## 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(n^{3}h^{3+ε}) time, for any constant ε > 0.

Originalsprache | englisch |
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Titel | 16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018 |

Herausgeber (Verlag) | Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH |

Seiten | 81-813 |

Seitenumfang | 733 |

Band | 101 |

ISBN (elektronisch) | 9783959770682 |

DOIs | |

Publikationsstatus | Veröffentlicht - 1 Jun 2018 |

Extern publiziert | Ja |

Veranstaltung | 16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018 - Malmo, Schweden Dauer: 18 Jun 2018 → 20 Jun 2018 |

### Konferenz

Konferenz | 16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018 |
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Land | Schweden |

Ort | Malmo |

Zeitraum | 18/06/18 → 20/06/18 |

## ASJC Scopus subject areas

- Software