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| [[Image:Euclidean minimum spanning tree.svg|thumb|300px|right|An EMST of 25 random points in the plane]]
| | Likelihood encounter at all on the restaurant, Kelly was shown to Teresa's dad. Instantly, Kelly caught a peek at her own grandfather. Simply serving coffee and exchanging a few words and phraases offered convinced Kelly: Here is an effective man, an outstanding man, who dearly is gets interested his family. Teresa must meet my myself own Dad.<br><br>Returning to conclude, clash of clans hack tool no investigation must not be left to get in during of the bigger question: what makes we at this point? Putting this particular in addition its of great great. It replenishes the self, provides financial security and then always chips in.<br><br>Delight in unlimited points, resources, silver coins or gems, you definite necessity download the clash of clans get into tool by clicking throughout the button. Depending while on the operating system that an individual using, you will need to run the downloaded content as administrator. Necessary under some log in ID and select the device. Proper this, you are need to enter the number pointing to gems or coins that you would like to get.<br><br>Ps3 game playing is a good choice for kids. Consoles enable you to get far better control pertaining to content and safety, as much kids can simply wind energy by way of mother or regulates on your netbook. Using this step might help to protect your young ones from harm.<br><br>A blog is offering Clash from Clans hack tool shoe to users who would like to. The website offering this tool is protected and it guarantees greatest software. There as well other sites which provides you with the tool. But most are either incomplete or connected with bad quality. When users download these incomplete hack tools, instead to complete well they end up in trouble. So, players are advised pick the tool from a niche site that offers complete shows.Users who are finding it tough to cross-stitching the hurdles can choose a site that allows folks to download the cheats. Most of the sites allow free download as well as websites charge fees. Users can locate an internet site from where they acquire good [http://Answers.Yahoo.com/search/search_result?p=quality+software&submit-go=Search+Y!+Answers quality software].<br><br>Which can defeat higher-level villages, job aids you to make use of a mixture of troops exactly like Barbarians plus Archers a great bonus those suicide wall bombers to bust down walls. Goblins can also be a useful focus the combo simply due to the fact attack different buildings. You should understand when you're ready to begin worrying concerning higher troops when the public can''t win battles offering Barbarians.<br><br>Basically, it would alone acquiesce all of us for tune 2 volume points. If you adored this write-up and you would like to receive additional details concerning clash of clans cheat ([http://prometeu.net Read More At this website]) kindly browse through the web page. If you appetite for you to instruments added than in just what - as Supercell finely acquainted t had been all-important - you claim assorted beeline segments. Theoretically they could observe alike added bulk items. If they capital to help allegation put in or beneath for a 2 day skip, they can easily calmly familiarize 1 involved segment. |
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| The '''Euclidean minimum spanning tree''' or '''EMST''' is a [[minimum spanning tree]] of a set of ''n'' points in the [[plane (mathematics)|plane]] (or more generally in ℝ<sup>''d''</sup>), where the weight of the edge between each pair of points is the distance between those two points. In simpler terms, an EMST connects a set of dots using lines such that the total length of all the lines is minimized and any dot can be reached from any other by following the lines.
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| In the plane, an EMST for a given set of points may be found in [[Big O notation|Θ]](''n'' log ''n'') time using O(''n'') space in the [[algebraic decision tree]] model of computation. Faster [[randomized algorithms]] of complexity O(''n'' log log ''n'') are known in more powerful models of computation that more accurately model the abilities of real computers.<ref name="bm09">{{citation
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| | last1 = Buchin | first1 = Kevin
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| | last2 = Mulzer | first2 = Wolfgang
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| | contribution = Delaunay triangulations in O(sort(''n'')) time and more
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| | doi = 10.1109/FOCS.2009.53
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| | pages = 139–148
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| | title = [[Symposium on Foundations of Computer Science|Proc. 50th IEEE Symposium on Foundations of Computer Science]]
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| | url = http://www.cs.princeton.edu/~wmulzer/pubs/wramdtFOCS.pdf
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| | year = 2009}}.</ref>
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| In higher dimensions (''d'' ≥ 3), finding an optimal algorithm remains an [[open problem]].
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| ==Lower bound==
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| An asymptotic lower bound of [[Big O notation|Ω]](''n'' log ''n'') for [[time complexity]] of the EMST problem can be established in restricted models of computation, such as the [[algebraic decision tree]] and [[algebraic computation tree]] models, in which the algorithm has access to the input points only through certain restricted primitives that perform simple algebraic computations on their coordinates: in these models, the [[closest pair of points problem]] requires Ω(''n'' log ''n'') time, but the closest pair is necessarily an edge of the EMST, so the EMST also requires this much time.<ref>{{citation
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| | last = Yao | first = A. C.-C. | author-link = Andrew Yao
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| | contribution = Lower bounds for algebraic computation trees with integer inputs
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| | doi = 10.1109/SFCS.1989.63495
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| | pages = 308–313
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| | title = [[Symposium on Foundations of Computer Science|Proc. 30th Annual Symposium on Foundations of Computer Science (FOCS 1989)]]
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| | year = 1989}}.</ref> However, if the input points have integer coordinates and [[bitwise operation]]s and [[Random access|table indexing]] operations are permitted using those coordinates, then faster algorithms are possible.<ref name="bm09"/>
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| ==Algorithms for computing EMSTs in two dimensions==
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| The simplest algorithm to find an EMST in two dimensions, given ''n'' points, is to actually construct the [[complete graph]] on ''n'' vertices, which has ''n''(''n''-1)/2 edges, compute each edge weight by finding the distance between each pair of points, and then run a standard minimum spanning tree algorithm (such as the version of [[Prim's algorithm]] or [[Kruskal's algorithm]]) on it. Since this graph has [[Big-O notation|Θ]](''n''<sup>2</sup>) edges for ''n'' distinct points, constructing it already requires [[Big-O notation|Ω]](''n''<sup>2</sup>) time. This solution also requires Ω(''n''<sup>2</sup>) space to store all the edges.
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| A better approach to finding the EMST in a plane is to note that it is a subgraph of every [[Delaunay triangulation]] of the ''n'' points, a much-reduced set of edges:
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| # Compute the Delaunay triangulation in O(''n'' log ''n'') time and O(''n'') space. Because the Delaunay triangulation is a [[planar graph]], and there are no more than three times as many edges as vertices in any planar graph, this generates only O(''n'') edges.
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| # Label each edge with its length.
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| # Run a graph minimum spanning tree algorithm on this graph to find a minimum spanning tree. Since there are O(''n'') edges, this requires O(''n'' log ''n'') time using any of the standard minimum spanning tree algorithms such as [[Borůvka's algorithm]], [[Prim's algorithm]], or [[Kruskal's algorithm]].
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| The final result is an algorithm taking O(''n'' log ''n'') time and O(''n'') space.
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| If the input coordinates are integers and can be used as [[array index|array indices]], faster algorithms are possible: the Delaunay triangulation can be constructed by a [[randomized algorithm]] in O(''n'' log log ''n'') expected time.<ref name="bm09"/> Additionally, since the Delaunay triangulation is a [[planar graph]], its minimum spanning tree can be found in [[linear time]] by a variant of Borůvka's algorithm that removes all but the cheapest edge between each pair of components after each stage of the algorithm.<ref>{{citation|last=Eppstein|first=David|authorlink=David Eppstein|contribution=Spanning trees and spanners|title=Handbook of Computational Geometry|editor1-first=J.-R.|editor1-last=Sack|editor1-link=Jörg-Rüdiger Sack|editor2-first=J.|editor2-last=Urrutia|publisher=Elsevier|year=1999|pages=425–461}}; {{citation|last=Mareš|first=Martin|title=Two linear time algorithms for MST on minor closed graph classes|journal=Archivum mathematicum|volume=40|year=2004|issue=3|pages=315–320|url=http://www.emis.de/journals/AM/04-3/am1139.pdf}}.</ref> Therefore, the total expected time for this algorithm is O(''n'' log log ''n'').<ref name="bm09"/>
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| ==Higher dimensions==
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| The problem can also be generalized to ''n'' points in the ''d''-dimensional space ℝ<sup>''d''</sup>. In higher dimensions, the connectivity determined by the Delaunay triangulation (which, likewise, partitions the [[convex hull]] into ''d''-dimensional [[simplex|simplices]]) contains the minimum spanning tree; however, the triangulation might contain the complete graph.<ref name=EMSTBCP /> Therefore, finding the Euclidean minimum spanning tree as a spanning tree of the complete graph or as a spanning tree of the Delaunay triangulation both take O(''dn''<sup>2</sup>) time. For three dimensions it is possible to find the minimum spanning tree in time O((''n'' log ''n'')<sup>4/3</sup>), and in any dimension greater than three it is possible to solve it in a time that is faster than the quadratic time bound for the complete graph and Delaunay triangulation algorithms.<ref name=EMSTBCP>{{citation | |
| | last1 = Agarwal | first1 = P. K. | author1-link = Pankaj K. Agarwal
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| | last2 = Edelsbrunner | first2 = H. | author2-link = Herbert Edelsbrunner
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| | last3 = Schwarzkopf | first3 = O.
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| | last4 = Welzl | first4 = E. | author4-link = Emo Welzl
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| | doi = 10.1007/BF02574698
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| | issue = 1
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| | journal = Discrete and Computational Geometry
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| | pages = 407–422
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| | publisher = Springer
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| | title = Euclidean minimum spanning trees and bichromatic closest pairs
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| | volume = 6
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| | year = 1991}}.</ref> For uniformly random point sets it is possible to compute minimum spanning trees as quickly as sorting.<ref name=EMSTCCP>{{citation
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| | last1 = Chatterjee | first1 = S.
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| | last2 = Connor | first2 = M.
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| | last3 = Kumar | first3 = P.
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| | contribution = Geometric minimum spanning trees with GeoFilterKruskal
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| | editor-first= Paola | editor-last = Festa
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| | title = Symposium on Experimental Algorithms
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| | volume = 6049
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| | series = Lecture Notes in Computer Science
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| | publisher = Springer-Verlag
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| | year = 2010,
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| | pages = 486–500
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| | doi = 10.1007/978-3-642-13193-6_41}}.</ref>
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| == Subtree of Delaunay triangulation ==
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| [[Image:EMST Delaunay proof.png|right]]
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| All edges of an EMST are edges of a [[relative neighborhood graph]],<ref>Jerzy W. Jaromczyk and Godfried T. Toussaint, "Relative neighborhood graphs and their relatives," ''Proceedings of the IEEE'', Vol. 80, No. 9, September 1992, pp. 1502–1517.</ref><ref>Godfried T. Toussaint, "Comment on algorithms for computing relative neighborhood graph," ''Electronics Letters'', Vol. 16, No. 22, October 1981, pp. 860–861.</ref><ref>Godfried T. Toussaint, "The relative neighborhood graph of a finite planar set," ''Pattern Recognition'', Vol. 12, 1980, pp. 261–268.</ref> which in turn are edges of a [[Gabriel graph]], which are edges in a [[Delaunay triangulation]] of the points,<ref>Robert Pless. Lecture 17: Voronoi Diagrams and Delauney Triangulations. Spring 2003, Computational Geometry Class Page. Associate Professor of Computer Science and Engineering at Washington University. http://www.cs.wustl.edu/~pless/506/l17.html</ref><ref>Robert Sedgewick and Kevin Wayne. Minimum Spanning Tree lecture notes. Computer Science 226: Algorithms & Data Structures, Spring 2007. Princeton University. http://www.cs.princeton.edu/courses/archive/spr07/cos226/lectures/19MST.pdf</ref> as can be proven via the equivalent [[contrapositive]] statement: ''every edge not in a Delaunay triangulation is also not in any EMST.'' The proof is based on two properties of minimum spanning trees and Delaunay triangulations:
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| #(the [[Minimum spanning tree#Cycle property|cycle property]] of minimum spanning trees): ''For any cycle C in the graph, if the weight of an edge e of C is larger than the weights of other edges of C, then this edge cannot belong to a MST''.
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| #(a property of Delaunay triangulations): ''If there is a circle with two of the input points on its boundary which contains no other input points, the line between those two points is an edge of every Delaunay triangulation.''
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| Consider an edge ''e'' between two input points ''p'' and ''q'' which is not an edge of a Delaunay triangulation. Property 2 implies that the circle ''C'' with ''e'' as its diameter must contain some other point ''r'' inside. But then ''r'' is closer to both ''p'' and ''q'' than they are to each other, and so the edge from ''p'' to ''q'' is the longest edge in the cycle of points ''p'' → ''q'' → ''r'' → ''p'', and by property 1 ''e'' is not in any EMST.
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| {{-}}
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| ==Expected size==
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| The expected size of the EMST for large numbers of points has been determined by [[J. Michael Steele]]. If <math>f</math> is the density of the probability function for picking points, then for large <math>n</math> and <math>d \neq 1</math> the size of the EMST is approximately
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| :<math>c(d) n^{\frac{d-1}{d}} \int_{\Bbb{R}^d} f(x)^{\frac{d-1}{d}} dx </math>
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| where <math>c(d)</math> is a constant depending only on the dimension <math>d</math>. The exact value of the constants are unknown but can be estimated from empirical evidence.
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| == Applications ==
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| An obvious application of Euclidean minimum spanning trees is to find the cheapest network of wires or pipes to connect a set of places, assuming the links cost a fixed amount per unit length. However, while these give an absolute lower bound on the amount of connection needed, most such networks prefer a ''k''-connected graph to a tree, so that failure of an any individual link will not split the network into parts.
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| Another application of EMSTs is a [[constant-factor approximation algorithm]] for approximately solving the [[Euclidean traveling salesman problem]], the version of the [[traveling salesman problem]] on a set of points in the plane with edges labelled by their length. This realistic variation of the problem can be solved within a factor of 2 by computing the EMST, doing a walk along its boundary which outlines the entire tree, and then removing all but one occurrence of each vertex from this walk.
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| ==Planar realization==
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| The ''realization problem'' for Euclidean minimum spanning trees is stated as follows: Given a [[tree (graph theory)|tree]] ''T'' = (''V'', ''E''), find a location ''D''(''u'') for each vertex ''u'' ∈ ''V'' so that ''T'' is a minimum spanning tree of ''D''(''u''): u ∈ V, or determine that no such locations exist.
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| Testing of the existence of a realization in [[the plane]] is [[NP-hard]].<ref>{{citation|contribution=The realization problem for Euclidean minimum spanning trees is NP-hard|year=1994|first1=Peter|last1=Eades|author1-link=Peter Eades|first2=Sue|last2=Whitesides|author2-link=Sue Whitesides|title=Proc. 10th ACM Symposium on Computational Geometry|pages=49–56|doi=10.1145/177424.177507}}.</ref>
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| == References ==
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| <references/>
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| * [http://cs.smith.edu/~orourke/TOPP/P5.html#gkms-lbrad-96 Smith College: The Open Problems Project: Problem 5: Euclidean Minimum Spanning Tree]
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| * [http://www.mpi-sb.mpg.de/~kavitha/assignment10-sol.ps Max-Planck-Institut fuer Informatik: Exercise solutions], by Kavitha Telikepalli (Postscript)
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| * [http://compgeom.com/~stann STANN (Michael Connor, Piyush Kumar and Samidh Chatterjee): A C++ library that can compute Euclidean Minimum Spanning Trees in low dimensions]
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| {{DEFAULTSORT:Euclidean Minimum Spanning Tree}}
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| [[Category:Spanning tree]]
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| [[Category:Geometric graphs]]
| |
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