Computing Balanced Islands in Two Colored Point Sets in the Plane

Oswin Aichholzer, Nieves Atienza, José M. Díaz-Báñez, Ruy Fabila-Monroy, David Flores-Peñaloza, Pablo Pérez-Lantero, Birgit Vogtenhuber, Jorge Urrutia Galicia

Research output: Contribution to journalArticleResearchpeer-review

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 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.
LanguageEnglish
Pages28 - 32
JournalInformation Processing Letters
Volume135
DOIs
StatusPublished - 2018

Keywords

  • Equipartition, Islands, Convex sets, Computational geometry

Cite this

Aichholzer, O., Atienza, N., Díaz-Báñez, J. M., Fabila-Monroy, R., Flores-Peñaloza, D., Pérez-Lantero, P., ... Urrutia Galicia, J. (2018). Computing Balanced Islands in Two Colored Point Sets in the Plane. Information Processing Letters, 135, 28 - 32. https://doi.org/10.1016/j.ipl.2018.02.008

Computing Balanced Islands in Two Colored Point Sets in the Plane. / Aichholzer, Oswin; Atienza, Nieves; Díaz-Báñez, José M.; Fabila-Monroy, Ruy; Flores-Peñaloza, David; Pérez-Lantero, Pablo; Vogtenhuber, Birgit; Urrutia Galicia, Jorge.

In: Information Processing Letters, Vol. 135, 2018, p. 28 - 32.

Research output: Contribution to journalArticleResearchpeer-review

Aichholzer, O, Atienza, N, Díaz-Báñez, JM, Fabila-Monroy, R, Flores-Peñaloza, D, Pérez-Lantero, P, Vogtenhuber, B & Urrutia Galicia, J 2018, 'Computing Balanced Islands in Two Colored Point Sets in the Plane', Information Processing Letters, vol. 135, pp. 28 - 32. https://doi.org/10.1016/j.ipl.2018.02.008
Aichholzer O, Atienza N, Díaz-Báñez JM, Fabila-Monroy R, Flores-Peñaloza D, Pérez-Lantero P et al. Computing Balanced Islands in Two Colored Point Sets in the Plane. Information Processing Letters. 2018;135:28 - 32. https://doi.org/10.1016/j.ipl.2018.02.008
Aichholzer, Oswin ; Atienza, Nieves ; Díaz-Báñez, José M. ; Fabila-Monroy, Ruy ; Flores-Peñaloza, David ; Pérez-Lantero, Pablo ; Vogtenhuber, Birgit ; Urrutia Galicia, Jorge. / Computing Balanced Islands in Two Colored Point Sets in the Plane. In: Information Processing Letters. 2018 ; Vol. 135. pp. 28 - 32.
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