Balanced Islands in Two Colored Point Sets in the Plane

Abstract

Let $S$ 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 $α\in \left [ 0,\frac{1}{2} \right ]$, a convex set containing exactly $\lceil αr\rceil$ red points and exactly $\lceil αb \rceil$ blue points of $S$; a convex set containing exactly $\left \lceil \frac{r+1}{2}\right \rceil$ red points and exactly $\left \lceil \frac{b+1}{2}\right \rceil$ blue points of $S$. Furthermore, we present polynomial time algorithms to find these convex sets. In the first case we provide an $O(n^4)$ time algorithm and an $O(n^2\log n)$ time algorithm in the second case. Finally, if $\lceil αr\rceil+\lceil αb\rceil$ is small, that is, not much larger than $\frac{1}{3}n$, we improve the running time to $O(n \log n)$.