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

Publikation: Beitrag in einer FachzeitschriftArtikelForschungBegutachtung

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.
Originalspracheenglisch
Seiten (von - bis)28 - 32
FachzeitschriftInformation Processing Letters
Jahrgang135
DOIs
PublikationsstatusVeröffentlicht - 2018

Schlagwörter

    Dies zitieren

    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, Jahrgang 135, 2018, S. 28 - 32.

    Publikation: Beitrag in einer FachzeitschriftArtikelForschungBegutachtung

    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, Jg. 135, S. 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 ; Jahrgang 135. S. 28 - 32.
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    title = "Computing 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 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.",
    keywords = "Equipartition, Islands, Convex sets, Computational geometry",
    author = "Oswin Aichholzer and Nieves Atienza and D{\'i}az-B{\'a}{\~n}ez, {Jos{\'e} M.} and Ruy Fabila-Monroy and David Flores-Pe{\~n}aloza and Pablo P{\'e}rez-Lantero and Birgit Vogtenhuber and {Urrutia Galicia}, Jorge",
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    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 -