texts in natural or programming languages for instance. Properties of DNA sequences and mathematical tools to study them are presented in [ Algorithms to detect palindromes have been studied in [ABG92] In this paper, we focus on DNA sequences only. The algorithm biocompress can also be used on RNA sequences but not on amino acid sequences of proteins. The sequences on which we realized compression tests are the following (see Figure 3) The complete genomes of two mitochondries: MIPACGA, the ....

A. Apostolico, D. Breslauer, and Z. Galil. Optimal parallel algorithms for periods, palindromes and squares. In unpublished, 1992.

....14 Assume P is a compressed pattern. Then we can compute in O( n log U )Eq(m) time the compressed representation of the set Periods(P ) The representation consists (in this case) of log U number of linear sets. Our algorithms for compressed palindromes and squares use ideas which apeared in [4]: palindromes are searched using periodicities implied by sequences of many palindromes which are close to each other and searching of squares is reduced to multiple application of pattern matching. First we consider a data structure for palindromes, consider only even length palindromes (the ....

....RightSq(i) can be considered similarly. Consider a square s which is in LeftSq(i) its larger part is in B = T [1 : a i Gamma1 ] and its size is between 2 k and 2 k 1 . The squares whose larger part is in C = T [a i Gamma1 1 : U ] can be processed in the same way. Similarly as in [4] we consider the sample v of size 2 k Gamma1 which is a suffix of B. Then s = uvw, for some u and w. An occurrence of the sample v is found in C by applying the pattern matching algorithm. The size of s is the distance between two occurrences. The parts u; w are found by finding the first ....

A. Apostolico, D. Breslauer, Z. Galil, Optimal parallel algorithms for periods, palindromes and squares, in ICALP'92, 296-307

....them. A significant amount of research has been done on each of these four notions: Periods: The preprocessing of the Knuth Morris Pratt algorithm [19] finds all periods of x in linear time in fact, all periods of every prefix of x. In parallel computation, Apostolico, Breslauer and Galil [2] gave an optimal O(log log n) time algorithm for finding all periods, where n is the length of x. Covers: Apostolico, Farach and Iliopoulos [4] introduced the notion of covers and described a linear time algorithm to test whether x is superprimitive or not (see also [7, 8, 17] Moore and ....

A. Apostolico, D. Breslauer and Z. Galil, Optimal parallel algorithms for periods, palindromes and squares, Proc. 19th Int. Colloq. Automata Languages and Programming, Lecture Notes in Computer Science 623 (1992), 296-307.

.... structures [277, 279, 280, 278, 282, 281] New algorithmic techniques for dynamic programming problems, with applications to sequence alignments in molecular biology, have been described in [208, 210, 209, 214, 215, 276] Parallel algorithms for many of the above problems have been developed in [35, 36, 33, 28, 25, 42, 32, 38, 37, 40, 27, 34, 30, 31, 26, 41, 29]. 3.3 Traces Traces theory studies the subsets of free partially commutative monoids (trace languages) This theory has been introduced in order to describe the behavior of concurrent processes. In particular, trace theory is interested to study trace languages recognized by finite automata. the ....

A. Apostolico, D. Breslauer, and Z. Galil. Optimal parallel algorithms for periods, palindromes and squares. In Proc. ICALP '92. Springer-Verlag, 1992.

....have different complexities. In this abstract we only discuss the first four problems. For all these eight problems there are wt optimal algorithms known on the CRCW PRAM : linear work and constant time for Problems 1 [15] and 3 [12] linear work and O(log log n) time for Problems 2 [6] 4 [12] 5 [4], 6 [13] 8 [4] O(n log n) work and O(log log n) time for Problem 7 [3] The Omega Gammae 4 log n) lower bounds for the time follow from [7] The Omega Gamma n log n) lower bound for the work of Problem 7 follows from [23] A logarithmic lower bound for all these problems on the CREW PRAM ....

....complexities. In this abstract we only discuss the first four problems. For all these eight problems there are wt optimal algorithms known on the CRCW PRAM : linear work and constant time for Problems 1 [15] and 3 [12] linear work and O(log log n) time for Problems 2 [6] 4 [12] 5 [4] 6 [13] 8 [4]; O(n log n) work and O(log log n) time for Problem 7 [3] The Omega Gammae 4 log n) lower bounds for the time follow from [7] The Omega Gamma n log n) lower bound for the work of Problem 7 follows from [23] A logarithmic lower bound for all these problems on the CREW PRAM follows immediately ....

A. Apostolico, D. Breslauer and Z. Galil, Optimal parallel algorithms for periods, palindromes and squares. Proc. 19th Int. Colloq. Automata Languages and Programming, Lecture Notes in Computer Science, Vol. 623, 1992, 296--307.

....Assume P is a compressed pattern. Then we can compute in O( n log U )Eq(m) time the compressed representation of the set Periods(P ) The representation consists (in this case) of log U number of linear sets. Our algorithms for compressed palindromes and squares use ideas which apeared in [4]: palindromes are searched using periodicities implied by sequences of many palindromes which are close to each other and searching of squares is reduced to multiple application of pattern matching. First we consider a data structure for palindromes, consider only even length palindromes (the ....

....RightSq(i) can be considered similarly. Consider a square s which is in LeftSq(i) its larger part is in B = T [1 : a i Gamma1 ] and its size is between 2 k and 2 k 1 . The squares whose larger part is in C = T [a i Gamma1 1 : U ] can be processed in the same way. Similarly as in [4] we consider the sample v of size 2 k Gamma1 which is a suffix of B. Then s = uvw, for some u and w. An occurrence of the sample v is found in C by applying the pattern matching algorithm. The size of s is the distance between two occurrences. The parts u; w are found by finding the first ....

A. Apostolico, D. Breslauer, Z. Galil, Optimal parallel algorithms for periods, palindromes and squares, in ICALP'92, 296-307

....on the right sides. The value of the last variable of the composition system S is the value of S and is denoted by val(S) Example. We have val(S) val(G) abbabababba in the following composition system S: A = a; B = b; C = A Delta B; D = B Delta C; E = C Delta D; F = D [2] Delta E [4] ; G = E Delta F . Lemma 3. For each string w given in LZ compressed form and a subword v of w we can construct a composition system generating v of size O(n Delta log(n) where n = jLZ(w)j, for each variable A the word val(A) is a subword of w. Proof. The proof will be given in the full ....

....= val(S) and the text T = val(S) ignoring the occurrence of the pattern at position 1. As a side effect we compute all suffixes of T which are prefixes of P. This determines all periods. ut 3 Compressed Palindromes and Squares Our algorithms for compressed palindromes and squares use ideas from [4]: palindromes are searched using periodicities implied by sequences of many palindromes which are close to each other and searching of squares is reduced to multiple application of pattern matching. First we consider a data structure for palindromes, consider only even length palindromes (the ....

A. Apostolico, D. Breslauer, Z. Galil, Optimal parallel algorithms for periods, palindromes and squares, in ICALP'92, pp. 296-307.

No context found.

A. Apostolico, D. Breslauer and Z. Galil, Optimal parallel algorithms for periods, palindromes and squares. Proc. 19th Int. Colloq. Automata Languages and Programming, Lecture Notes in Computer Science, Vol. 623, 1992, pp. 296--307.

....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 of periods and repetitions ....

A. Apostolico, D. Breslauer, and Z. Galil. Optimal Parallel Algorithms for Periods, Palindromes and Squares. In Proc. 19th International Colloquium on Automata, Languages, and Programming, number 623 in Lecture Notes in Computer Science, pages 296--307. Springer-Verlag, Berlin, Germany, 1992.

....infinite length on three letter alphabets that are square free, as shown by Thue [29, 30] at the beginning of the century. Since then, numerous works have been The results presented in this paper were were first reported at the 19th International Colloquium on Automata, Languages and Programming [3]. y Department of Computer Science, Purdue University, West Lafayette, IN 47907, USA and Dipartimento di Elettronica e Informatica, Universit a di Padova, Padova, Italy. Partially supported by NSF Grants CCR 89 00305 and CCR 92 01078, by NATO Grant CRG 900293, by the National Research Council of ....

....computed from the output of the string matching algorithm (which is a Boolean vector) using Fich, Ragde and Wigderson s [19] integer minima algorithm in constant time using O(l j ) operations. This idea has been successfully applied also in efficient parallel algorithms for other string problems [3, 9, 12]. If the fq i g sequence does not contain any elements, then the phase does not need to do anything. If there is one element q 1 , then the algorithm finds the family of repetitions that are associated with the difference q 1 Gamma P and certifies them to be squares as described above. The next ....

A. Apostolico, D. Breslauer, and Z. Galil. Optimal Parallel Algorithms for Periods, Palindromes and Squares. In Proc. 19th International Colloquium on Automata, Languages, and Programming, number 623 in Lecture Notes in Computer Science, pages 296--307. Springer-Verlag, Berlin, Germany, 1992.

....this paper is a factor of log n processors away from optimality. Note that there exists a trivial constant time superprimitivity testing algorithm that uses n 2 processors. The superprimitivity testing algorithm follows techniques that were used in solving several other parallel string problems [1, 2, 6, 10]. In particular, it uses the parallel string matching algorithm of Breslauer and Galil [8] as a procedure that solves several string matching problems simultaneously and then combines the results of the string matching problems into an answer to the superprimitivity problem. 1 The paper is ....

A. Apostolico, D. Breslauer, and Z. Galil. Optimal Parallel Algorithms for Periods, Palindromes and Squares. In Proc. 19th International Colloquium on Automata, Languages, and Programming. Springer-Verlag, Berlin, Germany, 1992. 296--307.

....bound that was given by Breslauer and Galil [10] for the string matching problem. Note that both problems can be solved even in constant time if more processors are available. The string prefix matching algorithm follows techniques that were used in solving several other parallel string problems [2, 3, 4, 7, 8, 11]. In particular, it uses the parallel string matching algorithm of Breslauer and Galil [9] as a procedure that solves several string matching problems simultaneously and then combines the results of the string matching problems into an answer to the string prefix matching problem. The paper is ....

A. Apostolico, D. Breslauer, and Z. Galil. Optimal Parallel Algorithms for Periods, Palindromes and Squares. In Proc. 19th International Colloquium on Automata, Languages, and Programming, number 623 in Lecture Notes in Computer Science, pages 296--307. Springer-Verlag, Berlin, Germany, 1992.

....the large number of processors leaves much to be desired. Fischer and Paterson [9] noticed that any string matching algorithm that finds all overhanging occurrences of a string in another can also find all initial palindromes. This observation has been used by Apostolico, Breslauer and Galil [1] to construct an optimal O(log log n) time parallel algorithm that finds all initial palindromes in strings over general alphabets, improving an O(log n) time nonoptimal algorithm of Galil [12] Breslauer and Galil [5] show that any parallel algorithm that finds initial palindromes in strings over ....

....a string A[1: l] is p. If A[1: l] occurs only at positions p 1 p 2 Delta Delta Delta p k of a string B and p k Gamma p 1 dl=2e, then the p i s form an arithmetic progression with difference p. Proof: A simple consequence of Lemma 3.2. See the paper by Apostolico, Breslauer and Galil [1]. 2 Lemma 5.3 The sequence fp i g, which is defined above, forms an arithmetic progression. Proof: The sequence fp i g lists the indices of all occurrences of a string of length 4l j in a string of length 6l j Gamma 1. By Lemma 5.2, the p i s form an arithmetic progression. 2 By the last ....

A. Apostolico, D. Breslauer, and Z. Galil. Optimal Parallel Algorithms for Periods, Palindromes and Squares. In Proc. 19th International Colloquium on Automata, Languages, and Programming, number 623 in Lecture Notes in Computer Science, pages 296--307. Springer-Verlag, Berlin, Germany, 1992.

....characters, which implies that there would be no such occurrence. Thus, the lower bound holds also for any such string matching algorithm. 2 3. 4 Finding All Periods In this section we present an O(log log n) time n log log n processor CRCW PRAM algorithm to compute all periods of a string [9, 27]. If the number of available processors is p we show that the algorithm takes O(d n p e log log d1 p=ne 2p) time. The period of a string is computed in linear time as a step in Knuth, Morris and Pratt s sequential string matching algorithm [66] and in optimal O(log log n) parallel time on a ....

A. Apostolico, D. Breslauer, and Z. Galil. Optimal Parallel Algorithms for Periods, Palindromes and Squares. In Proc. 19th International Colluquium on Automata, Languages, and Programming. Springer-Verlag, Berlin, Germany, 1992. To appear.

....bound that was given by Breslauer and Galil [8] for the string matching problem. Note that both problems can be solved even in a constant time if more processors are available. The string prefix matching algorithm follows techniques that were used in solving several other parallel string problems [1, 2, 5, 6, 9]. In particular, it uses the parallel string matching algorithm of Breslauer and Galil [7] as a procedure that solves several string matching problems simultaneously and then combines the results of the string matching problems into an answer to the string prefix matching problem. The paper is ....

A. Apostolico, D. Breslauer, and Z. Galil. Optimal Parallel Algorithms for Periods, Palindromes and Squares. In Proc. 19th International Colluquium on Automata, Languages, and Programming. Springer-Verlag, Berlin, Germany, 1992. 296--307.

....we mean computing min(p; r=4) When we say that we compute the witnesses of x, we mean computing the witnesses against all non periods i, 1 i r=4. The witnesses of x can be computed in optimal O(log log r) time by [2] It is possible to compute large periods and witnesses using techniques of [1]. But we will not need them here. Given two strings x and y, and a position i of y such that x does not occur at position i of y, a witness to non occurrence at i is a position w such that y[w] 6= x[w Gamma i] A substring of x of length i is called an i block of x. The positions of all strings ....

....for string matching leads to constant expected time randomized algorithms for several related problems. If we solve log m (or even more) string matching problems at the same time, the expected time is still a constant. This converts the algorithms for finding all periods, squares and palindromes [1] into constant expected time randomized algorithms. We believe that there may be more applications for our super fast deterministic sampling. DS can be considered as a deterministic fingerprint. No constant time algorithm is known for the conventional fingerprint computation on a CRCW PRAM. On the ....

A. Apostolico, D. Breslauer, and Z. Galil, Optimal parallel algorithms for periods, palindromes and squares, Proc. 19th Int. Colloq. Automata Languages and Programming, Lecture Notes in Computer Science, Vol. 623, 1992, pp. 296--307.

....are handled separately we are able to use periodicity properties of strings [18] in each segment. These properties let us represent and manipulate the output of some string matching problems efficiently. These ideas were successfully applied in several other parallel algorithms for string problems [1, 2, 7, 5, 6]. We denote by T j the time it takes to compute stage number j using P j processors. The number of operations at stage j will be denoted by O j = T j P j . We show later how to implement stage number j in T j = O(log log l j ) time and O j = l j operations using Breslauer and Galil s [8] ....

A. Apostolico, D. Breslauer, and Z. Galil. Optimal Parallel Algorithms for Periods, Palindromes and Squares. In Proc. 19th International Colloquium on Automata, Languages, and Programming, number 623 in Lecture Notes in Computer Science, pages 296--307. Springer-Verlag, Berlin, Germany, 1992.

....restriction on the alphabet size. However, the advantages of the algorithm become clear if the input strings are given in their coded form. Apostolico, Breslauer and Galil gave efficient parallel algorithms for testing if a string is square free and for finding all palindromes in a string [4, 5, 6]. Their algorithms share a similar structure, take O(log log n) time utilizing n log n= log log n processors, and rely on a procedure that is used to solve several string matching problems. Observing that it suffices to encode the input string only once and use the coded string as input to many ....

A. Apostolico, D. Breslauer, and Z. Galil. Optimal Parallel Algorithms for Periods, Palindromes and Squares. In Proc. 19th International Colloquium on Automata, Languages, and Programming, number 623 in Lecture Notes in Computer Science, pages 296--307. Springer-Verlag, Berlin, Germany, 1992.

....x. When we say that we compute witnesses of x, we will mean computing r 0 and for every 1 i r 0 a position w such that x[w] 6= x[w Gamma i] Such a w is called a witness of x against (the periodicity of) i [20] It is possible to compute large periods and witnesses using techniques of [5]. But we will not need them here. If position i of T is not an occurrence of P , a position w such that T [w] 6= P [w Gamma i] is called a witness to non occurrence i. Vishkin [21] introduced the notion of a deterministic sample, which is crucial for very fast optimal parallel search of the ....

....occurrences of P in T can be determined in O(1) time and O(n 2 ) work following preprocessing of P , which takes O(log log m) time and O(m 2 ) work. 6. Two Dimensional Witness Computation A natural approach to computing 2D witnesses is to follow the approach for computing string witnesses [8,5]. There, witnesses for a string s of length m are found in O(log log m) stages. Stage i finds witnesses for a prefix of s of length l i ; these witnesses are used in Stage i 1 to find witnesses for the prefix of s of length l i 1 in constant time. The lower bound of [9] shows that ....

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A. Apostolico, D. Breslauer and Z. Galil, Optimal parallel algorithms for periods, palindromes and squares. Proc. 19th Int. Colloq. Automata Languages and Programming, Lecture Notes in Computer Science, Vol. 623, 1992, pp. 296--307.

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A. Apostolico, D. Breslauer, and Z. Galil, #Optimal parallel algorithms for periods, palindromes and squares," Proc. of the Int. Colloq. on Automata, Languages, and Programming, 296#307, 1992.

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A. Apostolico, D. Breslauer and Z. Galil, Optimal parallel algorithms for periods, palindromes and squares. Proc. 19th Int. Colloq. Automata Languages and Programming, Lecture Notes in Computer Science, vol. 623, 1992, pp. 296--307.

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A. Apostolico, D. Breslauer and Z. Galil, Optimal parallel algorithms for periods, palindromes and squares. Proc. 19th Int. Colloq. Automata Languages and Programming , Lecture Notes in Computer Science, vol. 623, 1992, pp. 296--307.

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