Reyner, S. W.: An analysis of a good algorithm for the subtree problem, SIAM J. Computing, 6(4), 1977, 730--732.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to a set C of nodes, known as the context, and return another set of nodes. This set is given by fy j (x; y) 2 P (T ) x 2 Cg 3 Theorem 1 A simple XPath expression P can be evaluated on a tree T in time polynomial in the sizes of P and T . Proof: A modified version of the algorithm from [9] can be used. 3 Minimisation Without DTDs Let P and Q be simple XPath expressions. We say that P contains Q, written P Q, if for all trees T , P (T ) Q(T ) In addition, P is equivalent to Q, written P j Q, if P Q and Q P . Determining if P contains Q can be done by finding a ....

S. W. Reyner. An analysis of a good algorithm for the subtree problem. SIAM J. Computing, 6(4):730--732, December 1977.

....this eld is Subgraph Isomorphism (GT48 in [6] containing Clique, Complete Bipartite Subgraph or Hamiltonian as a special case. Subgraph Isomorphism is also NP complete for outerplanar graphs ( 21] sets of chains and forests ( 6] There exist polynomial algorithms if both graphs are trees ([19]) or 2 connected outerplanar ( 10] Another related NP complete problem is Largest Common Subgraph (GT49 in [6] It is only polynomial solvable if the graphs are trees. Levi and Luccio ( 9] proposed a heuristic which is also based on graph invariants. Further related NP complete problems are ....

Steven W. Reyner. An analysis of a good algorithm for the subtree problem. SIAM J. Comput., 6(4):730-732, 1977.

....y Computaci on, Universidad de la Habana. x Dipartimento di Informatica, Universit a di Pisa. 1 graph isomorphism. A nice alternative linear solution can be found in [1] Ch.3. Subtree isomorphism (m n) is more difficult. 1. Subtree isomorphism of rooted trees. After the pioneer work of [15], it was shown in [13] how to solve the problem for ordered trees in Theta(n) time, coding P and T as strings and applying a string matching algorithm to them. All known algorithms apply to ordered trees. In the present note we use a string representation of trees together with some advanced data ....

S.W. Reyner. An analysis of a good algorithm for the subtree problem. SIAM Journal on Computing 6 (1977) 730-732.

....1995 ] In the more general setting of simple conceptual graphs and conjunctive queries, testing for the existence of a homomorphism is an NP complete problem. In the restricted case of EL description trees, however, testing for the existence of a homomorphism can be realized in polynomial time [ Reyner, 1977; Baader et al. 1998 ] which shows that subsumption between EL concept descriptions is a tractable problem. Least common subsumer in EL The characterization of subsumption by homomorphisms allows us to characterize the lcs by the product of EL description trees. The product G Theta H of two ....

S. Reyner. An analysis of a good algorithm for the subtree problem. SIAM Journal of Computing, 6(4), 1977.

....w0 rw2ED w : 13 Since for each w 0 rw 2 ED there exists an j 2 f1; mg such that w = w j , is well defined, and by construction, it is a homomorphism from G(D) into G(A) with (w 0 ) v. 2 Whether there exists a homomorphism from a tree into a graph can be decided in polynomial time [8, 12]. Since G(C) and G(A) can be computed in polynomial time, we get Proposition 16 The instance problem for EL can be decided in polynomial time. 3.2 Computing the k approximations in EL In this section, we will show that, for an ELABox A and an individual a 2 Ind(abox) the k approximation of a ....

S.W. Reyner. An analysis of a good algorithm for the subtree problem. SIAM Journal of Computing, 6(4):730--732, December 1977.

....4 yields a homomorphism from G to GC . On the complexity of subsumption in EL For two EL concept descriptions C; D, subsumption C v D can be decided by (1) translating C; D into their corresponding EL description trees GC ; GD and (2) testing wether there exists a homomorphism from GD to GC . In [15], a polynomial time algorithm is introduced deciding wether there exists a homomorphism from a tree onto another tree. In [4] we have shown, that even for the DL ELIRO 1 (which extends EL by inverse roles, conjunction of roles, 8 Input: Two EL description trees H and G normal form. Output: ....

S.W. Reyner. An analysis of a good algorithm for the subtree problem. SIAM Journal of Computing, 6(4):730--732, December 1977.

....have binary r vertices without multi edges (by reduction from subgraph isomorphism (Garey and Johnson 1979) An injective projection from a tree to another tree is polynomially computable. The skeleton of an algorithm is described below. It is a trivial adaptation of the algorithm analyzed in (Reyner 1977). It involves the notion of bipartite graph matching: a matching of a bipartite graph is a set of its edges, such that no two edges share a common endpoint. Let T and T 0 be directed labelled trees. There is an injective generic morphism from T to T 0 iff given any vertex of T chosen as the ....

Reyner, S.W. (1977) An Analysis of a good Algorithm for the Subtree Problem. SIAM Journal on Computing, 6(4):130--132.

....cases of this problem come directly from those known for subgraph isomorphism [Garey Johnson 79, GT 48] In particular, when A and B are S trees, we have a polynomial algorithm that computes an injective morphism (or projection) from A to B. This algorithm is an extension of the one analyzed in [Reyner 77] On the other hand, when A is an S tree and B is any Sgraph, then the problem is still NP complete. Furthermore, counting the injective morphisms between two rooted S trees is an NP hard problem. Polynomial cases for equivalence come directly from projection. We do not know if there are ....

Reyner S.W., "An analysis of a good algorithm for the subtree problem", SIAM J. Computer, Vol. 6, N4, (December 1977), 730-732.

.... bibliography for various papers in which this technique is applied [14, 12, 13] and to one paper in particular for a detailed discussion of its use [13] 3 The basic techniques Our results are based on a dynamic programming technique first employed in a sequential algorithm of Matula and Reyner [25, 28, 32] for determining if one tree is a subgraph of another. In this section, we begin by outlining Matula s technique and then describe how it can be combined with Brent restructuring to yield a parallel algorithm. We then discuss modifications needed to handle topological embedding. Details of all ....

S. W. Reyner, An analysis of a good algorithm for the subtree problem, SIAM Journal on Computing 6, 4, (1977), pp. 730--732.

....requirement is that a node matches only a node of the same type (i.e. list or bag) For a bag node in te 1 , a complete bipartite match in te 2 is required, whereas for a list node in te 1 , a sublist match in te 2 is required. Since algorithms for finding subtree matches are well known [R77], we omit the detail. Assume that te 1 has n nodes and te 2 has m nodes. The time complexity of testing whether te 1 is weaker than te 2 is O(nm 1:5 ) or better, depending on how good an algorithm one has for a complete bipartite matching [R77] This complexity, however, does not affect the I O ....

....for finding subtree matches are well known [R77] we omit the detail. Assume that te 1 has n nodes and te 2 has m nodes. The time complexity of testing whether te 1 is weaker than te 2 is O(nm 1:5 ) or better, depending on how good an algorithm one has for a complete bipartite matching [R77]. This complexity, however, does not affect the I O cost because the testing is done im memory. 4 Efficiency We now study the efficiency of the algorithm. The efficiency depends not only on database size, but also on factors such as minimum support and pruning strategies. Therefore, it is ....

S.W. Reyner, "An analysis of a good algorithm for the subtree problem", SIAM J. Comput., Vol. 6, No. 4, December 1977

.... a; the value of a(v; w) is set to true if and only if P [v] has a root preserving embedding (of appropriate type for the problem at hand) in T [w] The next algorithm for unordered path inclusion was sketched in [Mat68] A more complete presentation and analysis of the algorithm can be found in [Rey77] and [Chu87] See also [vL90] Using bipartite matching as a subroutine of this algorithm is mentioned also in [HK73] Algorithm 4.14 [Mat68, Rey77] Unordered path inclusion algorithm. Input: Trees P = V; E; root(P ) and T = W; F; root(T ) Output: The nodes w of T such that there is a ....

....next algorithm for unordered path inclusion was sketched in [Mat68] A more complete presentation and analysis of the algorithm can be found in [Rey77] and [Chu87] See also [vL90] Using bipartite matching as a subroutine of this algorithm is mentioned also in [HK73] Algorithm 4. 14 [Mat68, Rey77] Unordered path inclusion algorithm. Input: Trees P = V; E; root(P ) and T = W; F; root(T ) Output: The nodes w of T such that there is a root preserving path embedding of P in T [w] Method: 1. for w : 1; n do 2. comment: Go through the target nodes bottom up; 3. Let w 1 ; ....

S. W. Reyner. An analysis of a good algorithm for the subtree problem. SIAM Journal on Computing, 6(4):730--732, December 1977.

....DLOG, NC, subtree isomorphism. 1 Introduction This note addresses the well known subtree isomorphism problem. In this problem the input consists of two (directed) trees T and T 0 , and one must determine whether T is isomorphic to any subtree of T 0 . The subtree isomorphism problem is in P [7, 9]. The best sequential algorithm for the problem requires O(n 2:5 ) time [7] Regarding the parallel complexity, in [8] an O(log n) time, n processor CRCW PRAM algorithm for the maximal subtree isomorphism problem is given. This is a restricted version of the subtree isomorphism problem in which ....

S. W. Reyner. An Analysis of a Good Algorithm for the Subtree Problem. SIAM Journal on Computing (1977) 6(4):730--732.

....be mapped into q) In 1977 Reyner claimed and in 1988 Verma and Reyner [21, 25] proved the following. Theorem 1.1 ( 25] Given an algorithm for bipartite matching that requires at most O(rs u ) operations, where u 1, the subtree algorithm requires at most O(nm u ) operations. Theorem 1. 2 ([20]) Given an algorithm for bipartite matching that requires at most O(rs) operations, the subtree algorithm will require at most O(nm ln n) operations. In this paper, we present some techniques and several master theorems that can be used to obtain fairly tight upper bounds on the functions T ....

....and since the algorithm for associative matching of linear terms is of the form given above [23] by Theorem 3.2 we have the stated result. Remark. This improves the bound given by Verma and Ramakrishnan [23] by a factor of log n, and the corollary below improves Theorem 1. 2 proved by Reyner in [20] by a factor of ln n. Corollary 6.3. If there is a bipartite matching algorithm of time complexity O(rs) where r sare the sizes of the vertex sets, then the algorithms for rooted subtree isomorphism, AC matching of linear terms, and rooted subgraph homeomorphism on trees are of time complexity ....

S. W. Reyner, An analysis of a good algorithm for the subtree problem, SIAM J. Comput., 6 (1977), pp. 730--732.

....time algorithms exist for the special case of finite rooted trees. Matula and Edmonds [12] describe once such technique, involving the solution of 2n 1 n 2 network flow problems, where n 1 and n 2 represent the number of vertices in the two graphs. The complexity was further reduced by Reyner [40] to O(n 1:5 1 n 2 ) assuming n 1 n2) through a reduction to the bipartite matching algorithm of Hopcraft and Karp [21] If we could transform our directed acyclic shock graphs to finite rooted trees, we could pursue a polynomial time solution to our problem. In the following subsections we ....

.... to a shock tree leads to a property that is invariant to any consistent permutation or reordering of its subtrees (submatrices) 3 After defining a suitable measure of shock distance between corresponding nodes in two shock trees, we present a novel modification to Reyner s algorithm [40] which combines coarse topological matching with shock distance to solve our largest isomorphic subgraph problem in polynomial time. 3.2 Shock Graphs to Shock Trees In this section we present a reduction that takes a DAG representing a shock graph to a unique vertex labeled rooted tree whose size ....

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S. W. Reyner. An analysis of a good algorithm for the subtree problem. SIAM J. Comput., 6:730--732, 1977.

....r 0 ) and from r 0 to LCA(x; r 0 ) to update the current subtree and the root r 0 . Since LCA(x; r 0 ) and these paths can be computed in O(h) time by simply following the parent pointers from each of the vertices x and r 0 , the lemma follows. The following lemma was established in [7]. Lemma 5 Given two rooted trees S and T , it can be determined in time O(st 1:5 ) if S occurs as a subtree of T , where s and t denote the number of vertices in S and T respectively. Using Lemmas 4 and 5, we know that each execution of step (D) takes O log n log log n Theta log 2 ....

....Corollary 1 LCST in bounded degree trees can be approximated in polynomial time to within a ratio of O(n(log log n) log 2 n) 3.4 Parallelization The derandomized algorithm above can in fact be implemented in NC. Only the NC implementation of Lemma 5 needs some explanation. The algorithm in [7] essentially proceeds as follows. Delete the roots of S and T to obtain rooted subtrees S 1 ; S 2 ; S p and T 1 ; T 2 ; T q . Form a p Theta q binary matrix M such that M ij = 1 if and only if S i is a rooted subtree of T j (determined recursively) Check if there is a matching in the ....

S.W. Reyner. An analysis of a good algorithm for the subtree problems. SIAM J. Comput., 6 (1977), pp 730--732.

....are trees, and the problem of matching such representations is of interest for pattern recognition. Applications in domains like computer vision [32, 47, 48, 52, 56] molecular biology [50] and natural language processing [35] abound, and many traditional, discrete algorithms have been developed [29, 31, 33, 46, 51]. On the other hand, no attempt has yet been made to approach such problems within a continuous framework, using analog continuoustime dynamics. The main difficulty is that it is not clear how to map the hierarchy embedded in the representations onto a flat optimization network. In this paper, ....

....Indeed, there is strong evidence that this cannot be the case, for otherwise the polynomial hierarchy would collapse [9, 49] The current belief is that the problem lies strictly between the P and NP complete classes. The subtree isomorphism problem, instead, can be solved in polynomial time [33, 46]. It may be argued therefore that trying to solve these problems by reducing it to the maximum clique problem is an altogether inappropriate choice. In contrast to them, in fact, finding just the cardinality of the maximum clique in a graph is known to be NP complete and, according to recent ....

S. W. Reyner. An analysis of a good algorithm for the subtree problem. SIAM J. Comput., 6:730--732, 1977.

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Reyner, S. W.: An analysis of a good algorithm for the subtree problem, SIAM J. Computing, 6(4), 1977, 730--732.

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S. W. Reyner. An Analysis of a Good Algorithm for the Subtree Problem. SIAM J. Comput., 6:730--732, 1977. J. Zhang et al.

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S. W. Reyner. An analysis of a good algorithm for the subtree problem. SIAM J. Comput., 6:730-732, 1977.

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S. W. Reyner. An Analysis of a Good Algorithm for the Subtree Problem. SIAM J. Comput., 6:730--732, 1977.

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Reyner, S. W.: An analysis of a good algorithm for the subtree problem, SIAM J. Computing, 6(4), 1977, 730--732.

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S.W. Reyner. An analysis of a good algorithm for the subtree problem. SIAM Journal on Computing 6 (1977) 730-732.

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S. W. Reyner, An analysis of a good algorithm for the subtree problem. SIAM Journal on Computing, 6:730--732, 1977.

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