Bichromatic separability with two boxes: A general approach

Let S be a point set in general position on the plane such that its elements are colored red or blue. We study the following problem: Remove as few points as possible from S such that the remaining points can be enclosed by two isothetic rectangles, one containing all the red points, the other all the blue points, and such that each rectangle contains only points of one color. We prove that this problem can be solved in O(n log n) time and O(n) space. We show how our techniques can be generalized to solve other variants of the given problem such as the 3-dimensional problem and the trichromatic problem.