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The Euclidean minimum spanning tree of a set of points is a subset of the Delaunay triangulation of the same points, [22] and this can be exploited to compute it efficiently. For modelling terrain or other objects given a point cloud, the Delaunay triangulation gives a nice set of triangles to use as polygons in the model. In particular, the ...
Euclidean minimum spanning tree of 25 random points in the plane. A Euclidean minimum spanning tree of a finite set of points in the Euclidean plane or higher-dimensional Euclidean space connects the points by a system of line segments with the points as endpoints, minimizing the total length of the segments. In it, any two points can reach ...
The Delaunay triangulation of a set of points in the plane contains the Gabriel graph, the nearest neighbor graph and the minimal spanning tree of . Triangulations have a number of applications, and there is an interest to find the "good" triangulations of a given point set under some criteria as, for instance minimum-weight triangulations .
A planar graph and its minimum spanning tree. Each edge is labeled with its weight, which here is roughly proportional to its length. A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. [1]
The Gabriel graph is a subgraph of the Delaunay triangulation. It can be found in linear time if the Delaunay triangulation is given. [4] The Gabriel graph contains, as subgraphs, the Euclidean minimum spanning tree, the relative neighborhood graph, and the nearest neighbor graph. It is an instance of a beta-skeleton.
However, it is not necessary to construct this graph in order to solve the optimization problem; the Euclidean minimum spanning tree problem, for instance, can be solved more efficiently in O(n log n) time by constructing the Delaunay triangulation and then applying a linear time planar graph minimum spanning tree algorithm to the resulting ...
Example of rectilinear minimum spanning tree from random points. In graph theory, the rectilinear minimum spanning tree (RMST) of a set of n points in the plane (or more generally, in ) is a minimum spanning tree of that set, where the weight of the edge between each pair of points is the rectilinear distance between those two points.
Their kinetic data structure handles () events, where m is the number of all changes to the Delaunay triangulation of the moving points. Their kinetic approach can work well for maintenance of the minimum spanning tree (MST) of a planar graph whose edge weights are changing as a continuous function of time.