Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

**Date and Time**: Tuesday, October 17, 2017, 12:15 pm

**Duration**: 30 minutes

**Location**: OAT S15/S16/S17

**Speaker**: Luis Barba

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 hull is based on the ``rubber band'' conception of the convex hull as a shortest curve that encloses a given set of sites. However, we show that 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 and to the geodesic hull inside a simple polygon. As a main result, we describe a class of polygonal domains that suffice to represent geodesic convex hulls 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 $O(n^3h^{3+\varepsilon})$ time.

This is joint work with M. Hoffmann, M. Korman and A. Pilz

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