Connecting colored point sets

We study the following Ramsey-type problem. Let S = B ∪ R be a two-colored set of n points in the plane. We show how to construct, in O(n log n) time, a crossing-free spanning tree T (B) for B, and a crossing-free spanning tree T (R) for R, such that both the number of crossings between T (B) and T (R) and the diameters of T (B) and T (R) are kept small. The algorithm is conceptually simple and is implementable without using any non-trivial data structure. This improves over a previous method in Tokunaga [16] that is less efficient in implementation and does not guarantee a diameter bound. Implicit to our approach is a new proof for the result in [16] on the minimum number of crossings between T (B) and T (R).