Extreme Distances in Multicolored Point Sets

Given a set of n colored points in some d-dimensional Euclidean space, a bichromatic closest (resp. farthest) pair is a closest (resp. farthest) pair of points of different colors. We present efficient algorithms to maintain both a bichromatic closest pair and a bichromatic farthest pair when the the points are fixed but they dynamically change color. We do this by solving the more general problem of maintaining a bichromatic edge of minimum (resp. maximum) weight in an undirected weighted graph with colored vertices, when vertices dynamically change color. We also give some combinatorial bounds on the maximum multiplicity of extreme bichromatic distances in the plane. Article Type Communicated by Submitted Revised regular paper M. T. Goodrich January 2003 March 2004 A preliminary version of this paper appeared in the Proceedings of the Second International Conference on Computational Science (Workshop on Computational Geometry and Applications CGA ’02), Amsterdam, April 2002. In Lecture Notes in Computer Science, Vol. 2331, 2002, 14–25. A. Dumitrescu and S. Guha, Extreme Distances, JGAA, 8(1) 27–38 (2004) 28