Connecting colored point sets

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...

Pattern Matching of Colored Point Sets in 3d

Balanced Islands in Two Colored Point Sets in the Plane

Let P be a set of n points in general position in the plane, r of which are red and b of which are blue. In this paper we prove that there exist: for every α ∈ [ 0, 1 2 ] , a convex set containing exactly dαre red points and exactly dαbe blue points of P ; a convex set containing exactly ⌈ r+1 2 ⌉ red points and exactly ⌈ b+1 2 ⌉ blue points of P . Furthermore, we present polynomial time algori...

Monochromatic empty triangles in two-colored point sets

Improving a result of Aichholzer et. al., we show that there exists a constant c > 0 satisfying the following condition. Any two-colored set of n points in general position in the plane has at least cn triples of the same color such that the triangles spanned by them contain no element of the set in their interiors.

On some partitioning problems for two-colored point sets

Let S be a two-colored set of n points in general position in the plane. We show that S admits at least 2 ⌊