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Convex Hulls in Polygonal Domains

Authors Luis Barba, Michael Hoffmann , Matias Korman, Alexander Pilz

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Author Details

Luis Barba
  • Department of Computer Science, ETH Zürich, Zürich, Switzerland
Michael Hoffmann
  • Department of Computer Science, ETH Zürich, Zürich, Switzerland
Matias Korman
  • Tohoku University, Sendai, Japan
Alexander Pilz
  • Department of Computer Science, ETH Zürich. Zürich, Switzerland

Funding

  • Korman, Matias: Supported in part by MEXT KAKENHI Nos. 17K12635, 15H02665, and 24106007.
  • Pilz, Alexander: Supported by a Schrödinger fellowship of the Austrian Science Fund (FWF): J-3847-N35.

Cite AsGet BibTex

Luis Barba, Michael Hoffmann, Matias Korman, and Alexander Pilz. Convex Hulls in Polygonal Domains. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SWAT.2018.8

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 different, more direct generalization of convexity, where a set X is geodesically convex if it contains all geodesics between every pair of points x,y in X. The corresponding geodesic convex hull presents a few surprises, and turns out to behave quite differently compared to the classic Euclidean setting or to the geodesic hull inside a simple polygon. We describe a class of geometric objects that suffice 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(n^3h^{3+epsilon}) time, for any constant epsilon > 0.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • geometric graph
  • polygonal domain
  • geodesic hull
  • shortest path

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