Effective Use of Geometric Information for Clustering and Related Topics

Tetsuo ASANO

IEICE TRANSACTIONS on Information and Systems   Vol.E83-D    No.3    pp.418-427
Publication Date: 2000/03/25
Online ISSN: 
Print ISSN: 0916-8532
Type of Manuscript: INVITED SURVEY PAPER
Category: Algorithms for Geometric Problems
bipartite graph,  coloring,  computational geometry,  diameter,  duality transform,  geometric clustering,  intercluster distance,  maximum spanning tree,  separability,  Voronoi dia-gram,  

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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 be NP-complete even for k = 3. The case of k=2, that is, bipartition problem is solvable in polynomial time. On 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 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.