Solving the chromatic cone clustering problem via minimum spanning sphere

Abstract

In this paper, we study the following Chromatic Cone Clustering (CCC) problem: Given n point-sets with each containing k points in the first quadrant of the d-dimensional space R d , find k cones apexed at the origin such that each cone contains at least one distinct point (i.e., different from other cones) from every point-set and the total size of the k cones is minimized, where the size of a cone is the angle from any boundary ray to its center line. CCC is motivated by an important biological problem and finds applications in several other areas. Our approaches for solving the CCC problem relies on solutions to the Minimum Spanning Sphere (MinSS) problem for point-sets. For the MinSS problem, we present two (1 + ε)-approximation algorithms based on core-sets and ε-net respectively. With these algorithms, we then show that the CCC problem admits (1 + ε)-approximation solutions for constant k. Our results are the first solutions to these problems. © 2011 Springer-Verlag.