construct a Euclidean minimum spanning tree
construct a minimum spanning tree for specified norm
EuclideanMinimumSpanningTree( P, opts )
GeometricMinimumSpanningTree( P, norm, opts )
Matrix or list of lists representing set of points
positive number, infinity, or Euclidean
(optional) one or more options as specified below
method : one of Kruskal or Prim.
Specifies the algorithm to use in constructing the minimum spanning tree. The default is Kruskal's algorithm.
For more information, see GraphTheory[MinimalSpanningTree].
triangulation : list of three-element lists.
Supply a previously computed Delaunay triangulation of P. The input must be a valid Delaunay triangulation in the format returned by ComputationalGeometry[DelaunayTriangulation]: a list of three-element lists of integers, representing triangles in a triangulation of P.
vertices : list of integers, strings or symbols
Specifies the vertices to be used in the generated graph.
weighted : true or false
If weighted=true, the result is a weighted graph whose edge weights correspond to the distance between points using the specified norm. Default is false.
The EuclideanMinimumSpanningTree(P, opts) command returns a minimum spanning tree for the graph generated from the point set P.
The GeometricMinimumSpanningTree(P, norm, opts) command returns a minimum spanning tree for the graph generated from the point set P using the norm norm.
The parameter P must be a Matrix or list of lists representing a set of points.
The parameter norm must be a positive number or one of the symbols Euclidean or infinity. This specifies the norm to be used in computing distances.
For more information on norms, see LinearAlgebra[Norm].
Let P be a set of points in n dimensions, let p and q be arbitrary points from P, and let dist⁡p,q be the Euclidean distance between p and q.
The Euclidean minimum spanning tree is simply a minimum-weight spanning tree for the complete weighted graph on P with the weight of the edge between points p and q defined to be dist⁡p,q.
For any norm ρ on P, the minimum spanning tree for norm ρ is a minimum-weight spanning tree for the complete weighted graph on P with the weight of the edge between points p and q defined to be ρ⁡p−q.
The Euclidean minimum spanning tree has the following relationships with other graphs:
The Euclidean minimum spanning tree on P is a subgraph of the relative neighborhood graph on P.
The Euclidean minimum spanning tree on P is a subgraph of the Urquhart graph on P.
Generate a set of random two-dimensional points and draw the Euclidean minimum spanning tree.
points≔9.8501769734180382.975030438619586.067018374966383.318865936399664.374679554674173.867160763967357.36705572946662.3439977588303123.623426484493352.687336738732847.002754735000322.245948836755274.921349155896362.047182022071892.151343470907396.310726263708048.231962435594463.756326714414190.944187743180533.8527464913022⋮⋮60 × 2 Matrix
EMST≔Graph 1: an undirected weighted graph with 60 vertices and 59 edge(s)
Now draw the rectilinear minimum spanning tree (corresponding to the 1-norm) on the same data.
RMST≔Graph 2: an undirected weighted graph with 60 vertices and 59 edge(s)
The GraphTheory[GeometricGraphs][EuclideanMinimumSpanningTree] and GraphTheory[GeometricGraphs][GeometricMinimumSpanningTree] commands were introduced in Maple 2020.
For more information on Maple 2020 changes, see Updates in Maple 2020.
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