TY - JOUR

T1 - Computing Balanced Islands in Two Colored Point Sets in the Plane

AU - Aichholzer, Oswin

AU - Atienza, Nieves

AU - Díaz-Báñez, José M.

AU - Fabila-Monroy, Ruy

AU - Flores-Peñaloza, David

AU - Pérez-Lantero, Pablo

AU - Vogtenhuber, Birgit

AU - Urrutia Galicia, Jorge

PY - 2018

Y1 - 2018

N2 - 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 present algorithms to find convex sets containing a balanced number of red and blue points. We provide an $O(n^4)$ time algorithm that for a given $alpha in left [ 0,12 right ]$ finds a convex set containing exactly $lceil alpha r red points and exactly $lceil alpha b blue points of $S$. If $lceil alpha rlceil alpha b is not much larger than $13n$, we improve the running time to $O(n log n)$. We also provide an $O(n^2log n)$ time algorithm to find a convex set containing exactly $left lceil r+12right red points and exactly $left lceil b+12right blue points of $S$, and show that balanced islands with more points do not always exist.

AB - 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 present algorithms to find convex sets containing a balanced number of red and blue points. We provide an $O(n^4)$ time algorithm that for a given $alpha in left [ 0,12 right ]$ finds a convex set containing exactly $lceil alpha r red points and exactly $lceil alpha b blue points of $S$. If $lceil alpha rlceil alpha b is not much larger than $13n$, we improve the running time to $O(n log n)$. We also provide an $O(n^2log n)$ time algorithm to find a convex set containing exactly $left lceil r+12right red points and exactly $left lceil b+12right blue points of $S$, and show that balanced islands with more points do not always exist.

KW - Equipartition, Islands, Convex sets, Computational geometry

U2 - 10.1016/j.ipl.2018.02.008

DO - 10.1016/j.ipl.2018.02.008

M3 - Article

VL - 135

SP - 28

EP - 32

JO - Information Processing Letters

JF - Information Processing Letters

SN - 0020-0190

ER -