Balanced Islands in Two Colored Point Sets in the Plane

TL;DR

This paper proves the existence of convex sets with balanced red and blue points in a plane and provides efficient algorithms to find such sets, with improved running times for smaller sets.

Contribution

It establishes the existence of balanced convex sets in two-colored point sets and introduces polynomial-time algorithms with optimized performance for specific cases.

Findings

Existence of convex sets with specified red and blue point counts.

Algorithms with $O(n^4)$ and $O(n^2 ext{log} n)$ running times for finding these sets.

Improved $O(n ext{log} n)$ algorithm when the set size is small.

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 $\alpha \in \left [ 0,\frac{1}{2} \right ]$, a convex set containing exactly $\lceil \alpha r\rceil$ red points and exactly $\lceil \alpha b \rceil$ 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 $\lceil \alpha r\rceil+\lceil \alpha b\rceil$ is small, that is, not much larger than $\frac{1}{3}n$, we improve the running time to $O(n \log n)$.