Euclidean minimum spanning tree + Relative neighborhood graph + Delaunay triangulation

1.Algorithms for computing EMSTs:

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:

1. 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.
2. Label each edge with its length.
3. 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.

The final result is an algorithm taking O(n log n) time and O(n) space.

--wiki

2.Relative neighborhood graph

In computational geometry, the relative neighborhood graph (RNG) is an undirected graph defined on a set of points in the Euclidean plane by connecting two points p and q by an edge whenever there does not exist a third point r that is closer to both p and q than they are to each other.

 

3. Delaunay triangulation:

http://www.cse.unsw.edu.au/~lambert/java/3d/delaunay.html

这些知识自己还没有消化,等以后计算几何的时候再回头搞定 :）