G. Brodal, R. Lyngso, Ch. Pedersen, J. Stoye. Finding Maximal Pairs with Bounded Gap. Journal of Discrete Algorithms, 1:134-149, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in [19] The exercise follows the basic approach of [19] Homework 5 Homework 5 covers results from three papers. The application of level linked (a; b) trees for nger searches was given in [9] The use of ecient merging of search trees for nding regularities in strings was explored in [3], where the regularities found are pairs with bounded gap which are an generalization of tandem repeats. The approach of extracting all tandem repeats from the set of all branching tandem repeats was taken in [21] The algorithm in [21] for computing all tandem repeats is simpler than the approach ....

G. S. Brodal, R. B. Lyngs, C. N. S. Pedersen, and J. Stoye. Finding maximal pairs with bounded gap. In Proceedings of the 10th Annual Symposium on Combinatorial Pattern Matching (CPM), volume 1645 of Lecture Notes in Computer Science, pages 134-149, 1999.

....at least twice in a word. More about nding repeated subwords in a word can be found in [Gus97] An intermediate problem, occurring for example in molecular biology applications, consists in nding subwords repeated within some speci ed distance. This problem has been studied in a recent paper [BLPS99] More precisely, the problem considered was to nd all subwords uvu, where the size of v, called the gap, belongs to a speci ed interval. Using su x trees together with binary search trees, it has been shown in [BLPS99] that all such pairs (of occurrences of u) can be found in time O(n log n ....

....some speci ed distance. This problem has been studied in a recent paper [BLPS99] More precisely, the problem considered was to nd all subwords uvu, where the size of v, called the gap, belongs to a speci ed interval. Using su x trees together with binary search trees, it has been shown in [BLPS99] that all such pairs (of occurrences of u) can be found in time O(n log n S) where S is the size of the output. In this paper, we consider a restricted version of this problem, when v has a xed size, and show that all repeated occurrences of u with a gap equal to r can be found in time O(n ....

G. Brodal, R. Lyngs, Ch. Pedersen, and J. Stoye. Finding maximal pairs with bounded gap. In M. Crochemore and M. Paterson, editors, Proceedings of the 10th Annual Symposium on Combinatorial Pattern Matching, volume 1645 of Lecture Notes in Computer Science. Springer-Verlag, 1999.

....aa is also maximal repeat since t:aa:a and a:aa:c are in p, and nally a is maximal, since t:a:g and a:a:a (or t:a:g and a:a:c ) appear in W . These maximal repeats are of main importance in biological sequence analysis and many studies have been performed to obtain ecient algorithms to nd them [6, 3, 7]. In [6] page 144, an algorithm based on a sux tree [9, 8] is proposed to enumerate in O(jpj z) all the maximal repeats of p, where z is the number or reported results. In [3] an algorithm Equipe g enome, cellule et informatique, Universit e de Versailles, 45 avenue des Etats Unis, 78035 ....

....in biological sequence analysis and many studies have been performed to obtain ecient algorithms to nd them [6, 3, 7] In [6] page 144, an algorithm based on a sux tree [9, 8] is proposed to enumerate in O(jpj z) all the maximal repeats of p, where z is the number or reported results. In [3], an algorithm Equipe g enome, cellule et informatique, Universit e de Versailles, 45 avenue des Etats Unis, 78035 Versailles Cedex, E mail: raffinot genome.uvsq.fr 1 also based on sux tree is given to enumerate in O(jpj log jpj z) all the maximal pairs with bounded gaps, i.e. the distance ....

G. S. Brodal, R. B. Lyngs, C. N. S. Pedersen, and J. Stoye. Finding maximal pairs with bounded gap. In M. Crochemore and M. Paterson, editors, Proceedings of the 10th Annual Symposium on Combinatorial Pattern Matching, number 1645 in Lecture Notes in Computer Science, pages 134-149, Warwick University, UK, 1999. Springer-Verlag, Berlin.

....an exercise for the reader to find an efficient algorithm for the construction of the table of all such tandem indices. The particularization of the problem to the tandem index of occurrences of the same pattern, which is in fact a relaxed square detection problem, has also been studied recently [49]. A. Amir et al. 10] have used tries to organize and speed up the discovery of association rules in a typical data base, the entries of which are sets of attributes. The first step consists in transforming each record into a string by numbering the different attributes. Next, every set is ....

G.S. Brodal, R. Lyngso, C.N.S. Pedersen and J. Stoye. Finding Maximal Pairs with Bounded Gap. Proc. 10th Combinatorial Pattern Matching, 342--351. Springer Verlag LNCS volume 1645 (1999).

....The exercise follows the basic approach of [19] Takehome Final Exam The takehome exam covers results from three papers. The application of level linked (a; b) trees for nger searches was given in [9] The use of ecient merging of search trees for nding regularities in strings was explored in [3], where the regularities found are gaps with bounded gap which are an generalization of tandem repeats. The approach of extracting all tandem repeats from the set of all branching tandem repeats was taken in [21] The algorithm in [21] for computing all tandem repeats is simpler than the approach ....

G. S. Brodal, R. B. Lyngs, C. N. S. Pedersen, and J. Stoye. Finding maximal pairs with bounded gap. In Proceedings of the 10th Annual Symposium on Combinatorial Pattern Matching (CPM), volume 1645 of Lecture Notes in Computer Science, pages 134-149, 1999.

....at least twice in a word. More about nding repeated subwords in a word can be found in [Gus97] An intermediate problem, occurring for example in molecular biology applications, consists in nding subwords repeated within some speci ed distance. This problem has been studied in a recent paper [BLPS99] More precisely, the problem considered was to nd all subwords uvu, where the size of v, called the gap, belongs to a speci ed interval. Using sux Part of this work has been done during the rst author s visit of LORIA INRIA Lorraine in AugustOctober 1999, within a joint project of the ....

....and Applied Mathematics, Moscow University, 119899 Moscow, Russia, e mail: roman vertex.inria.msu.ru z LORIA INRIA Lorraine, 615, rue du Jardin Botanique, B.P. 101, 54602 Villers l es Nancy, France, e mail: kucherov loria.fr 1 trees together with binary search trees, it has been shown in [BLPS99] that all such pairs (of occurrences of u) can be found in time O(n log n S) where S is the size of the output. In this paper, we consider a restricted version of this problem, when v has a xed size, and show that all repeated occurrences of u with a gap equal to r can be found in time O(n ....

G. Brodal, R. Lyngs, Ch. Pedersen, and J. Stoye. Finding maximal pairs with bounded gap. In M. Crochemore and M. Paterson, editors, Proceedings of the 10th Annual Symposium on Combinatorial Pattern Matching, volume 1645 of Lecture Notes in Computer Science. Springer-Verlag, 1999.

.... node is a general technique for finding regularities in a string, e.g. for finding squares in a string (or tandem repeats) 2, 18] for finding maximal quasi periodic substrings, i.e. substrings that can be covered by a shorter substring, 1, 6] and for finding maximal pairs with bounded gap [4]. All these problems can be solved using this technique in time log n) Other applications are listed by Gusfield in [10, Chapter 7] A crucial component of our algorithm is the representation of a leaf list by a collection of search trees, such that the leaf list of a node in the su#x tree of ....

G. S. Brodal, R. Lyngs, C. N. S. Pedersen, and J. Stoye. Finding maximal pairs with bounded gap. Journal of Discrete Algorithms, Special Issue of Matching Patterns, 1(1):77--104, 2000.

.... each node is a general technique for finding regularities in a string, e.g. for finding squares in a string (or tandem repeats) 2,17] for finding maximal quasi periodic substrings, i.e. substrings that can be covered by a shorter substring, 1,6] and for finding maximal pairs with bounded gap [4]. All these problems can be solved using this technique in time log n) Other applications are listed by Gusfield in [10, Chapter 7] A crucial component of our algorithm is the representation of a leaf list by a collection of search trees, such that the leaf list of a node in the su#x tree of ....

G. S. Brodal, R. Lyngs, C. N. S. Pedersen, and J. Stoye. Finding maximal pairs with bounded gap. Journal of Discrete Algorithms, Special Issue of Matching Patterns, 1(1):77--104, 2000.

....version of the well known smaller half trick used in the algorithms in [6, 11, 24] for nding tandem repeats. The stronger version of the smaller half trick is hinted at in [20, Exercise 35] and stated in Lemma 6. 2 In [21, Chapter 5] it is used in the analysis of nger searches, and in [9] it is used in the analysis and formulation of an algorithm to nd all maximal pairs with bounded gap in a string. Recently, and independent of our work, Iliopoulos and Mouchard in [15] report an algorithm with running time O(n log n) for nding all maximal quasiperiodic substrings in a string of ....

G. S. Brodal, R. B. Lyngs, C. N. S. Pedersen, and J. Stoye. Finding maximal pairs with bounded gap. In Proceedings of the 10th Annual Symposium on Combinatorial Pattern Matching (CPM), volume 1645 of Lecture Notes in Computer Science, pages 134-149, 1999.

....if the characters to the immediate left (right) of the two occurrences of the substring are different. A pair is maximal if it is both left and right maximal. The A preliminary version of this paper appeared in the Proceedings of the 10th Annual Symposium on Combinatorial Pattern Matching [3]. Partially supported by the ESPRIT Long Term Research Programme of the EU under project number 20244 (ALCOM IT) y Basic Research in Computer Science, Centre of the Danish National Research Foundation. z Deutsches Krebsforschungszentrum (German Cancer Research Center) J. of Discrete ....

G. S. Brodal, R. B. Lyngs, C. N. S. Pedersen, and J. Stoye. Finding maximal pairs with bounded gap. In Proceedings of the 10th Annual Symposium on Combinatorial Pattern Matching (CPM), volume 1645 of Lecture Notes in Computer Science, pages 134--149, 1999.

....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, ....

....version of the well known smaller half trick used in the algo2 rithms in [6, 11, 26] for nding tandem repeats. The stronger version of the smaller half trick is hinted at in [22, Exercise 35] and stated in Lemma 6. In [23, Chapter 5] it is used in the analysis of nger searches, and in [9] it is used in the analysis and formulation of an algorithm to nd all maximal pairs with bounded gap in a string. Recently, and independent of our work, Iliopoulos and Mouchard in [16] reported to have an algorithm for nding all maximal quasiperiodic substrings in a string of length n. They ....

G. S. Brodal, R. B. Lyngs, C. N. S. Pedersen, and J. Stoye. Finding maximal pairs with bounded gap. In Proceedings of the 10th Annual Symposium on Combinatorial Pattern Matching (CPM), volume 1645 of Lecture Notes in Computer Science, pages 134-149, 1999.

....stronger version of the well known smaller half trick used in the algorithms in [6, 11, 24] for nding tandem repeats. The stronger version of the smaller half trick is hinted at in [20, Exercise 35] and stated in Lemma 6. In [21, Chapter 5] it is used in the analysis of nger searches, and in [8] it is used in the analysis and formulation of an algorithm to nd all maximal pairs with bounded gap in a string. Recently, and independent of our work, Iliopoulos and Mouchard in [15] report an algorithm with running time O(n log n) for nding all maximal quasiperiodic substrings in a string of ....

G. S. Brodal, R. B. Lyngs, C. N. S. Pedersen, and J. Stoye. Finding maximal pairs with bounded gap. In Proceedings of the 10th Annual Symposium on Combinatorial Pattern Matching (CPM), volume 1645 of Lecture Notes in Computer Science, pages 134-149, 1999.

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G. S. Brodal, R. B. Lyngs, C. N. S. Pedersen, and J. Stoye. Finding maximal pairs with bounded gap. Technical Report RS-99-12, BRICS, April 1999.

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G. Brodal, R. Lyngso, Ch. Pedersen, J. Stoye. Finding Maximal Pairs with Bounded Gap. Journal of Discrete Algorithms, 1:134-149, 2000.

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Brodal, G.S., R. B. Lyngso, C. N. Storm Pedersen and Jens Stoye.: Finding Maximal pairs with bounded gaps. Journal of Discrete Algorithms, Vol. 1(1), pages 77-104, 2000.

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