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| {{graph search algorithm}}
| | == Barbour Veste and don let yourself fail the weekend == |
| In [[computer science]], '''Prim's algorithm''' is a [[greedy algorithm]] that finds a [[minimum spanning tree]] for a [[Connected graph|connected]] [[Weighted graph|weighted]] [[undirected graph]]. This means it finds a subset of the [[edge (graph theory)|edge]]s that forms a [[Tree (graph theory)|tree]] that includes every [[Vertex (graph theory)|vertex]], where the total weight of all the [[graph theory|edges]] in the tree is minimized. The algorithm was developed in 1930 by [[Czech people|Czech]] mathematician [[Vojtěch Jarník]] and later independently by [[computer scientist]] [[Robert C. Prim]] in 1957 and rediscovered by [[Edsger Dijkstra]] in 1959. Therefore it is also sometimes called the '''DJP algorithm''', the '''Jarník algorithm''', or the '''Prim–Jarník algorithm'''.
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| Other algorithms for this problem include [[Kruskal's algorithm]] and [[Borůvka's algorithm]]. These algorithms find the minimum spanning forest in a possibly disconnected graph. By running Prim's algorithm for each [[Connected component (graph theory)| connected component]] of the graph, it can also be used to find the minimum spanning forest.
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| [[File:Prim's algorithm.svg|left|120px|thumb|Prim's algorithm starting at vertex A. In the second step, BD is chosen to add to the tree instead of AB arbitrarily, as both have weight 2. Afterwards, AB is excluded because it is between two nodes that are already in the tree.]]
| | <li>[http://gkirohamo.egloos.com/9146012/ http://gkirohamo.egloos.com/9146012/]</li> |
| == Description ==
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| ===Informal===
| | <li>[http://netburst.org/index.php?site=polls&pollID=3 http://netburst.org/index.php?site=polls&pollID=3]</li> |
| {|
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| | <li>[http://test.bpmn.info/forum/read.php?2,272020 http://test.bpmn.info/forum/read.php?2,272020]</li> |
| # Initialize a tree with a single vertex, chosen arbitrarily from the graph.
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| # Grow the tree by one edge: Of the edges that connect the tree to vertices not yet in the tree, find the minimum-weight edge, and transfer it to the tree.
| | <li>[http://globaldialogue2013.go.tz/gdforum/activity http://globaldialogue2013.go.tz/gdforum/activity]</li> |
| # Repeat step 2 (until all vertices are in the tree).
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| |}
| | <li>[http://www.juegosetnicos.com.ar/spip.php?article87&lang=ru/ http://www.juegosetnicos.com.ar/spip.php?article87&lang=ru/]</li> |
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| ===Technical===
| | </ul> |
| If a graph is empty then we are done immediately. Thus, we assume otherwise.
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| The algorithm starts with a tree consisting of a single vertex, and continuously increases its size one edge at a time, until it spans all vertices.
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| {|
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| * Input: A non-empty connected weighted graph with vertices ''V'' and edges ''E'' (the weights can be negative).
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| * Initialize: ''V''<sub>new</sub> = {''x''}, where ''x'' is an arbitrary node (starting point) from ''V'', ''E''<sub>new</sub> = {}
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| * Repeat until ''V''<sub>new</sub> = ''V'':
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| ** Choose an edge {''u'', ''v''} with minimal weight such that ''u'' is in ''V''<sub>new</sub> and ''v'' is not (if there are multiple edges with the same weight, any of them may be picked)
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| ** Add ''v'' to ''V''<sub>new</sub>, and {''u'', ''v''} to ''E''<sub>new</sub>
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| * Output: ''V''<sub>new</sub> and ''E''<sub>new</sub> describe a minimal spanning tree
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| |}
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| == Time complexity ==
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| [[File:MAZE 30x20 Prim.ogv|thumb|Prim's algorithm has many applications, such as in the [[maze generation|generation]] of this maze, which applies Prim's algorithm to a randomly weighted [[grid graph]].]]
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| {| class="wikitable"
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| ! Minimum edge weight data structure !! Time complexity (total)
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| |-
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| | [[adjacency matrix]], searching || O(<nowiki>|</nowiki>V<nowiki>|</nowiki><sup>2</sup>)
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| |-
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| | [[binary heap]] and [[adjacency list]] || O((<nowiki>|</nowiki>V<nowiki>|</nowiki> + <nowiki>|</nowiki>E<nowiki>|</nowiki>) log <nowiki>|</nowiki>V<nowiki>|</nowiki>) = O(<nowiki>|</nowiki>E<nowiki>|</nowiki> log <nowiki>|</nowiki>V<nowiki>|</nowiki>)
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| |-
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| | [[Fibonacci heap]] and [[adjacency list]] || O(<nowiki>|</nowiki>E<nowiki>|</nowiki> + <nowiki>|</nowiki>V<nowiki>|</nowiki> log <nowiki>|</nowiki>V<nowiki>|</nowiki>)
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| |}
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| A simple implementation using an [[adjacency matrix]] graph representation and searching an array of weights to find the minimum weight edge to add requires [[Big-O notation|O]](<nowiki>|</nowiki>V<nowiki>|</nowiki><sup>2</sup>) running time. Using a simple [[binary heap]] data structure and an [[adjacency list]] representation, Prim's algorithm can be shown to run in time [[Big-O notation|O]](<nowiki>|</nowiki>E<nowiki>|</nowiki> log <nowiki>|</nowiki>V<nowiki>|</nowiki>) where <nowiki>|</nowiki>E<nowiki>|</nowiki> is the number of edges and <nowiki>|</nowiki>V<nowiki>|</nowiki> is the number of vertices. Using a more sophisticated [[Fibonacci heap]], this can be brought down to [[Big-O notation|O]](<nowiki>|</nowiki>E<nowiki>|</nowiki> + <nowiki>|</nowiki>V<nowiki>|</nowiki> log <nowiki>|</nowiki>V<nowiki>|</nowiki>), which is [[Asymptotic computational complexity|asymptotically faster]] when the graph is [[Dense graph|dense]] enough that <nowiki>|</nowiki>E<nowiki>|</nowiki> is [[Big-O notation#Family_of_Bachmann.E2.80.93Landau_notations|ω]](<nowiki>|</nowiki>V<nowiki>|</nowiki>).
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| [[File:Prim's_algorithm_proof.svg|thumb|150px|Demonstration of proof. In this case, the graph ''Y<sub>2</sub>'' = ''Y<sub>1</sub>'' − ''f'' + ''e'' is already equal to ''Y''. In general, the process may need to be repeated.]] | |
| == Proof of correctness ==
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| Let ''P'' be a connected, weighted [[graph theory|graph]]. At every iteration of Prim's algorithm, an edge must be found that connects a vertex in a subgraph to a vertex outside the subgraph. Since ''P'' is connected, there will always be a path to every vertex. The output ''Y'' of Prim's algorithm is a [[Tree (graph theory)|tree]], because the edge and vertex added to tree ''Y'' are connected. Let ''Y<sub>1</sub>'' be a minimum spanning tree of graph P. If ''Y<sub>1</sub>''=''Y'' then ''Y'' is a minimum spanning tree. Otherwise, let ''e'' be the first edge added during the construction of tree ''Y'' that is not in tree ''Y<sub>1</sub>'', and ''V'' be the set of vertices connected by the edges added before edge ''e''. Then one endpoint of edge ''e'' is in set ''V'' and the other is not. Since tree ''Y<sub>1</sub>'' is a spanning tree of graph ''P'', there is a path in tree ''Y<sub>1</sub>'' joining the two endpoints. As one travels along the path, one must encounter an edge ''f'' joining a vertex in set ''V'' to one that is not in set ''V''. Now, at the iteration when edge ''e'' was added to tree ''Y'', edge ''f'' could also have been added and it would be added instead of edge ''e'' if its weight was less than ''e'', and since edge ''f'' was not added, we conclude that
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| :<math>w(f) \ge w(e).</math>
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| Let tree ''Y<sub>2</sub>'' be the graph obtained by removing edge ''f'' from and adding edge ''e'' to tree ''Y<sub>1</sub>''. It is easy to show that tree ''Y<sub>2</sub>'' is connected, has the same number of edges as tree ''Y<sub>1</sub>'', and the total weights of its edges is not larger than that of tree ''Y<sub>1</sub>'', therefore it is also a minimum spanning tree of graph ''P'' and it contains edge ''e'' and all the edges added before it during the construction of set ''V''. Repeat the steps above and we will eventually obtain a minimum spanning tree of graph ''P'' that is identical to tree ''Y''. This shows ''Y'' is a minimum spanning tree.
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| == See also ==
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| * [[Kruskal's algorithm]]
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| == References ==
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| {{Reflist}}
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| * V. Jarník: ''O jistém problému minimálním'' [About a certain minimal problem], Práce Moravské Přírodovědecké Společnosti, 6, 1930, pp. 57–63. (in Czech)
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| * R. C. Prim: ''Shortest connection networks and some generalizations''. In: ''Bell System Technical Journal'', 36 (1957), pp. 1389–1401
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| * [[David Cheriton|D. Cheriton]] and [[Robert Tarjan|R. E. Tarjan]]: ''Finding minimum spanning trees''. In: ''[[SIAM Journal on Computing]]'', 5 (Dec. 1976), pp. 724–741
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| * [[Thomas H. Cormen]], [[Charles E. Leiserson]], [[Ronald L. Rivest]], and [[Clifford Stein]]. ''[[Introduction to Algorithms]]'', Third Edition. MIT Press, 2009. ISBN 0-262-03384-4. Section 23.2: The algorithms of Kruskal and Prim, pp. 631–638.
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| ==External links==
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| {{commons category|Prim's Algorithm}}
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| * [http://scanftree.com/Data_Structure/prim's-algorithm Prim's algorithm with 'C' implementation]
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| * [http://students.ceid.upatras.gr/~papagel/project/prim.htm Animated example of Prim's algorithm]
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| * [http://www.mincel.com/java/prim.html Prim's Algorithm (Java Applet)]
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| * [http://people.csail.mit.edu/rivest/programs.html Minimum spanning tree demonstration Python program by Ronald L. Rivest]
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| * [http://code.google.com/p/annas/ Open Source Java Graph package with implementation of Prim's Algorithm]
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| * [http://code.google.com/p/ngenerics/ Open Source C# class library with implementation of Prim's Algorithm]
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| * [https://github.com/monmohan/mgraphlib/ Open Source Java graph library with implementation of Prim's Algorithm]
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| [[Category:Graph algorithms]]
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| [[Category:Spanning tree]]
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| [[Category:Articles containing proofs]]
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Barbour Veste and don let yourself fail the weekend
Kickstarter makes a 5% commission on every successful project and no one can donate to a failed project. Plus, it looks bad when you're trying to convince the world that anyone can fund anything through crowdfunding and you point out that 56% of . I was then given a fulltime position in an industry that I love and I couldn be happier. Words can't describe the growth this company has experienced during my time here.
IT DOES WORK! I been through it. I can honestly say if you follow along with everything they tell you to do, and commit, and don let yourself Barbour Veste fail the weekend, you come out transformed forever. Yes Facebook is the new thing and i think it is a good thing depending on that person's maturity. I also think that these kids should not be on facebook for hours on end.
When Plasma cells are charged, they emit invisible UV light. That strikes the red, green and blue phosphors on the back of the display, creating the pixels that form the image you see on the screen. His statement says in part, "JCOPE found that there was no impropriety whatsoever with regard to those matters. It cleared Mr.
People who knew him, notably, often Pandora Bijoux report a total awe at the power of his personality and mind, a power that overwhelmed them, catalyzed some of Nike Free Run 2 Pas Cher their greatest creativity and effort, inspired them them with its focus and capacity to find the point, the consequence, the animating vision sought in any effort. There is no question that Jobs was a rare sort of individual, one whom I credit with dramatically improving human access to creativitysupporting computation (among other feats that matter to me a great deal).
However, Ibn Sina's journey was not complete because he was not completely secluded from danger. After a brief stop in Qazvin, Ibn Sina continued to Hamadan where he made close friendly ties with the leader Shams alDawla. The last week or so they tyres are slippy, wheel spin and and I don't feel as confident. Checked the depth and guess what 3mm on the fronts! Up until this point they have been pretty good and trust me I work them hard, but i can't see me keeping them on the car much below 3mm..
While blessed in some ways with tackling the show's flashier and arguably meatier role, Lunettes De Soleil Ray Ban Burstein has also been handed the greater challenge, especially for those who still carry the indelible memory of Matt's original portrayer, Judd Hirsch. But he never falters, not just in making the part his own, but also in letting Matt's pain always simmer just beneath the jovial surface at least until the moment when it's forced to come to a slow, inevitable boil..