New results on the coarseness of bicolored point sets
Abstract
Let S be a 2-colored (red and blue) set of n points in the plane. A subset I of S is an island if there exits a convex set C such that I=C S. The discrepancy of an island is the absolute value of the number of red minus the number of blue points it contains. A convex partition of S is a partition of S into islands with pairwise disjoint convex hulls. The discrepancy of a convex partition is the discrepancy of its island of minimum discrepancy. The coarseness of S is the discrepancy of the convex partition of S with maximum discrepancy. This concept was recently defined by Bereg et al. [CGTA 2013]. In this paper we study the following problem: Given a set S of n points in general position in the plane, how to color each of them (red or blue) such that the resulting 2-colored point set has small coarseness? We prove that every n-point set S can be colored such that its coarseness is O(n1/4 n). This bound is almost tight since there exist n-point sets such that every 2-coloring gives coarseness at least (n1/4). Additionally, we show that there exists an approximation algorithm for computing the coarseness of a 2-colored point set, whose ratio is between 1/128 and 1/64, solving an open problem posted by Bereg et al. [CGTA 2013]. All our results consider k-separable islands of S, for some k, which are those resulting from intersecting S with at most k halfplanes.