Computing the Geodesic Centers of a Polygonal Domain

We present an algorithm that computes the geodesic center of a given polygonal domain. The running time of our algorithm is $O(n^{12+\epsilon})$ for any $\epsilon>0$, where $n$ is the number of corners of the input polygonal domain. Prior to our work, only the very special case where a simple polygon is given as input has been intensively studied in the 1980s, and an $O(n \log n)$-time algorithm is known by Pollack et al. Our algorithm is the first one that can handle general polygonal domains having one or more polygonal holes.

[1]  Joseph S. B. Mitchell,et al.  Two-point Euclidean shortest path queries in the plane , 1999, SODA '99.

[2]  Joseph S. B. Mitchell,et al.  Shortest paths among obstacles in the plane , 1993, SCG '93.

[3]  Bernard Chazelle,et al.  A theorem on polygon cutting with applications , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[4]  Prosenjit Bose,et al.  A Linear-Time Algorithm for the Geodesic Center of a Simple Polygon , 2015, Discret. Comput. Geom..

[5]  Sang Won Bae,et al.  The Geodesic Diameter of Polygonal Domains , 2010, ESA.

[6]  Micha Sharir,et al.  Computing the geodesic center of a simple polygon , 1989, Discret. Comput. Geom..

[7]  Subhash Suri,et al.  Matrix searching with the shortest path metric , 1993, SIAM J. Comput..

[8]  Sure.  Computing Geodesic Furthest Neighbors in Simple Polygons * , 2003 .

[9]  Joseph S. B. Mitchell,et al.  Geometric Shortest Paths and Network Optimization , 2000, Handbook of Computational Geometry.

[10]  G. Toussaint,et al.  Computing the Geodesic Center of a Simple Polygon , 1987 .

[11]  M. Sharir,et al.  New bounds for lower envelopes in three dimensions, with applications to visibility in terrains , 1993, SCG '93.

[12]  Vladlen Koltun.  Almost tight upper bounds for lower envelopes in higher dimensions , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[13]  Subhash Suri,et al.  An Optimal Algorithm for Euclidean Shortest Paths in the Plane , 1999, SIAM J. Comput..