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Topic: Transitive closure

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Transitive relation

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	Transitive Closure and Reduction
		Transitive closure is fundamental in propagating the consequences of modified attributes of a graph G.
		Transitive reduction (also known as minimum equivalent digraph) is essentially the inverse operation of transitive closure, namely reducing the number of edges while maintaining identical reachability properties.
		The transitive closure of G is identical to the transitive closure of the transitive reduction of G.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE163.HTM (1011 words)

	Transitive closure - Wikipedia, the free encyclopedia
		In mathematics, the transitive closure of a binary relation R on a set X is the smallest transitive relation on X that contains R.
		Similarly, the class L is first-order logic with the commutative, transitive closure.
		The transitive reduction of a relation R is the smallest relation having the transitive closure of R as its transitive closure.
	en.wikipedia.org /wiki/Transitive_closure (523 words)

	Welcome LP
		For example, "descendent of" is a typical transitive relation: if "Marry" is a decscendent of "Paul" and "Paul" is a descendent of "Jim" than clearly "Marry" is a descendent of "Jim".
		By the same token, many famous relations are not transitive: eg in general "mother of" is not a transitive relation: { "Lynn" is the mother "Marry" and "Marry" is a the mother of "Paul" } does not entail that "Lynn" is the mother of "Paul".
		Hence, "successor of" is the transitive closure of "immediate successor of".
	www.tutor.ms.unimelb.edu.au /transitivity/transitivity.html (414 words)

	Three Variants for -Moves
		Moreover, the transitive closure algorithm is fired only once (for the full graph), whereas the following two variants call a specialised transitive closure algorithm possibly many times.
		Instead of computing the transitive closure of a given graph, this algorithm only computes the closure for a given set of states.
		The approach in which the transitive closure is computed for one state at a time is defined by the following definition of the epsilon_closure function.
	odur.let.rug.nl /~vannoord/papers/fsmnlp98/node4.html (504 words)

	Galaxy (Site not responding. Last check: 2007-10-08)
		Transitive closure is the concept of clarifying the existence of meaningful connection or relationship (direct or indirect) between two system elements.
		The dynamic aspect of the dynamic transitive closure analysis is simply that the analysis is conducted dynamically, by progressively eliminating from consideration direct relationships which do not meet minimum criteria.
		The smaller values in the dtca.dat file identify the presence of close edges which must be removed to break transitive closure, indicating a strong co-correlation present in the overall graph.
	www.osc.edu /hpc/software/apps/galaxy.shtml (451 words)

	Analysis of Algorithms: Lecture 19
		Then the "additive closure" of, for example, { 2 }, would be the set of even numbers, the additive closure of { 1, -1 } would be the set of integers, and so forth.
		The transitive closure of a graph is the result of adding the fewest possible edges to the graph such that it is transitive.
		At the end, the transitive closure is a graph with a complete subgraph (a clique) involving vertices 1, 2, 3, and 4.
	camino.rutgers.edu /ut/utsa/cs3343/lecture19.html (1479 words)

	Transitive Closure on the Instruction Systolic Array
		Furthermore, a generalization of the transitive closure algorithm is implemented to solve other path problems, such as the shortest path problem.
		Systolic solutions for the transitive closure and the shortest path problem are presented by Kung, Lo and Lewis [3].
		The problem of computing the transitive closure of a graph may be viewed as a special case of the algebraic path problem.
	www.inf.fh-flensburg.de /lang/papers/trans/transcl.htm (3699 words)

	Transitive closure in relational database systems (Site not responding. Last check: 2007-10-08)
		The computation of the transitive closure as a binary relation is a common requirement for many applications.
		However, it is provable that transitive closure is not possible in traditional SQL [1] and hence are a problem in relational database management systems.
		Another idea would simply be to store the transitive closure in a separate table and to try to keep this table consistent with the actual graph table during updates.
	www.stefan-wagner.info /cs/trans_clos.php (1488 words)

	1.4.5 Transitive Closure and Reduction (Site not responding. Last check: 2007-10-08)
		Problem: For transitive closure, construct a graph G'=(V,E') with edge (i,j) \in E' iff there is a directed path from i to j in G.
		For transitive reduction, construct a small graph G'=(V,E') with a directed path from i to j in G' iff (i,j) \in E.
		Transitive closure is fundamental in propagating the consequences of modified attributes of a graph G.
	www.cs.sunysb.edu /~algorith/files/transitive-closure.shtml (251 words)

	Transitive set - Wikipedia, the free encyclopedia
		The transitive closure of a set A is the smallest (with respect to inclusion) transitive set B which contains A.
		Transitive classes are often used for construction of interpretations of set theory in itself, usually called inner models.
		An ordinal number may be defined as a transitive set whose members are also transitive.
	en.wikipedia.org /wiki/Transitive_set (200 words)

	Transitive closure
		Finding the transitive closure of a directed graph is an important problem in many computational tasks.
		It is required, for instance, in the reachability analysis of transition networks representing distributed and parallel systems and in the construction of parsing automata in compiler construction.
		Recently, efficient transitive closure computation has been recognized as a significant subproblem in evaluating recursive database queries, since almost all practical recursive queries are transitive.
	www.cs.hut.fi /~enu/tc.html (480 words)

	Algorithms::Graphs::TransitiveClosure - Calculate the transitive closure. (Site not responding. Last check: 2007-10-08)
		The subroutine floyd_warshall takes a directed graph, and calculates its transitive closure, which will be returned.
		Note than in specific cases, when the graph can be embedded on surfaces of bounded genus, or in the case of sparse connection matrices, faster algorithms than cubed in the number of vertices exist.
		The space used by this algorithm is at most quadratic in the number of vertices, which is optimal as the resulting transitive closure can have a quadratic number of edges.
	www.foad.org /~abigail/Perl/Algorithms/Graphs/TransitiveClosure.html (496 words)

	Transitive Closure of a Graph - Warshall's Algorithm
		To find all such round about or via-paths there is a procedure to be followed, known as the transitive closure, which results in a matrix showing all those via-paths, if ever exist.
		This way of finding the transitive closure was derived by Warshall and thus the name, Warshall's algorithm.
		Well, there are other ways to find the transitive closure but Warshall has actually simplified them and hence I have referred that only.
	datastructures.itgo.com /graphs/transclosure.htm (347 words)

	Transitive closure (Site not responding. Last check: 2007-10-08)
		The program calculates transitive closure of a relation represented as an adjacency matrix.
		For calculating transitive closure it uses Warshall's algorithm.
		Output: The adjacency matrix T of the transitive closure of R. Procedure:
	www.cs.nmsu.edu /~ipivkina/TransClosure/index.html (246 words)

	Reference Counting Can Handle Cycles
		For each object y in the transitive closure of x: if y's current reference count exceeds its temporary reference count, then it must have a link from "outside", so it is live by the previous axiom.
		All objects in the transitive closure of x which were not demonstrated to be live by the previous step, are dead.
		Since the transitive closure of these objects is contained within the transitive closure of x, we do not need to recurse the procedure: these decrements cannot affect the live/dead status of any object under consideration.
	c2.com /cgi/wiki?ReferenceCountingCanHandleCycles (565 words)

	Transitive Closure
		For example, Fosco Baggins is the father of Drogo, and Drogo is the father of Frodo; therefore, Fosco is an ancestor of Frodo by transitivity.
		More formally, transitive closure is an extension or superset of a binary relation such that whenever (a,b) and (b,c) are in the extension, (a,c) is also in the extension.
		For example, the transitive closure of a BOM input is all pairs of containing/contained items, be they contained directly—e.g., if a cake contains cream, return (cake, cream)—or indirectly—e.g., if a cake contains cream and cream contains sugar, a cake transitively contains sugar; return (cake, cream), (cream, sugar), (cake, sugar).
	www.sqlmag.com /Article/ArticleID/46117/Transitive_Closure.html (1314 words)

	A transitive closure function for XPath (Site not responding. Last check: 2007-10-08)
		xlinkit and my thesis, I have had to deal with several cases where I had to compute the transitive closure of elements in XML files inside an xpath expression.
		My thesis is to do with consistency checking, and one of the rules in the UML is that no class can be a superclass and a subclass of another at the same time.
		node 1 node 2 node 3 node 4
	www.cs.ucl.ac.uk /staff/c.nentwich/closure (696 words)

	[No title]
		Specifying the base_interfaces as I suggest would match the operations and attributes fields of the same structs, which are already explicitly specified to contain all operations and attributes respectively from the transitive closure of the inheritance graph.
		The operations and attributes fields of the FullInterfaceDescription structure include descriptions of all of the operations and attributes in the transitive closure of the inheritance graph of the interface being described." to add a sentence defining the base_interfaces member, like this: "The describe_interface operation returns a FullInterfaceDescription describing the interface, including its operations and attributes.
		The operations and attributes fields of the FullInterfaceDescription structure include descriptions of all of the operations and attributes in the transitive closure of the inheritance graph of the interface being described.
	www.omg.org /issues/issue9140.txt (633 words)

	Boost Graph Library: Transitive Closure
		This maps each vertex in the input graph to the new matching vertices in the output transitive closure graph.
		The set of vertices adjacent to v in the transitive closure G* is the same as the successor set of v in the original graph G.
		Computing the transitive closure is equivalent to computing the successor set for every vertex in G.
	www.boost.org /libs/graph/doc/transitive_closure.html (599 words)

	Detecting transitive behavior in reversible one dimensional cellular automata
		As in the connectivity relation defined in Table 1, we can do the transitive closure of the transition relation.
		Using the principal diagonal and the transitive closure of the transition relation, we have the following result:
		In this way, we have defined simple matrix methods that using the properties of block permutations and transitive closures detect periodical and transitive behavior.
	delta.cs.cinvestav.mx /~mcintosh/comun/summer2000/seck/node12.html (490 words)

	Efficient transitive closure computation in large digraphs (Site not responding. Last check: 2007-10-08)
		Two new transitive closure algorithms that are based on detecting the strong components are presented.
		The representation generalizes a previous method for compressing the transitive closure of an acyclic graph.
		The experiments also indicate that with the interval representation and the new algorithms, the transitive closure can be computed typically in time linear to the size of the input graph.
	www.cs.hut.fi /~enu/thesis.html (256 words)

	Transitive Closure by Graph Powering (Site not responding. Last check: 2007-10-08)
		The transitive closure T(G) of a given graph G connects vertices u and v iff there is a path in G from u to v.
		This animation finds the transitive closure of a graph by taking its adjacency matrix and raising it to the nth power, where n is the number of vertices in G. When we raise the graph to the kth power, we add exactly the edges which represent paths of length k in the original graph.
		A computationally cheaper way to find transitive closure is to use Warshall's algorithm, but graph powering also allows us to count how many paths there are of different lengths.
	www.cs.sunysb.edu /~skiena/combinatorica/animations/graphpower.html (198 words)

	Per subset and per state
		In both of the two integrated approaches, the subset construction algorithm is initialised with an agenda containing a single subset which is the
		Note that we make sure that the transitive closure computation is only performed once for each input state, by memorising the closure function; the full computation is memorised as well.
		The motivation for the per state variant is the insight that in this case the closure algorithm is called at most Q times.
	odur.let.rug.nl /vannoord/papers/cl00/fsmnlp/node9.html (396 words)

	[Abstract] Transitive Closure Revisited
		In this paper, we propose a new algorithm for computing transitive closures.
		The main idea behind this is tree labeling and graph decomposition, based on which the transitive closure of a directed graph can be computed in O(eb) time and O(nb) space, where n is the number of the nodes of the graph, e is the numbers of the edges, and b is the graph's breadth.
		Especially, this method hints a new way to speed up the computation of recursion in relational databases.
	www.actapress.com /Abstract.aspx?paperId=15319 (111 words)

	Method and apparatus for estimating transitive closure and reachability (US5752241)
		Method and apparatus for estimating transitive closure and reachability
		The invention relates to method and apparatus for computing transitive closure and reachability in directed graphs.
		Edith Cohen, "Estimating the Size of the Transitive Closure in Linear Time", Proceedings 35th Annual Symposium on Foundations of Computer Science, pp.
	www.delphion.com /details?pn=US05752241__ (436 words)

	[exslt] exslt - transitive closure (Site not responding. Last check: 2007-10-08)
		I cannot live with at least two of those.
		here is a abstracted snippet of an application of my transitive closure function closure(foo,$myset[@variable=current()/@variable]) i.e.
		So: - I need the current node to be set, otherwise I cannot refer to the current node that the transitive closure is visiting (since all relative expressions are relative to the context node, which is a 'candidate next node').
	lists.fourthought.com /pipermail/exslt/2001-May/000141.html (240 words)

	Parallel Transitive Closure and Point Location in Planar Structures
		``Parallel Transitive Closure and Point Location in Planar Structures,'' SIAM Journal on Computing, 20(4), August 1991, 708-725.
		A shortened version appears in ``Optimal Parallel Algorithms for Transitive Closure and Point Location in Planar Structures,'' Proceedings of the 1st Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA '89), Sante Fe, N. M., June 1989, 399-408.
		Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar st-graphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs.
	www.cs.duke.edu /~jsv/Papers/catalog/node172.html (150 words)

	transitive reduction (Site not responding. Last check: 2007-10-08)
		Definition: The transitive reduction of a directed graph G is the directed graph G' with the smallest number of edges such that for every path between vertices in G, G' has a path between those vertices.
		HTML page formatted Mon Sep 11 09:46:08 2006.
		Paul E. Black, "transitive reduction", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology.
	www.nist.gov /dads/HTML/transitiveReduction.html (111 words)

	Threads, Locking, Parallelism, and Concurrency
		Lower Bounds for Dynamic Transitive Closure, Planar Point Location, and Parentheses Matching
		Fully Dynamic Transitive Closure in Plane Dags with One Source and One Sink
		This list was kindly provided by Terry Lambert.
	people.freebsd.org /~fsmp/SMP/threads.html (91 words)

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