Cole, R., Hariharan, R., Indyk, P.: Tree pattern matching and subset matching in deterministic # ### # # # - time, 10th Symposium on Discrete Algorithms, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....patterns beyond those obtained from set and sequence mining. 8. RELATED WORK Tree mining, being an instance of frequent structure mining, has obvious relation to association [3] and sequence [4] mining. Frequent tree mining is also related to tree isomorphism [18] and tree pattern matching [8]. Given a pattern tree P and a target tree T , with P # T , the subtree isomorphism problem is to decide whether P is isomorphic to any subtree of T , i.e. there is a one to one mapping from P to a subtree of T , that preserves the node adjacency relations. In tree pattern matching the ....

R. Cole, R. Hariharan, and P. Indyk. Tree pattern matching and subset matching in deterministic o(n log n)-time. In 10th Symposium on Discrete Algorithms, 1999.

....and Thorup [17] The rst algorithm with a subquadratic time complexity is due to Kosaraju [10] who proposed an algorithm running in O(n m 3=4 polylog(m) time. This result has been improved successively by Dubiner, Galil and Magen [7] Cole and Hariharan [4] and Cole, Hariharan and Indyk [5], who achieved an almost linear O(n log n) time complexity. However, we can remark that these algorithms are quite sophisticated and complex to implement. If we recall that the naive algorithm is simple to implement and has a good behavior on the average, the theoretically optimal algorithms ....

R. Cole, R. Hariharan, and P. Indyk. Tree pattern matching and subset matching in deterministic O(n log n)-time. In ACM-SIAM Symposium on Discrete Algorithms, SODA'99, pages 245-254. Assoc. Comput. Mach. Press, 1999.

....that the matching function would then be injective. There is an obvious algorithm which solves this problem in O(mn) time. Improving this bound was a long time open problem, rst solved in [15] to attain a bound of O(nm 0:75 polylog(m) To our knowledge, the best algorithm is O(nlog m)[7]. 4 a a c a a x 1 x 3 x 2 p p Figure 3: A tree pattern p of arity 3, with result nodes marked, and its translation to a boolean pattern p, used in Proposition 1. The second related problem was de ned in [14] as unordered tree inclusion. The simplest statement of the problem is: ....

R. Cole, R. Harihan, and P. Indyk. Tree pattern matching and subset matching in deterministic o(nlog3n) time. Proc. 10th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 245-254, 1999.

....that the matching function would then be injective. There is an obvious algorithm which solves this problem in O(mn) time. Improving this bound was a long time open problem, rst solved in [15] to attain a bound of O(nm 0:75 polylog(m) To our knowledge, the best algorithm is O(nlog 3 m)[7]. The second related problem was de ned in [14] as unordered tree inclusion. The simplest statement of the problem is: given a pattern and input tree, can the pattern tree be obtained from the input tree by node deletions. It turns out that this problem is equivalent to evaluating a pattern in ....

R. Cole, R. Harihan, and P. Indyk. Tree pattern matching and subset matching in deterministic o(nlog3n) time. Proc. 10th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 245-254, 1999.

....0, matches Myriapoda and matches a path of the data tree. This query nds all the phylogenetic trees in TreeBASE that contain the query tree. 2.4 Related Approaches 2.4. 1 Approximate Embedding Queries Ho man and O Donnell [49] and later Ramesh and Ramakrishnan [81] and Cole et al. [28] presented algorithms for nding the occurrences of a wildcard free ordered query tree Q in an ordered data tree D. In an ordered tree, the order among siblings matters. Both Q and D are ordered and the occurrences of Q in D refer to those subtrees of D that can be obtained from Q by attaching ....

R. Cole, R. Hariharan, and P. Indyk. Tree pattern matching and subset matching in deterministic O(n log 3 n)-time. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pages 245-254, 1999.

....branches in T . We say that a tree S = N s ; B s ) is a subtree of T , denoted as S T , if and only if N s N , and for all branches b = x; y) 2 B s , x is 2 S1 1 1 2 S2 2 2 1 2 0 1 2 3 1 0 2 2 T (a tree in D) n = 6, s = 6, 6] n = 5, s = 5, 5] n = 4, s = 4, 4] n = 3, s =[3, 3] n = 2, s = 2, 3] n = 1, s = 1, 5] n = 0, s = 0, 6] support = 1 weighted support = 1 S3 2 (not a subtree) 1 3 1 0 string = 0 1 1 2 1 2 1 2 1 string = 1 1 1 2 1 weighted support = 2 support = 1 T s String Encoding: 0 1 3 1 1 2 1 1 2 1 1 2 1 Figure 1: An Example Tree with ....

.... say that a tree S = N s ; B s ) is a subtree of T , denoted as S T , if and only if N s N , and for all branches b = x; y) 2 B s , x is 2 S1 1 1 2 S2 2 2 1 2 0 1 2 3 1 0 2 2 T (a tree in D) n = 6, s = 6, 6] n = 5, s = 5, 5] n = 4, s = 4, 4] n = 3, s = 3, 3] n = 2, s = [2, 3] n = 1, s = 1, 5] n = 0, s = 0, 6] support = 1 weighted support = 1 S3 2 (not a subtree) 1 3 1 0 string = 0 1 1 2 1 2 1 2 1 string = 1 1 1 2 1 weighted support = 2 support = 1 T s String Encoding: 0 1 3 1 1 2 1 1 2 1 1 2 1 Figure 1: An Example Tree with Subtrees and their ....

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R. Cole, R. Hariharan, and P. Indyk. Tree pattern matching and subset matching in deterministic o(n log 3 n)- time, 1999.

....is slightly more general because a non leaf vertex x of P may have smaller degree than the corresponding vertex y of S, but the children of x must be put into correspondence with the leftmost children of y. The algorithm requires O( n m 0:5 poly log(m) time. A new approach presented in [5] attains a time bound of O(n log 3 n) thus improving over the result of [6] for m not too small. The use of different data structures leads to very good practical performances [2] although the required time is quadratic in the worst case. 2 In a recent work [3] the problem is made much more ....

....W [y k ] represent the children of v in the order in which they are encountered scanning A, then C[x] points to the list y 1 ; y k . In figure 1 such lists are columns of integers. For example vertex e is represented in W [2] and has children d; b; a, respectively represented in W [3] W [5]; W [11] Scanning A we first encounter A[1] 11, and since the parent of W [11] is in position F [11] 2 we append 11 to the list pointed by C[2] Later we encounter A[5] 5 and A[8] 3, both with parent in 2, and append 5 and 3 to the list C[2] Note that the order of the list elements ....

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R. Cole, R. Hariharan, and P. Indyk. Tree pattern matching and subset matching in deterministic O(n log 3 n time. Proc. ACM-SIAM Symp. on Discrete Algorithms, SODA'99. ACM Press (1999) 245-254.

....with mismatches and don t cares problem) our time complexity matches the currently best known algorithm [1] 2 Definitions and Preliminaries Cole and Hariharan [5] introduced subset matching and proposed a near linear time randomized algorithm for the problem. This was subsequently improved in [6] to a deterministic algorithm. We introduce the approximate version with don t care symbols. Let Sigma be our alphabet and let = 2 Sigma. will denote our don t care symbol. Let S 1 ; S 2 Sigma or S 1 = or S 2 = Denote mis(S 1 ; S 2 ) ae jS 1 Gamma (S 1 S 2 )j; if S 1 ; S ....

....mismatches and don t care s problem. Abrahamson [1] developed a divide and conquer algorithm that solves that problem in time O(n p m log m) Variations on the Abrahamson approach were used in [2, 3] Here we develop a new variation of this technique. 1 3 The Algorithm We will assume, as in [5, 6], that w.l.o.g. n = 2m and 2m s s 0 4m. This is true since the text may be cut into n=m overlapping segments, each having length 2m. For each segment, the sets may be broken down into smaller sets such that the sum of their elements will range between 2m and 4m elements. Intuition: Let S 1 ....

R. Cole, R. Harihan, and P. Indyk. Tree pattern matching and subset matching in deterministic o(n log 3 n) time. Proc. 10th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 245--254, 1999.

....contains only linear variables, is known under the name tree matching. In early 80 s, a simple practical solution has been proposed [HO82] More recently, a series of work has been done to nd the most ecient (in the worst case) algorithm for tree matching. We refer to the latest achievement [CHI99] which proposes an O(n log 3 n) deterministic algorithm, where n is the size of the tree (assumed to be bigger than the size of the pattern) The algorithm (as well as previously proposed theoretically ecient algorithms) is however rather complicated and dicult to implement, and the problem of ....

R. Cole, R. Hariharan, and P. Indyk. Tree pattern matching and subset matching in deterministic o(n log 3 n)-time. In Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms, Baltymore, Maryland, January 17-19, 1999, pages 245-254. ACM, SIAM, 1999.

.... work whose extended abstracts have appeared earlier in the Proceedings of the 29th ACM Symposium on Theory of Computing, 1997 [3] Proceedings of the 38th IEEE Symposium on Foundations of Computer Science, 1997 [12] and Proceedings of the 10th ACM SIAM Symposium on Discrete Algorithms, 1999 [4]. Courant Institute, New York University, cole cs.nyu.edu. This work was supported in part by NSF grants CCR9202900, CCR9503309, CCR9800085. Indian Institute of Science, Bangalore, ramesh csa.iisc.ernet.in. This work was done in part while visiting NYU. This work was supported in part by NSF ....

R. Cole, R. Hariharan, P. Indyk. Tree pattern matching and subset matching in deterministic O(n log m) time. Proceedings of the 10th ACM-SIAM Symposium on Discrete Algorithms, 1999, pp. 245-254.

....algorithm for the Subset Matching problem running , where s is the sum of the sizes of all the pattern and text sets. Subsequently, Indyk [8] gave a deterministic algorithm for the Subset Matching problem running (n s)m q log m (1 o(1) Finally, Cole, Hariharan and Indyk [3] gave a deterministic algorithm running in time O and a randomized algorithm running in time O (n s) The above algorithms will be described in a companion paper [4] It follows that there is a deterministic algorithm running in time O(n log m) and a randomized algorithm ....

R. Cole, R, Hariharan, P. Indyk. Tree pattern matching and subset matching in deterministic O(n log m) time. Proceedings of the 10th ACM-SIAM Symposium on Discrete Algorithms, 1999, pp. 245-254.

.... Sigma, return a binary array o[0; n Gamma 1] such that o[i] 1 if and only if p[j] ae t[i j mod n] for all j 2 f0; m Gamma 1g. Recent work of Cole and Hariharan [CH97] showed that this problem can be solved in (randomized) O(n log 3 m) time. Cole, Hariharan, and Indyk [CHI99] have shown that the above algorithm can be derandomized to run in O(n log 3 m) time. We present Algorithm 2D Concentric Pattern Matching which solves the approximate noisy pattern matching problem in almost quadratic time by reducing it to multiple instances of the subset matching problem. The ....

R. Cole, R. Hariharan, and P. Indyk. Tree pattern matching and subset matching in deterministic O(n log 3 m) Time. In This proceedings, 1999.

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Cole, R., Hariharan, R., Indyk, P.: Tree pattern matching and subset matching in deterministic # ### # # # - time, 10th Symposium on Discrete Algorithms, 1999.

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R. Cole, R. Hariharan, and P. Indyk, "Tree Pattern Matching and Subset Matching in Deterministic on log

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R. Cole, R. Hariharan, and P. Indyk. Tree pattern matching and subset matching in deterministic O(n log m) time. In Proc. 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'99), pp. 245254, 1999.

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R. Cole, R. Hariharan, and P. Indyk. Tree pattern matching and subset matching in deterministic O(n log n time. Proc. ACM-SIAM Symp. on Discrete Algorithms, SODA'99. ACM Press (1999) 245-254.

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R. Cole, R. Hariharan, and P. Indyk. Tree pattern matching and subset matching in deterministic O(n log m)time. In Proc. 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'99), pp. 245254, 1999.

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