TY - GEN

T1 - Convex hulls in polygonal domains

AU - Barba, Luis

AU - Hoffmann, Michael

AU - Korman, Matias

AU - Pilz, Alexander

N1 - Funding Information:
Supported in part by MEXT KAKENHI Nos. 17K12635, 15H02665, and 24106007. 2 Supported by a Schrödinger fellowship of the Austrian Science Fund (FWF): J-3847-N35.
Publisher Copyright:
© Luis Barba, Michael Hoffmann, Matias Korman, and Alexander Pilz.

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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.

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.

KW - Geodesic hull

KW - Phrases geometric graph

KW - Polygonal domain

KW - Shortest path

UR - http://www.scopus.com/inward/record.url?scp=85049044687&partnerID=8YFLogxK

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U2 - 10.4230/LIPIcs.SWAT.2018.8

DO - 10.4230/LIPIcs.SWAT.2018.8

M3 - Conference contribution

AN - SCOPUS:85049044687

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 81

EP - 813

BT - 16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018

A2 - Eppstein, David

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018

Y2 - 18 June 2018 through 20 June 2018

ER -