Convex hulls in polygonal domains

Luis Barba, Michael Hoffmann, Matias Korman, Alexander Pilz

Publikation: Beitrag in Buch/Bericht/KonferenzbandBeitrag in einem KonferenzbandForschungBegutachtung

Abstract

We study generalizations of convex hulls to polygonal domains with holes. Convexity in Euclidean space is based on the notion of shortest paths, which are straight-line segments. In a polygonal domain, shortest paths are polygonal paths called geodesics. One possible generalization of convex hulls is based on the “rubber band” conception of the convex hull boundary as a shortest curve that encloses a given set of sites. However, it is NP-hard to compute such a curve in a general polygonal domain. Hence, we focus on a di erent, more direct generalization of convexity, where a set X is geodesically convex if it contains all geodesics between every pair of points x, y ∈ X. The corresponding geodesic convex hull presents a few surprises, and turns out to behave quite di erently compared to the classic Euclidean setting or to the geodesic hull inside a simple polygon. We describe a class of geometric objects that su ce to represent geodesic convex hulls of sets of sites, and characterize which such domains are geodesically convex. Using such a representation we present an algorithm to construct the geodesic convex hull of a set of O(n) sites in a polygonal domain with a total of n vertices and h holes in O(n3h3+ε) time, for any constant ε > 0.

Originalspracheenglisch
Titel16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018
Herausgeber (Verlag)Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH
Seiten81-813
Seitenumfang733
Band101
ISBN (elektronisch)9783959770682
DOIs
PublikationsstatusVeröffentlicht - 1 Jun 2018
Extern publiziertJa
Veranstaltung16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018 - Malmo, Schweden
Dauer: 18 Jun 201820 Jun 2018

Konferenz

Konferenz16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018
LandSchweden
OrtMalmo
Zeitraum18/06/1820/06/18

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    Barba, L., Hoffmann, M., Korman, M., & Pilz, A. (2018). Convex hulls in polygonal domains. in 16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018 (Band 101, S. 81-813). Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH. https://doi.org/10.4230/LIPIcs.SWAT.2018.8

    Convex hulls in polygonal domains. / Barba, Luis; Hoffmann, Michael; Korman, Matias; Pilz, Alexander.

    16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018. Band 101 Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH, 2018. S. 81-813.

    Publikation: Beitrag in Buch/Bericht/KonferenzbandBeitrag in einem KonferenzbandForschungBegutachtung

    Barba, L, Hoffmann, M, Korman, M & Pilz, A 2018, Convex hulls in polygonal domains. in 16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018. Bd. 101, Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH, S. 81-813, Malmo, Schweden, 18/06/18. https://doi.org/10.4230/LIPIcs.SWAT.2018.8
    Barba L, Hoffmann M, Korman M, Pilz A. Convex hulls in polygonal domains. in 16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018. Band 101. Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH. 2018. S. 81-813 https://doi.org/10.4230/LIPIcs.SWAT.2018.8
    Barba, Luis ; Hoffmann, Michael ; Korman, Matias ; Pilz, Alexander. / Convex hulls in polygonal domains. 16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018. Band 101 Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH, 2018. S. 81-813
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    AB - We study generalizations of convex hulls to polygonal domains with holes. Convexity in Euclidean space is based on the notion of shortest paths, which are straight-line segments. In a polygonal domain, shortest paths are polygonal paths called geodesics. One possible generalization of convex hulls is based on the “rubber band” conception of the convex hull boundary as a shortest curve that encloses a given set of sites. However, it is NP-hard to compute such a curve in a general polygonal domain. Hence, we focus on a di erent, more direct generalization of convexity, where a set X is geodesically convex if it contains all geodesics between every pair of points x, y ∈ X. The corresponding geodesic convex hull presents a few surprises, and turns out to behave quite di erently compared to the classic Euclidean setting or to the geodesic hull inside a simple polygon. We describe a class of geometric objects that su ce to represent geodesic convex hulls of sets of sites, and characterize which such domains are geodesically convex. Using such a representation we present an algorithm to construct the geodesic convex hull of a set of O(n) sites in a polygonal domain with a total of n vertices and h holes in O(n3h3+ε) time, for any constant ε > 0.

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