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Topic: Topological sorting

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	Topological sorting - Wikipedia, the free encyclopedia
		In graph theory, a topological sort of a directed acyclic graph (DAG) is a linear ordering of its nodes which is compatible with the partial order R induced on the nodes where x comes before y (xRy) if there's a directed path from x to y in the DAG.
		The canonical application of topological sorting is in scheduling a sequence of jobs.
		The usual algorithm for topological sorting has running time linear in the number of nodes plus the number of edges (Θ(V+E)).
	en.wikipedia.org /wiki/Topological_sorting (343 words)

	[No title]
		Given such a set of jobs with associated priorities, the (topological sorting) problem is to find a sequence of performing the n jobs so that no priority is violated.
		Topological sorting is a seemingly rather difficult problem in general, as n can be quite large, and there might be a rather complicated set of priorities.
		For topological sorting, we do an out-traversal, and keep track of the reverse-order of the order in which the vertices are explored in the associated digraph D. That the reverse-order exploration number gives a proper sequencing of jobs follows immediately from the following Proposition.
	www.ececs.uc.edu /~jpaul/472/lec3.html (803 words)

	XForms Computes
		The topological sort is a method for sequencing the members of a set S such that all constraints in a relation are satisfied.
		For the purpose of applying the topological sort, the computational dependencies in an XForms instance can be represented as a directed graph, or digraph, in which there is a vertex for each instance node and for each desired property of each instance node.
		The sequence of vertices produced by the topological sort dictates a valid order of XForms recalculation in which an expression is re-evaluated after all expressions on whose results it depends and before any expressions that depend on its result.
	pacificcoast.net /~lightning/XFormsComputes_IASTEDFinalManuscript.htm (5412 words)

	Data Structures Using C: CHAPTER 11: TOPOLOGICAL SORTING: AN ARCHETYP
		Topological sorting requires ranking a set of objects subject to constraints on the resultant topology--that is, on the placement of the objects.
		A topological sort is a ranking of the n objects of S that is consistent with the given partial order.
		Suppose a solution to the topological sort problem had been constructed by determining what object should be at the bottom of the ranking.
	www.cis.temple.edu /~wolfgang/cis551/Korsh_CH11/chap11.htm (8161 words)

	[No title]
		The animation illustrates the topological sorting of a directed graph.
		Topological sorting sorts all nodes in sequence so that all children of a node are positioned behind the node in the list.
		Topological sorting can also be used to generate a time plan, a possible sequence of activities which considers all requirements (availability of material - storage - processing - sale).
	www.animal.ahrgr.de /en/Animation55.html (211 words)

	Topic #30: Directed Acyclic Graphs
		The property of dags which is optimal for these problems is their ability to be topologically sorted using depth-first search.
		A topological sort is an ordering of vertices in a dag such that, if there is path from node u to node v, then v appears after u in the ordering.
		A topological sort of a graph can be viewed as an ordering of its vertices along a horizontal line so that all directed edges go in one direction.
	www.cs.mcgill.ca /~cs251/OldCourses/1997/topic30 (1764 words)

	Graph-3 (Site not responding. Last check: 2007-10-14)
		A digraph can be topologically sorted as far as there is no cycle in the graph.
		Topological sorting is usually done by repeatedly mark or remove vertices with no (unvisited) successor from the graph.
		Another solution for topological sorting is to repeatedly removing vertices with no successors, then reverse the list of vertices.
	www.cis.temple.edu /~pwang/223-DS/Lecture/Graph-3.htm (380 words)

	1.3.1 Sorting (Site not responding. Last check: 2007-10-14)
		Sorting is the first step in solving a host of other algorithm problems.
		Sorting is also used to illustrate the standard paradigms of algorithm design.
		The result is that most programmers are familiar with many different sorting algorithms, which sows confusion as to which should be used for a given application.
	www.cs.sunysb.edu /~algorith/files/sorting.shtml (187 words)

	Topological sorting: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-14)
		(a topological sort of a directed acyclic graph directed acyclic graph quick summary:
		In mathematics, a directed acyclic graph, also called a dag, dag, or acyclic directed graph, is a directed graph with no dicycles; that...
		The usual algorithm for topological sorting has running time linear in the number of nodes plus the number of edges (Θ Big O notation quick summary:
	www.absoluteastronomy.com /encyclopedia/t/to/topological_sorting.htm (853 words)

	Topological Sorting
		A topological sort of a directed acyclic graph is an ordering on the vertices such that all edges go from left to right.
		Only an acyclic graph can have a topological sort, because a directed cycle must eventually return home to the source of the cycle.
		Topological sorting proves very useful in scheduling jobs in their proper sequence, as discussed in catalog Section
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK2/NODE70.HTM (299 words)

	1.4.2 Topological Sorting (Site not responding. Last check: 2007-10-14)
		Topological sorting orders the vertices and edges of a DAG in a simple and consistent way and hence plays the same role for DAGs that depth-first search does for general graphs.
		Topological sorting can be used to schedule tasks under precedence constraints.
		These precedence constraints form a directed acyclic graph, and any topological sort (also known as a linear extension) defines an order to do these tasks such that each is performed only after all of its constraints are satisfied.
	www.cs.sunysb.edu /~algorith/files/topological-sorting.shtml (243 words)

	CMSC-341 Fall 2002 Project 5 (Site not responding. Last check: 2007-10-14)
		Recall from class that a strongly connected component in a directed graph is a subset of the vertices such that a path exists from each element of that set to every other element.
		Also, a topological sort of a directed graph is an ordering of the vertices such that if (u, v) is an edge in the graph then vertex u occurs before vertex v in the ordering.
		If the graph has cycles, it cannot be topologically sorted, but some strongly connected component will contain more than one node.
	userpages.umbc.edu /~lbotto1/341/341-Proj5.shtml (1297 words)

	Topological sorting
		We will assume that the objects to be sorted are numbered from 1 to N in any order.
		The input to a topological sorting algorithm is a directed graph with no cycles - i.e.
		Now we remove this object from the input set of objects, the resulting array is again partially ordered, and the process can be repeated until the whole set has been sorted.
	www.geocities.com /SiliconValley/Garage/3323/aat/a_topo.html (362 words)

	[No title]
		Topological sorting maintains a set of nodes with indegree zero, which is initialized to the set of DAG leaves.
		The topological sorting algorithm repeatedly selects a DAG node from the zero-indegree set, appends it to the current schedule, removes it from the DAG and updates the zero-indegree set accordingly (see Figure 3).
		However, the exhaustive enumeration of topological sorts implicitly builds a tree-like representation of all schedules of the DAG, called the selection tree (see Figure 4).
	www.ida.liu.se /~andbe/codegen/codegen.php (3768 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		When the bag is empty, one topological sort has been found.
		Return to the previous version of the bag and select an item which had not been selected first, if there is any, to add to the sorted list.
		When all are discarded, all sorts have been found.¡´ 2´ ðH ð@ ð0“”Þ½h¿ÿ ?ð ÿÿÌff3333ÌÿÌfî9ï éðáð<ðyð( ð#éé ð<ðA¢ ð< ð0$"Ç¿¿Àÿð@pàà ðá¨¯Assignment: Create a job that has 5 or 6 tasks with some precedence rules.
	www.tcnj.edu /~vanhise/honr280/sorting.ppt (191 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		The subsegments have indices 0 to n-1 This was all in Steven's talk, by the way, but he didn't give the details of topological sorting because it is a standard algorithm used worldwide by spreadsheet programs!
		I say ``nominally'' because as Steven explained in his talk, the value might be -1 or 2 in case of rounding error.
		Let subs[n] be the array of new subsegments, indexed by topological sort.
	www.cs.miami.edu /~vjm/csc647/overlay.txt (619 words)

	ACMSOLVER: The ACM-ICPC International Collegiate Programming Contest Training Web Site - Valladolid Online Judge (Site not responding. Last check: 2007-10-14)
		Sorting is commonly used as the initialization step for another algorithm, such as: Binary Search.
		All sorting methods naturally involve a lot of moving around of the things being sorted.
		The bad side of this sorting algorithm is this algorithm need 2*the size of initial array to perform it's task.
	www.daffodilvarsity.edu.bd /acmsolver/Wc3526149007f8.htm (1065 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		We now we give a very nice illustation of solving a seemingly difficult problem with ease given an appropriate numbering generated by depth-first traversal of a digraph having no cycles.
		Suppose you have a sequence of jobs, again indexed by 0, 1,2,..., n-1, and you also have a priority relation between jobs, i.e., some jobs must be done before others.
		The following dag illustrates the reverse-order exploration numbering, and resulting topological sorting.
	www.ececs.uc.edu /~jpaul/471/chap11b.html (1822 words)

	Sorting
		Regarding the post from two years ago about topological sorting (Archive), here is another approach that I came up with.
		Topological sort refers to printing the nodes in a graph such that you print a node before you print any nodes that reference that node.
		Here's the driver code, that sorts the elements according to their weight.
	www.dpawson.co.uk /xsl/rev2/sort2.html (471 words)

	Directed Acyclic Graphs (DAGS) (Site not responding. Last check: 2007-10-14)
		Topological sorting can obviously be useful in the management of construction and manufacturing tasks.
		There may be several different topological sorts for a given DAG, but there must be at least one.
		There are two obvious strategies for topological sorting.
	www.csse.monash.edu.au /~lloyd/tildeAlgDS/Graph/DAG (720 words)

	Surfing With a Linker Alien
		This variable suppressed topological sorting and reverted to the original breadth first sorting.
		This fall back was provided in case applications were found to be dependent upon breadth first sorting, or in case bugs existed in the topological sort mechanism.
		The runtime linker has sorted this family and is in the process of firing the.init's.
	blog.sun.com /roller/page/rie/20050927 (1922 words)

	Strategic Directions in Computational Geometry Working Group Report
		Approximations in the execution of the constructions give rise to approximate results, which may nevertheless be entirely acceptable as long as the maximum error does not exceed the resolution required by the application (in all cases, some more or less coarse grid).
		Similarly, detailed analyses of sorting and hashing are taught in specialized algorithms courses.
		The crucial difference is that sorting and hashing are widely applied in other disciplines (e.g., systems programming), and are presented as fl-box tools in non-algorithmic courses.
	cs.brown.edu /people/rt/sdcr/report/report.html (7069 words)

	5. Query Transformations
		For example, there are any number of reasonable ways for a user to express the topological sorting query in Figure 1(c), including representing the input graph as a list of edges, an adjacency matrix, or a list of neighbors.
		The corresponding topological sorting checklet, on the other hand, might expect the input graph only in a matrix form.
		For example, given the topological sorting query in Figure 6(b)
	www.cs.arizona.edu /people/collberg/Research/Publications/CollbergProebsting2000a/main/node11.html (414 words)

	[No title]
		15-451 Algorithms 10/09/01 Graph algorithms I: * Topological sorting * MST - Prim * MST - Kruskal * Intro to union-find ========================================================================= A lot of algorithmic problems can be modeled as problems on graphs.
		Today we will talk about a couple basic ones and we will continue talking about graph algorithms for a lot of the rest of the course.
		Algorithm: Sort edges by length and examine them from shortest to longest.
	www.cs.cmu.edu /afs/cs/academic/class/15451-f01/www/lectures/lect1009 (696 words)

	CS 341 Assignment #5 (Site not responding. Last check: 2007-10-14)
		A directed graph without cycles can be organized in a linear order called a topological order.
		In a topological order, vertex x precedes vertex y if there is a directed edge from x to y in the graph.
		Create a C++ application that will topologically sort a graph representing the prerequisite structure of the CS courses at Western.
	www.cs.pdx.edu /~annieg/cs341winter01/assignments/assignment5.html (204 words)

	1. Checklets: Result Checkers in AgoVista
		Figure 2 (d), finally, shows a particularly interesting checklet for topological sorting.
		It is therefore not possible for the checklet to simply run a topological sorting procedure on the input graph and compare the resulting list of nodes with the output list given in the query.
		Rather, the checklet must, as shown in Figure 2 (d), first check that every node in the input graph occurs in the output node list, and then check that if node f comes before node t in the output list then there is no path
	www.cs.arizona.edu /~collberg/Research/Publications/CollbergProebsting2000a/main/node6.html (323 words)

	Zeolites, Frameworks, Coordination Sequences & Encyclopedia of Integer Sequences
		In the FOCUS method, coordination sequences are used for identifying and sorting of frameworks.
		The notion of CS was formally introduced by Brunner & Laves (1971) in order to investigate the topological identity of frameworks and of atomic positions within a framework.
		The CS is a number sequence in which the k-th term is the number of atoms in "shell" k that are bonded to atoms in "shell"k-1.
	cci.lbl.gov /~rwgk/EIS/CS.html (580 words)

	Nordic Journal of Computing Bibliography (Site not responding. Last check: 2007-10-14)
		The generalized topological sorting problem takes as input a positive integer
		Given a generalized topological sorting problem, we want to compute a solution, if one exists, and also to test the uniqueness of a given solution.
		The best previous algorithm for the generalized topological sorting problem computes a solution, if one exists, and tests its uniqueness in
	www.cs.helsinki.fi /njc/References/hagerupm1994:38.html (174 words)

	05: Combinatorics
		A graph is a set V of vertices and a set E of edges -- pairs of elements of V. This simple definition makes Graph Theory the appropriate language for discussing (binary) relations on sets, which is clearly a broad topic; a more detailed description is available on the index page for Graph Theory.
		Among the topics of interest are topological properties such as connectivity and planarity (can the graph be drawn in the plane?); counting problems (how many graphs of a certain type?); coloring problems (recognizing bipartite graphs, the Four-Color Theorem); paths, cycles, and distances in graphs (can one cross the Köningsberg bridges exactly once each?).
		There is a significant number of graph-theoretic topics which are the object of complexity studies in computation (e.g.
	www.math.niu.edu /~rusin/known-math/index/05-XX.html (1978 words)

	SWAT.html
		I/O efficient topological sorting has been a long-standing open problem, and our results help to further progress work on this problem.
		Note that this bound matches the best known bound for topological sorting a general digraph using DFS.
		While our results don’t beat the previously known algorithm of [7], we provide further insight into the external-memory topological sorting problems.
	www.uwyo.edu /bshader/swat.html (599 words)

	Order Relation (Site not responding. Last check: 2007-10-14)
		It is based on the fact that a finite poset has at least one minimal element.
		The basic idea of the topological sorting is to first remove a minimal element from the given poset, and then repeat that for the resulting set until no more elements are left.
		The following algorithm formally describes the topological sorting.
	www.cs.odu.edu /~toida/nerzic/content/relation/order/order.html (1120 words)

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