Barba, Luis ;
Hoffmann, Michael ;
Korman, Matias ;
Pilz, Alexander
Convex Hulls in Polygonal Domains
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 straightline 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 NPhard 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.
BibTeX  Entry
@InProceedings{barba_et_al:LIPIcs:2018:8834,
author = {Luis Barba and Michael Hoffmann and Matias Korman and Alexander Pilz},
title = {{Convex Hulls in Polygonal Domains}},
booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)},
pages = {8:18:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770682},
ISSN = {18688969},
year = {2018},
volume = {101},
editor = {David Eppstein},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8834},
URN = {urn:nbn:de:0030drops88343},
doi = {10.4230/LIPIcs.SWAT.2018.8},
annote = {Keywords: geometric graph, polygonal domain, geodesic hull, shortest path}
}
2018
Keywords: 

geometric graph, polygonal domain, geodesic hull, shortest path 
Seminar: 

16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

Issue date: 

2018 
Date of publication: 

2018 