2026-07-24T18:10:38Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:8834 2024-03-06T10:42:45Z ddc:004 open_access

Convex Hulls in Polygonal Domains Barba, Luis Hoffmann, Michael Korman, Matias Pilz, Alexander geometric graph polygonal domain geodesic hull shortest path 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Luis Barba and Michael Hoffmann and Matias Korman and Alexander Pilz 2018 Is Part Of LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SWAT.2018.8 urn:nbn:de:0030-drops-88343 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.8 eng https://creativecommons.org/licenses/by/3.0/legalcode