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Topic: Minimal spanning tree

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	Minimum spanning tree - Biocrawler (Site not responding. Last check: 2007-10-29)
		Given a connected, undirected graph, a spanning tree of that graph is a subgraph which is a tree and connects all the vertices together.
		The fastest minimum spanning tree algorithm to date was developed by Bernard Chazelle, and based on Borůvka's.
		A related graph is the k-minimum spanning tree (k-MST) which is the tree that spans some subset of k vertices in the graph with minimum weight.
	www.biocrawler.com /encyclopedia/Minimal_spanning_tree (649 words)

	Geometry in Action: Minimum Spanning Trees (Site not responding. Last check: 2007-10-29)
		Pope use geometric minimum spanning trees to model locality of particle interactions in turbulent fluid flows.
		Minimal spanning tree analysis of fungal spore spatial patterns, C.
		A minimal spanning tree analysis of the CfA redshift survey.
	www.ics.uci.edu /~eppstein/gina/mst.html (289 words)

	Random minimal spanning tree - Wikipedia, the free encyclopedia
		In mathematics, random minimal spanning tree, or random MST, is a model (actually two related models) for a random tree (see also minimal spanning tree).
		It might be compared against the uniform spanning tree, a different model for a random tree which has been researched much more extensively.
		This tree (denote it by T) is also a spanning tree for the unweighted graph G.
	en.wikipedia.org /wiki/Random_minimal_spanning_tree (285 words)

	minimal spanning trees and Delaunay triangulations
		Minimal spanning trees ---------------------- A spanning tree S of a graph G is a connected acyclic subgraph of G that includes all of G's nodes.
		A minimal spanning tree of the graph is a graph whose cost is the minimal spanning-tree cost.
		The minimal spanning tree of a set of points is the minimal spanning tree of the complete graph of those points, that is, the graph in which there is one arc connecting each pair of nodes.
	lists.canonical.org /pipermail/kragen-tol/1999-July/000436.html (1373 words)

	ECE242S - Algorithms and Data Structures
		Kruskal invented this algorithm in 1956, for determining the minimal spanning tree of a graph.
		Prim invented this algorithm in 1957, for determining the minimal spanning tree of a graph.
		It is clear that both these algorithms produce acyclic spanning trees as their solutions, containing one fewer edges than the number of nodes in the graph.
	www.ecf.toronto.edu /apsc/courses/ece242/2003fall/notes/MCST_examples/notes_graph.htm (1056 words)

	JAVA code for Grand Tour and MST - Software documentation (Site not responding. Last check: 2007-10-29)
		Minimal Spanning Tree If you click the top left button on the grand tour window, the minimal spanning tree edges will be drawn on the window connecting points.
		Also the empirical cumulative distribution of the minimal spanning tree edge lngths will be plotted on the new second graph.
		When the minimal spanning tree is constructed, the observation numbers will be printed next to the points on the main grand tour graph.
	www.public.iastate.edu /~dicook/research/Sungsoo.Kim/gt-mst.html (328 words)

	CS 207
		A minimal spanning tree is one whose total edge weights is the minimum of all the possible spanning trees for the graph.
		ShortestDist is the shortest distance from a node in the tree to the one in the fringe.
		The next node to be added to the tree will be node 3 since it is closest of any of the nodes in the fringe to any of the nodes in the tree.
	www.cs.uah.edu /~rcoleman/Graphs/MinSpanTree.html (1016 words)

	Suprema 2001: Graphs and Complexity - Trees, Forests, Cycles and Connected Graphs (Site not responding. Last check: 2007-10-29)
		A Tree is a connected graph that contains no cycles Figure 5 below shows an example of a tree, but figure 6 is not a tree since it contains a cycle.
		A minimal spanning tree is a tree that contains every node of a given connected graph, with the edges of the tree being actual edges of the graph, and the sum of the weights on the edges is minimal.
		Minimal spanning trees are not unique, a graph can have many of them, or it may even have none.
	www.doc.ic.ac.uk /~atr99/suprema/trees.html (270 words)

	Minimal Spanning Trees/Shortest Path Tree
		There are two main approaches to finding a minimal-sum-of-weights spanning tree (minimal, rather than minimum, since there may be several trees with the same smallest sum of weights): we will examine the nearest-neighbor method, which is closely related to the technique for finding a shortest path in a weighted graph.
		We'll generate a spanning tree by accumulating a set of connected edges, in such a way as to cause the sum of the edge weights to be as small as possible (for a given weighted graph).
		For each vertex, there should be a status field (tree, fringe, or unseen are the three possible values), a field for the nearest neighbor in the tree (for fringe vertices), and the weight on that edge.
	www.cs.rutgers.edu /~kaplan/503/handouts/nnmst.html (928 words)

	ECE242S - Algorithms and Data Structures
		Kruskal invented this algorithm in 1956, for determining the minimal spanning tree of a graph.
		It is clear that both these algorithms produce acyclic spanning trees as their solutions, containing one fewer edges than the number of nodes in the graph.
		Two operations of concern in computing a minimal spanning tree are to find if two vertices are connected by an edge (ie is (u,v) in E), and to enumerate all the adjacent vertices of a given vertex.
	www.ecf.utoronto.ca /apsc/courses/ece242/2003fall/notes/MCST_examples/notes_graph.htm (1056 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		A spanning tree is a tree that connects all of the nodes of a network (again, with no loops).
		A Minimal Spanning Tree is the spanning tree for a given network which represents the least total distance (or cost, the two are often related in network problems) required to connect all the nodes.
		An application of a minimal spanning tree is the generation of a network of paved roads linking several rural towns where it is permissible to travel through one or more towns to get to a destination but the goal is to minimize the miles of pavement required.
	web.umr.edu /~meinert/emgt382/lec18.doc (485 words)

	COMP 228 Final Exam, Winter 2005
		The minimal spanning tree of a graph is the set of edges of minimal weight that connect all the nodes in the graph.
		A spanning tree of minimal weight, where the weight is the sum of the weights of the edges in the tree.
		Minimal Spanning Tree Edges: [a, c, 4], [b, g, 1], [c, b, 2], [c, d, 3], [c, f, 10], [d, e, 6].
	www.yukoncollege.yk.ca /~ttopper/COMP228/exam.html (1384 words)

	Prim's algorithm
		This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized.
		The output Y of Prim's algorithm is a tree, because the edge and vertex added to Y are connected to other vertices and edges of Y and at no iteration is a circuit created since each edge added connects two vertices in two disconnected sets.
		Therefore, Y is a spanning tree for P.
	www.brainyencyclopedia.com /encyclopedia/p/pr/prim_s_algorithm.html (589 words)

	The Minimum Spanning Tree Problem
		A spanning tree for a graph G is a subgraph of G that contains every vertex of G and is a tree.
		A minimal spanning tree for a weighted graph is a spanning tree that has the least possible total weight compared to all other spanning trees for the graph.
		Another application of the minimum spanning tree is an approximate solution to the traveling salesman problem (recall that finding the best solution to this problem is N-P hard).
	people.uncw.edu /tompkinsj/133/graphTheory/mst.htm (633 words)

	Checkpoint - Course 2, Unit 5 (Site not responding. Last check: 2007-10-29)
		Minimal spanning trees and shortest paths are both the "shortest" in some sense, which could mean cheapest, or least in some way.
		But a minimal spanning tree is a shortest tree that includes all vertices as opposed to a shortest path that generally does not include all vertices.
		In one case, you want a minimal spanning tree, and in the other, you want a minimal spanning circuit (starts and ends at the same vertex).
	www.wmich.edu /cpmp/parentsupport/checkpoints/c2u5.html (703 words)

	Class notes CS251B -- Winter 1997
		Spanning tree: a free tree on V (thus having V-1 edges that are a subset ofE).
		Minimum Spanning tree: the spanning tree with minimal total weight, where the weights of the edges picked are summed to obtain a total weight..
		In this graph, the red lines are the minimum spanning tree and the fl lines are the edges which exist but are not part of the minimum spanning tree..
	www.cs.mcgill.ca /~cs251/OldCourses/1997/topic28 (1859 words)

	Dotzel: Minimal Spanning Tree (Forest) in Oberon-2
		Trees and forests are graphs of a very special kind that play an important role in many applications.
		An undirected graph G is a tree if and only if there is exactly one path between every pair of vertices in G. Since n-1 edges are the fewest possible to connect n vertices and a tree with n vertices contains exactly n-1 edges, trees can be thought of as graphs that are minimally connected.
		Such a tree is called a minimum spanning tree." written by Helmut Sperber, Sep-1982.
	www.modulaware.com /mdlt55.htm (1206 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		The minimal spanning tree is one such a subtree.
		Since minimal spanning trees connect multivariate points they exist in the dimension of the data set.
		One difference is that the genes have been clustered into 6 groups and single plots shows the spanning trees in color for the 6 groups.
	www.galaxy.gmu.edu /~dcarr/eda/assign/a12_transformations.ssc (1841 words)

	Configuring Spanning Tree (Site not responding. Last check: 2007-10-29)
		Spanning tree is a link management protocol that provides path redundancy while preventing undesirable loops in the network.
		Spanning tree defines a tree with a root switch and a loop-free path from the root to all switches in the extended Layer 2 network.
		Note If you disable spanning tree for a TrBRF using the set spantree disable command, the logical ports of the TrBRF are placed in the forwarding state regardless of the state you configured using the set spantree portstate command.
	www.cisco.com /univercd/cc/td/doc/product/lan/cat5000/rel_5_2/config/spantree.htm (4569 words)

	American Mathematical Society :: Feature Column
		The goal is to try to find a spanning tree of the graph which has the property that the sum of the weights of the edges in the tree is a minimum.
		The meaning of being a spanning tree is that the tree includes all of the vertices of the original graph.
		In this model we seek that spanning tree of the original graph such that the sum of the weights on the edges of the spanning tree T together with the sum of the weights at the vertices of T is a minimum.
	www.ams.org /featurecolumn/archive/trees.html (4709 words)

	Local Steiner Improvement
		We claim that this is not overly restrictive since dangling lines are never found in shortest spanning trees and there is little difference in spanning trees for a set of points containing two which share a coordinate and a set where the points nearly share a coordinate.
		For example, the tree in figure 1c can be constructed by drawing the edge with a horizontal line through a, connecting this to the point b with the edge , and then filling in the edge .
		Since we began with a particular tree (the minimal spanning tree) and required a smaller tree to be found before iterating once more, this is a finite process.
	www.cs.engr.uky.edu /~lewis/research/Papers/Steiner-Local/local.html (4174 words)

	Graphs in Image Analysis - Introduction
		A minimal spanning tree is a spanning tree of a graph, such that the sum of all edge costs is minimal.
		In general, the minimal spanning tree of an image graph is not unique.
		The minimal spanning tree can be implemented for image graphs with 4- or 8-neighborhood.
	www.prip.tuwien.ac.at /Research/Graphs/mst/node04.html (111 words)

	Steiner Points and Trees (Site not responding. Last check: 2007-10-29)
		Example: the minimal spanning tree connecting the three vertices of a unit equilateral triangle has length 2.
		An additional point that reduces the length of the spanning tree is called a steiner point.
		In a minimal steiner tree, the steiner points of degree 3 all have angles of 120°.
	www.mathreference.com /gph,stein.html (544 words)

	EIMM_C07 (Site not responding. Last check: 2007-10-29)
		s you have learned, a minimum spanning tree of a connected weighted graph is a spanning tree that has the least possible weight.
		In this case, a minimum spanning tree is simply a spanning tree with the least possible total length.
		Use the applet to create a minimal spanning tree with 4 vertices, of which 3 have degree 1.
	www.prenhall.com /divisions/esm/app/ph-la/excursions/html/EIMM_C07.html (248 words)

	Minimal Spanning Trees
		A spanning tree is a tree that includes all of the nodes in a graph.
		Finally, for our purposes, a minimal spanning tree is the spanning tree the sum of whose edge weights is less than or equal to the sum of the weights of any other spanning tree for this graph.
		Notice, by the way that, for physical travel, trees are not the solution since they not consider the distance to return from any dead end branch of the tree.
	www.delphiforfun.org /Programs/Math_Topics/MinimalSpanningTrees.htm (1122 words)

	Online Algorithm for Computing a Minimal Spanning Tree
		When a point from an MST is deleted, we are left with d - 1 subtrees, where d is the degree of the vertex being deleted.
		[For minimal spanning trees in the plane based on Euclidean distance, d is bounded by the constant 6, and therefore this is O(n).
		We have proven that AddToMST(T,x) returns a minimal spanning tree of the set S + {x} where S is the set of the nodes in T. We have also demonstrated that AddToMST runs in time O(kn), where k is the number of near points of x.
	cgm.cs.mcgill.ca /~soss/geometry/online_mst.html (3554 words)

	BioMed Central \| Full text \| Iterative class discovery and feature selection using Minimal Spanning Trees
		A minimal spanning tree (MST) is a tree that connects all points such that the sum of the length of the edges is a minimum.
		Minimal Spanning Trees were initially proposed for clustering by Zahn [15].
		Iteration between minimal spanning tree cluster analysis and feature selection is used to converge onto partitions that form well separated clusters and gene subsets that support these partitions.
	www.biomedcentral.com /1471-2105/5/126 (4484 words)

	section short
		In addition, algorithms for finding a minimal spanning tree are also described, which are useful for trimming a graph to a subgraph which leaves all nodes connected, but eliminates ``unnecessary'' connections.
		Definition: A spanning tree for a connected graph G is a non-rooted tree containing the nodes of the graph and a subset of the arcs of G.
		A minimal spanning tree is a spanning tree of least weight of a simple, weighted, connected graph G.
	nku.edu /~longa/classes/2002spring/mat385/days/day20/short/short.html (644 words)

	[No title]
		This is because G will have at least one spanning tree (just remove edges of G one at a time until all cycles have been removed but the vertices are still connected).
		In fact, we could list all possible spanning trees of G. Then a minimum spanning tree is any one of these trees which has minimal weight.
		Suppose E1 is a subset of E with the property that E1 is a subset of the edges in a minimal spanning tree T for G. Let V1 be the set of vertices incident with edges in E1.
	www.ececs.uc.edu /~cpurdy/lec21.html (1031 words)

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