Effective use of geometric information for clustering and related topics
This paper surveys how geometric information can be effectively used for efficient algorithms with focus on clustering problems. Given a complete weighted graph G of n vertices, is there a partition of the vertex set into k disjoint subsets so that the maximum weight of an innercluster edge (whose two endpoints both belong to the same subset) is minimized? This problem is known to Le NP-complete even for k = 3. The case of k = 2, that is, bipartition problem is solvable in polynomial time. Oil the other hand, in geometric setting where vertices are points in the plane and weights of edges equal the distances between corresponding points, the same problem is solvable in polynomial time even for k greater than or equal to 3 as far as k is a fixed constant. For the: case k = 2, effective use of geometric property of an optimal solution leads to considerable improvement on the computational complexity. Other related topics are: also discussed.
bipartite graph coloring computational geometry diameter duality transform geometric clustering intercluster distance maximum spanning tree separability voronoi diagram