K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM J. Computing, 18(6):1245-1252, December 1989. 65  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of a subject tree S. Aside its theoretical interest (it is a natural generalization of the classical pattern matching in words (WPM) problem) it has applications in several areas of computer science such as design of programming languages [8, 1] theorem proving or computational molecular biology [14, 18]. We can formally de ne this problem in the following way. The tree pattern matching problem. Let S and P be two arbitrary ordered trees, of respective sizes (number of vertices) n and m, whose vertices are labelled with the symbols of an alphabet . For a vertex v of S, there is an occurrence ....

K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM J. Comput., 18(6):1245-1262, 1989.

....In addition, diff based utilities do not recognize hierarchically structured data and are unable to discover movement of data from one location to another. Due to these shortcomings, we are adapting existing research on finding minimum cost edit distances of structured data (see, for example, [5, 23]) This line of research transforms the data into a tree structure. An edit script can change tree A into tree B with a sequence of inserts, deletes and moves of A s nodes so that it looks like B in both shape and content. A minimum cost edit script is one that is least expensive with respect to ....

K. Zhang and D. Shasha, "Simple fast algorithms for the editing distance between trees and related problems," SIAM Journal of Computing, vol. 18, pp. 1245-1262, 1989.

....to thank the Spanish CICyT for partial support of this work through project TIC2000 1703 CO3 02. 2 Such algorithms can e ciently nd the k NN when the points are represented by structures like strings, trees or graphs and the distance functions can be some variants of the edit distance ( 9] [10]) Many general metric space k NN fast search algorithms have been developed trough these years for the special case where k = 1 (Fukunaga and Narendra s [3] Kalantary and McDonald s [5] AESA [8] LAESA [7] TLAESA [6] One of these algorithms has been extended for the general case ....

Zhang, K., Shasha, D.: Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal of Computing (1989) 18 12451262

....to each element and to each attribute. We do not represent the actual values of the elements or attributes in the tree we are only interested in the structural properties of the XML file. 2. 3 Related Work There is considerable previous work on finding edit distances between trees [5 8, 13 17]. Most algorithms in this category are direct descendants of the dynamic programming techniques for finding the edit distance between strings [12] The basic idea in all of these tree edit distance algorithms is to find the cheapest sequence of edit operations that can transform one tree into ....

....in the tree. The work by Chawathe in [5] utilizes these same edit operations and restrictions, but is targeted for situations when external memory is needed to calculate the edit distance. There are several other approaches that allow insertion and deletion of single nodes anywhere within a tree [14 17]. Expanding upon these more basic operators, Chawathe, et al. 7] define a move operator that can move a subtree as a single edit operation, and in subsequent work [6] copying (and its inverse, gluing) of subtrees is allowed. These two operations bear some resemblance to the insert subtree and ....

K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal of Computing, 18(6):1245--1262, December 1989.

....small. 41 General survey on data clustering and classification can be found in [AHD96] Here, we focus on works related to distance metrics for non traditional data types such as trees. The most well known distance measure for trees is Tree Edit Distance and has been extensively studied (e.g. [Tai79, ZS89, CRG96, BCD95, WDC01, CAM02, GJK02, CTZ01]) In tree edit distance problem, a distance between two trees is defined as the summation of the costs needed to convert the source tree to target tree using the pre defined set of edit operations such as insert or delete. Since general tree edit distance problem (also known as Tree to Tree ....

....needed to convert the source tree to target tree using the pre defined set of edit operations such as insert or delete. Since general tree edit distance problem (also known as Tree to Tree editing problem or Tree to Tree Correction problem) for unordered trees is known as NP hard, works in [Tai79, ZS89, CRG96, WDC01] instead try to find an e#cient algorithm for limited cases such as ordered binary trees or trees with additional constraints. The best known algorithm for computing tree edit distance between two ordered trees is by Zhang and Shasha [ZS89] and has the time complexity of roughly O(n ) where ....

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K. Zhang and D. Shasha. "Simple Fast Algorithms for the Editing Distance Between Trees and Related Problems". SIAM J. Comput., 18(6):1245--1262, Dec. 1989. 181

....for secondary structures. The reasons for this are several, but prominent among them are probably that a number of algorithms associated with RNA secondary structure, e.g. predicting secondary structure [167, 112] computing the full partition function [101] comparing secondary structures [161], simultaneous alignment and structure prediction of RNA sequences [131, 45] and stochastic models for RNA secondary structures [128, 38, 76] are unable to handle structures containing pseudoknots. Another reason is that the possible presence of a given pseudoknot involves spatial constraints ....

K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal on Computing, 18(6):1245--1262, Dec. 1989.

....practices. A search of scienti c data available on the Web (e.g. 3] shows that archiving is a ubiquitous problem. Even databases of physical constants [16] are less constant than one might naively imagine. A popular approach to keep all versions of data is to use di based technique [24, 5, 10, 19, 20, 6]. A sequence of edit scripts is stored so that one can roll back to any version. There are two problems with this approach. First, as a document goes through many versions, it becomes increasingly costly to recover an old version by undoing the sequence of edit scripts. The second issue is ....

....concentrate on comparisons between our archive and incremental di approach. The experimental results with cumulative di s are shown in [2] In addition to the choice of incremental di s or cumulative di s, we also have a choice of tree di s or line di s. Tree di s have been extensively studied [10, 11, 19, 20, 6] and we used XML Di [7] which is implemented for XML and is downloadable from the Web. However, when compared with line di , XML Di in OMIM Data V1 incremental diffs gzip(V1 incremental diffs) # of versions 0.00 20.00 40.00 60.00 80.00 100.00 SwissProt Data V1 successive diffs ....

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal of Computing, 18(6):1245-1262, 1989.

....j) separately. This subtree distances can be obtained as a byproduct of computing treedist(i 1 , j 1 ) Using the above ideas we can design a dynamic program that forestdist and treedist mutually call each other, and finally compute treedist( T 1 ] T 2 ) You may see the whole program in [18]. An improved version 6 of this algorithm for the case that each of edit operation has unit cost is presented in [19] Also, the problem of finding a parallel algorithm for this problem is considered in [17] The idea of this algorithm again is lemmas 3. Author in this paper has shown that this ....

K. Zhang, D. Shasha, "Simple fast algorithm for the editing distance between trees and related problems", SIAM J. Comput. 18(6), 1989, 1245-1262.

....children should be considered ordered or unordered. The first non exponential algorithm for solving the ordered tree di#erencing problem is due to [Tai79] who also introduced the concept of edit distance to measure the di#erence between two trees [ZS97] As the majority of subsequent work (e.g. [ZS89, ZS90, CRGW96] on tree di#erencing is based on this distance, we shall examine it in some detail. At the same time we will define the related concepts of edit script and matchings between trees. CHAPTER 4. WORK ON SYNCHRONIZATION, MERGING AND DIFFERENCING 37 b c d e b R b c d e b R b c d a e b R b c ....

....transforms T 1 into T 2 and thus the edit distance is 2. 4.4.2 Edit distance and scripts Suppose T 1 and T 2 are trees and that we have defined a set of tree edit operations. For instance, the edit operations may be the commonly used node insert, delete and update operations (see e.g. Tai79, ZS89] delete(x) Delete the node x, and insert the children of x between the left and right neighbors of x. insert(x, y, n, s, t) Insert the node x as the n:th child of node y, making the children s trough t of y children of x. This is the inverse of the delete( operation. update(x, y) Updates ....

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[chap14] Zhang K. and Shasha D. "Simple fast algorithms for the editing distance between trees and related problems" SIAM J Computing

....isomorphism of non rooted trees. After the pioneer works in [14, 4] a recent paper [16] shows how to solve the problem in O(nm 1:5 = log m) time. ffl Approximate matching. The problem of approximate tree matching asks for a definition of distance between trees. This was cleverly treated in [17] for rooted ordered trees, where the operations of label change, vertex insertion and vertex deletion were defined. The distance between two trees S, T was then defined as the minimum total cost of a sequence of such operations to transform S into T . The transformation, together with some ....

....of label change, vertex insertion and vertex deletion were defined. The distance between two trees S, T was then defined as the minimum total cost of a sequence of such operations to transform S into T . The transformation, together with some relevant variations of the problem, was constructed in [17] in more than quadratic time. Approximate matching of a pattern P with subtrees of T , often called tree pattern matching, was more widely studied. All the works mentioned below apply to rooted ordered trees. 1. Approximate subtree matching with equal vertex degree. The seminal paper is [9] ....

K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal on Computing 18 (1989) 1245-1262. 8

....be matched. Since XML documents can be represented as trees, it is a natural idea to utilize tree to tree correction techniques [10, 14, 17] to detect changes in XML documents. Zhang and Shasha proposed a fast algorithm to find the minimum cost editing distance between two ordered labeled trees [20]. Given two ordered trees T 1 and T 2 , in which each node has an associated label, their algorithm finds an optimal edit script in time O( T 1 T 2 min depth(T 1 ) leaves(T 1 ) min depth(T 2 ) leaves(T 2 ) which is the best known result for the general tree to tree correction ....

....[5] computes the difference between two XML documents. First, it computes hash values for the nodes of both documents using DOMHash [11] and then reduces the size of the two trees by removing identical subtrees (i.e. ones with identical hash values) Second, it uses Zhang and Shasha s algorithm [20] to generate the difference (in terms of an editing script) between the two simplified trees. While using DOMHash to filter out identical subtrees can dramatically reduce the size of the two trees, its use conflicts with the cost model employed by Zhang and Shasha s algorithm. Thus, XMLTreeDiff ....

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K. Zhang and D. Shasha, "Simple Fast Algorithms for the Editing Distance between Trees and Related Problems", SIAM Journal of Computing, 18(6): 1245-1262, 1989.

.... in computational biology, methods for measuring the similarity between ordered labeled trees of bounded degree can be used in the comparison of RNA secondary structures [2, 4, 9] The problem also occurs in evolutionary trees comparison, organic chemistry, pattern recognition, and image clustering [2, 4, 8, 12]. The similarity between two labeled trees can be de ned in various ways analogous to the ways of de ning the similarity between two sequences [5, 7, 8] For example, one can look for the largest maximum agreement subtree, the largest common subgraph, the smallest common supertree, the minimum ....

....two labeled trees can be de ned in various ways analogous to the ways of de ning the similarity between two sequences [5, 7, 8] For example, one can look for the largest maximum agreement subtree, the largest common subgraph, the smallest common supertree, the minimum tree edit distance etc. [2 4, 7, 10, 12]. In [2] Jiang et al. generalized the concept of an alignment between sequences to include labeled trees as follows. An insert operation on a labeled tree adds a new node u which is labeled by a blank symbol (space) not belonging to . The operation either (1) turns the current root of the ....

K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal of Computing 18, 6 (1989), pp. 1245{ 1262.

....in the leftmost diagram of Fig. 1, an example of mapping between two trees. The rightmost diagram includes a sequence of edit operations not constituting a mapping. 3 The Zhang and Shasha s Algorithm We have based our work in the Zhang and Shasha s tree pattern matching algorithm, introduced in [8] and extended with advanced matching features in [9] A major characteristic of this algorithm is its bottom up oriented approach. Given the l keyroots(T ) the set of all nodes in T which have a left sibling plus the root, 3 a b c d e f g a b c d e g f a b c d e f g a b c d e g f Fig. 1. An ....

....j 1 l(j ) 1 . l(j) j l(j) 1 tree(j) 1 1 forestdist(l(i ) l(i) 1, l(j ) l(j) 1) l(i) l( or l(j) l( i 1 1 j 1 1 Fig. 3. The forest distance in Zhang and Shasha s algorithm 5 4 Relating Parsing and Approximate Tree Matching The major question of Zhang and Shasha s algorithm [8], is the tree distance algorithm itself. However, parsing and tree to tree correction are topologically related and it is necessary to understand the mechanisms that cause the phenomenon of tree duplication to get the best performance. A major factor to take into account is the syntactic ....

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Zhang, K., Shasha, D.: Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal on Computing (1989) 18 1245-1262.

....f(P; D; s)g where S is the set of all possible vldc substitutions, and (P; D; s) is the distance ( P ; D) being P the result of apply the substitution s to P . As a consequence, no cost is induced by vldc substitutions. 3 Approximate vldc tree matching The major question of Zhang et al. in [9] [10] is the tree distance algorithm itself. However, parsing and tree to tree correction are topologically related and, a b c d e f g a b c d e g f a b c d e g a b c d e g f a b d g a b c d e g f a b c d e g a b c d e g f a b d g a b c d e g f Mapping without VLDC Mapping with a Path VLDC ....

Zhang, K., Shasha, D.: Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal on Computing (1989) 18 1245-1262.

....represents the immediate enclosure relation between the two edges (see fig. 1) a b c d d = parent(a) Figure 1: Secondary structure of an RNA sequence Consequently, much of the work on comparing the secondary structures of two RNA strings have been modeled as problems of comparing two trees [7, 22, 25, 26]. In this paper, we study several problems in computing the similarity between two RNA strings that take into consideration both the primary sequence and secondary base pairing information provided with the strings. We also investigate the problem of inferring the secondary structure of an RNA ....

....our formulation of the prediction problem appears increasingly relevant. Related work Comparison methods. First we review work on comparison methods developed to estimate distances between RNA secondary structures. Since secondary structures can be represented as trees, there are several papers [7, 22, 25, 26] addressing comparisons of trees. Tree edits are discussed and efficient algorithms are derived in [22, 25, 26] while a new notion of tree alignment is proposed and algorithms developed in [7] Even though these comparison methods compute distances only between secondary structures, the worst case ....

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K. Zhang and D. Shasha, "Simple fast algorithms for the editing distance between trees and related problems, SIAM J. Comput. 18, 12451262 (1989).

....can be solved efficiently. For example, if the only edit operations are insertions and deletions of subtrees, Sel77] presents an efficient solution that is similar in spirit to the algorithm in [WF74] Another formulation, using insertion, deletion, and label update operations is studied in [ZS89] which presents a dynamic programming algorithm to solve the problem. The algorithm can be further improved if we assume all edit operations to have unit cost [SZ90] Our algorithm for change detection in ordered trees, presented in Chapter 4, differs from prior work such as [ZS89] in three ....

....is studied in [ZS89] which presents a dynamic programming algorithm to solve the problem. The algorithm can be further improved if we assume all edit operations to have unit cost [SZ90] Our algorithm for change detection in ordered trees, presented in Chapter 4, differs from prior work such as [ZS89] in three major ways: First, we use a different set of 2.1. CHANGE DETECTION 15 tree edit operations. In particular, in addition to node insertions, deletions, and label updates, we also permit subtree moves. As we will see in Chapter 4, subtree moves significantly improve the usability of the ....

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal of Computing, 18(6):1245--1262, 1989.

....of E(T ) to be the union, over all special subtrees T 0 of T , of the special substrings of T 0 with respect to P (T 0 ) Lemma 3. The number of relevant substrings of T is at most 2jT j log jT j Proof. The proof consists in combining a slight modi cation of Lemma 7 of Zhang and Shasha [10] with our Lemma 2. The analogue of Zhang and Shasha s lemma states that X special subtree T 0 jT 0 j = X v2T collapsed depth of v (1) To prove this equality, note that for each node v, the number of special subtrees T 0 containing v is the collapsed depth of v. Thus v contributes the ....

K. Zhang and D. Shasha, \Simple fast algorithms for the editing distance between trees and related problems, SIAM Journal on Computing 18 (1989), pp. 1245-1262.

....and edge deletions seem to be rather natural to de ne near symmetry. In our setting, if a graph cannot be turned into a symmetric one, then the degree of symmetry is zero. Our das problems are related to graph isomorphism problems. Such problems include tree inclusion [12] and edit distance [20], which have applications to analyzing molecular structures in biology. Given two labelled trees A and B, tree inclusion is to determine whether A can be obtained from B by contracting nodes, whereas edit distance is to determine the minimum number of changes, contracting (or its dual) ....

....be the ordered forest obtained from F by (a) reversing the order of the trees in F , and (b) reversing the order of the children of v for each node v in F . For any ordered tree T , T (i) be the subtree of T rooted at v i , where v 1 ; v 2 ; v n is the postordering of T . Fact 3. 1 (see [13, 20]) For any ordered forests F 1 and F 2 , dist ec (F 1 ; F 2 ) can be computed in polynomial time. Theorem 3.2. das ec for ordered trees can be solved in polynomial time. Proof: Clearly, one of the following two cases holds for any axially symmetric tree T . Case 1: The root of T has 2k 1 ....

K. Zhang, and D. Shasha, Simple Fast Algorithms for the Editing Distance between Trees and Related Problems, SIAM Journal Computing 18(6): 1245-1262, 1989. 12

....trees T 1 and T 2 : T 1 ; T 2 ) minf (M)jM is a mapping from T 1 to T 2 g: 4 Simple Dynamic Programming Algorithm Some more de nitions are necessary before we can actually start with the algorithm. Figure 6 visualizes some of these de nitions. T T[1. 3] T[1. 5] T[1] T[5] T[3] T[2] T[4] T[6] T[1] T[3] T[2] T[4] T[1] T[2] T[3] T[5] Figure 6: Forests 5 De nitions. As de ned earlier, T [i] refers to the ith node in the left to right postorder numbering. As our algorithm will compute not only distances between trees, but also between ordered forests, we de ne T [i: j] to ....

....; T 2 ) minf (M)jM is a mapping from T 1 to T 2 g: 4 Simple Dynamic Programming Algorithm Some more de nitions are necessary before we can actually start with the algorithm. Figure 6 visualizes some of these de nitions. T T[1. 3] T[1. 5] T[1] T[5] T[3] T[2] T[4] T[6] T[1] T[3] T[2] T[4] T[1] T[2] T[3] T[5] Figure 6: Forests 5 De nitions. As de ned earlier, T [i] refers to the ith node in the left to right postorder numbering. As our algorithm will compute not only distances between trees, but also between ordered forests, we de ne T [i: j] to be the forest of nodes ....

K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM J. Comput., 18:1245-1262, 1989.

.... are disjoint) The pseudoknot restriction is thus crucial in algorithms for e.g. structure prediction (Eddy et al. 1994; Knudsen et al. 1999; Nussinov et al. 1980; Sakakibara et al. 1994; Zuker et al. 1981) partition function computations (McCaskill, 1990) comparing secondary structures (Zhang et al. 1989), and simultaneous alignment and structure prediction of RNA sequences (Gorodkin et al. 1997; RNA PSEUDOKNOT PREDICTION IN ENERGY BASED MODELS 3 ) Hairpin loop Hairpin loop Bulge Multibranched loop Internal loop External base I Stacked pairs z Helix FIG. 1. An example ....

Zhang, K. and Shasha, D. (1989). Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal on Computing, 18(6):12451262.

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal of Computing,18(6):1245-1262, 1989

....these structures by labeled ordered trees. Following the problem considered, this tree representation can be rough (considering only the structural patterns) or re ned until an exact coding of the structure is obtained. After some preliminary de nitions and the description of the Zhang Shasha [ZS89] tree edit algorithm, which is on the one hand the reference when dealing with ordered labeled trees comparison, and on the other hand the starting point of our work, this article will present an exact analysis of its complexity. The purpose of this work is also to lead us to a better ....

.... based on principles that are frequently used for sequence comparison, especially the global alignment (Needleman Wunsch [NW70] and the notion of edit distance or Levenshtein distance rst introduced in [Lev66] This article mainly focus on the complexity analysis of the Zhang Shasha algorithm [ZS89], which is the reference when dealing with ordered labeled trees comparison, and will be divided in three parts. We will rstly give some preliminaries concerning the ways to get di erent tree representations from an RNA secondary structure. As a base for our analysis, we shall then present this ....

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. In SIAM Journal on Computing, 18(6), pages 1245-1262, 1989. 16

....S, how to compute a tree with the largest number of edges which is obtainable through a sequence of edge contractions from both restrictions T 1 jA and T 2 jA for some subset A S. An edge contraction is shown in Figure 1(b) and is also referred to as deletion of an internal node in tree edit [20]. Let s call the second problem maximum agreement subtree with edge contractions (MAST EC) Here we settle these two problems by showing that MAST for three unbounded degree trees cannot be approximated within ratio 2 log n in polynomial time for any 1, unless NP DTIME[2 polylog n ] ....

....the two resulting trees T 0 1 and T 0 2 have the same structure, i.e. they are identical if the labels are ignored, and then overlaying T 0 1 on T 0 2 . Here inserting a new node u under an existing internal node v means we make u the parent of some children of v and then u a child of v [20]. Thus, each node in the alignment A is labeled with a pair of (possibly null) symbols, one from T 0 1 and the other from T 0 2 . A score scheme is de ned for each pair of labels. The value of the alignment A is the sum of the scores of all pairs of opposing labels. An optimal alignment is one ....

K. Zhang and D. Shasha, Simple fast algorithms for the editing distance between trees and related problems, SIAM J. Comput. 18, 1245-1262, 1989. 19

.... nodes) We believe the presented techniques can also contribute to comparison and search of 2 D and 3 D (macro)molecules in protein and DNA structures [14] Comparison to Past Research This paper generalizes the work on the edit distance between strings [6, 11, 13, 16, 20, 21, 25] and trees [19, 28, 29]. Various kinds of constrained and generalized edit distance on strings and trees have been developed [1, 9, 10, 17, 27] Our degree 2 distance, when applied to unordered trees, is a restricted form of the constrained distance previously reported in [27] When applied to ordered trees, the ....

K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM J. Comput., 18(6):1245--1262, Dec. 1989.

.... algorithms is O(jP j Theta jDj Theta min(depth(P ) leaves(P ) Theta min(depth(D) leaves(D) where jP j and jDj are the number of nodes respectively of the pattern P and the data tree D) the same as for the best approximate tree matching algorithm without VLDC s previously reported in [25]. This work was supported in part by the National Science Foundation under Grants IRI 8901699 and CCR 9103953, by the Office of Naval Research under Grants N00014 90 J 1110 and N00014 91 J 1472, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0046373, by the New ....

....that converts one tree to the other [19] In [21] Tai presented an algorithm to solve this problem in time O(jT 1 j Theta jT 2 j Theta (depth(T 1 ) 2 Theta (depth(T 2 ) 2 ) where jT 1 j and jT 2 j are the number of nodes of trees T 1 and T 2 respectively. More recently, Zhang and Shasha [25] developed a faster algorithm that computes the distance between two trees in time O(jT 1 j Theta jT 2 j Theta min(depth(T 1 ) leaves(T 1 ) Theta min(depth(T 2 ) leaves(T 2 ) and space O(jT 1 j Theta jT 2 j) Using suffix trees, they developed a fast parallel algorithm for the unit cost ....

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM J. Computing, 18(6):1245--1262, Dec. 1989.

....biology and natural language processing, including the representation of images [12] patterns [2, 10] and secondary structures of RNA [14] They are frequently used in other disciplines as well. A large amount of work has been performed for comparing two trees based on various distance measures [4, 9, 11, 21, 25]. 16, 19, 27] recently generalized one of the most commonly used distance measures, namely the edit distance, for both rooted and unrooted unordered trees. These works laid out a foundation that is useful for comparing graphs [15, 24] In this paper we extend the previous work by considering the ....

....nodes numbered i to j inclusive (see Fig. 3) If i j, then T [i: j] The definition of mappings for ordered forests is the same as for trees. Let F 1 and F 2 be two forests. The distance from F 1 to F 2 , denoted Delta(F 1 ; F 2 ) equals the cost of a minimum cost mapping from F 1 to F 2 [25]. Let F = T [i: j] A set S of nodes of F is said to be a set of consistent subtree cuts in F if (i) t[p] 2 S implies that i p j, and (ii) t[p] t[q] 2 S implies that neither is an ancestor of the other in F . We use Cut(F; S) to represent the sub forest F with subtree removals at all nodes in ....

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal on Computing, 18(6):1245--1262, Dec. 1989.

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM J. Computing, 18(6):1245-1252, December 1989. 65

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K. Zhang and D. Shasha. 1989. Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal on Computing, 18(6):1245--1262.

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K. Zhang and D. Shasha, "Simple fast algorithms for the editing distance between trees and related problems" SIAM Journal of Computing, Vol 18-6, (1989), p. 12451262.

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal on Computing, 18(6):1245--1262, 1989. 312

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM J. Comput., 18(6):1245--1262, 1989.

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K. Zhang and D. Shasha, "Simple fast algorithms for the editing distance between trees and related problems" SIAM Journal of Computing, Vol 18-6, (1989), p. 1245-1262.

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM J. Comput., 18(6):1245--1262, 1989.

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal of Computing, 18(6):1245--1262, 1989.

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K. Zhang and D. Shasha, "Simple Fast Algorithms for the Editing Distance between Trees and Related Problems," SIAM J. Computing, vol. 18, no. 6, pp. 1245-1262, 1989.

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal of Computing, Vol 18-6 (1989), p. 1245-1262.

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K. Zhang, R. Stgatman, and D. Shasha. Simple fast algorithm for the editing distance between trees and related problems. SIAM Journal on Computing, 18(6):1245--1262, 1989.

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal on Computing, 18(6):1245--1262, 1989.

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K. Zhang, D. Shasha, "Simple, Fast Algorithm For the Editing Distance Between Trees and Related Problems", SIAM Journal on Computing (18), 1989, pp. 1245-1262.

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal on Computing, 18(6):1245--1262, December 1989. 10

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM J. Computing, 18(6):1245--1262, December 1989.

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K. Zhang, D. Shasha. "Simple fast algorithms for the editing distance between trees and related problems". SIAM Journal of Computing, vol. 18, p.1245-1262, December 1989.

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal on Computing 18 (1989) 1245-1262. 18

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal on Computing, 18(6):1245-1262, 1989.

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K. Zhang and D. Shasha, Simple fast algorithms for the editing distance between trees and related problems. SIAM J. of Comp., 18:1245--1262, 1989.

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Kaizhong Zhang and Dennis Shasha. Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal of Computing, 18:1245--1262, 1989.

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K. Zhang and D. Shasha. Simple Fast Algorithms for the Editing Distance Between Trees and Related Problems. SIAM J. Comput., 18(6):1245--1262, 1989. 6

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K. Zhang, R. Stgatman, and D. Shasha. Simple fast algorithm for the editing distance between trees and related problems. SIAM Journal on Computing, 18(6):1245--1262, 1989.

No context found.

K. Zhang and D. Shasha, `Simple fast algorithms for the editing distance between trees and related problems', SIAM J. Computing, 18, (6), 1245--1262 (1989).

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K. Zhang and D. Shasha. Simple fast algorithms for the editing distance between trees and related problems. In SIAM Journal on Computing, volume 18, pages 1245--1262. 1989.

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