Partitioning Colored Point Sets into Monochromatic Parts

Abstract

It is shown that any two-colored set of n points in general position in the plane can be partitioned into at most $\left\lceil {\tfrac{{n + 1}} {2}} \right\rceil$ monochromatic subsets, whose convex hulls are pairwise disjoint. This bound cannot be improved in general. We present an O(n log n) time algorithm for constructing a partition into fewer parts, if the coloring is unbalanced, i.e., the sizes of the two color classes differ by more than one. The analogous question for k-colored point sets (k > 2) and its higher dimensional variant are also considered.