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
SN - 0020-0190
VL - 135
SP - 28
EP - 32
JO - Information Processing Letters
JF - Information Processing Letters
ER -