@article{b3e30aa4e3464b8194b1e500b7ea935e,

title = "Geodesic-preserving polygon simplification",

abstract = "Polygons are a paramount data structure in computational geometry. While the com- plexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reex vertices of the polygon. In this paper, we give an easy-to- describe linear-time method to replace an input polygon P by a polygon P' such that (1) P' contains P, (2) P0 has its reex vertices at the same positions as P, and (3) the number of vertices of P' is linear in the number of reex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both P and P', our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reex vertices rather than on the size of P. We describe several of these applications (including linear-time post-processing steps that might be necessary).",

keywords = "Polygon, Voronoi diagram, geodesic, geodesic hull, polygonal domain, shortest path, simple polygon",

author = "Oswin Aichholzer and Thomas Hackl and Matias Korman and Alexander Pilz and Birgit Vogtenhuber",

note = "Funding Information: Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon P by a polygon P′ such that (1) P′ contains P, (2) P′ has its reflex vertices at the same positions as P, and (3) the number of vertices of P′ is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both P and P′, our algorithm can be used as a preprocessing step for several algorithms and makes their running time ∗Research supported by the ESF EUROCORES programme EuroGIGA - ComPoSe, Austrian Science Fund (FWF): I 648-N18 and grant EUI-EURC-2011-4306. T.H. supported by the Austrian Science Fund (FWF): P23629-N18 {\textquoteleft}Combinatorial Problems on Geometric Graphs{\textquoteright}. M.K. received support of the Secretary for Universities and Research of the Ministry of Economy and Knowledge of the Government of Catalonia and the European Union. Part of this work has been done while A.P. was recipient of a DOC-fellowship of the Austrian Academy of Sciences at the Institute for Software Technology, Graz University of Technology, Austria. A preliminary version of this paper appeared in the proceedings of ISAAC 2013. Publisher Copyright: {\textcopyright} 2014 World Scientific Publishing Company.",

year = "2014",

month = dec,

day = "26",

doi = "10.1142/S0218195914600097",

language = "English",

volume = "24",

pages = "307--323",

journal = "International Journal of Computational Geometry and Applications",

issn = "0218-1959",

publisher = "World Scientific Publishing Co. Pte Ltd",

number = "4",

}