Abstract
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 \(\mathcal{P}\) by a polygon \(\mathcal{P}'\) such that (1) \(\mathcal{P}'\) contains \(\mathcal{P}\), (2) \(\mathcal{P}'\) has its reflex vertices at the same positions as \(\mathcal{P}\), and (3) the number of vertices of \(\mathcal{P}'\) 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 \(\mathcal{P}\) and \(\mathcal{P}'\), our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of \(\mathcal{P}\).
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 ‘Combinatorial Problems on Geometric Graphs’. 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. A.P. is recipient of a DOC-fellowship of the Austrian Academy of Sciences at the Institute for Software Technology, Graz University of Technology, Austria.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aichholzer, O., Hackl, T., Korman, M., Pilz, A., Vogtenhuber, B.: Geodesic-preserving polygon simplification. ArXiv e-prints (2013), arXiv:1309.3858
Aichholzer, O., Korman, M., Pilz, A., Vogtenhuber, B.: Geodesic order types. Algorithmica (to appear, 2013)
Aichholzer, O., Miltzow, T., Pilz, A.: Extreme point and halving edge search in abstract order types. Computational Geometry: Theory and Applications 46(8), 970–978 (2013)
Arkin, E.M., Chiang, Y.-J., Held, M., Mitchell, J.S.B., Sacristan, V., Skiena, S., Yang, T.-H.: On minimum-area hulls. Algorithmica 21(1), 119–136 (1998)
Aronov, B.: On the geodesic Voronoi diagram of point sites in a simple polygon. In: SoCG, pp. 39–49 (1987)
Aronov, B., Fortune, S., Wilfong, G.T.: The furthest-site geodesic Voronoi diagram. Discrete and Computational Geometry 9, 217–255 (1993)
Bern, M.W., Eppstein, D.: Mesh generation and optimal triangulation. In: Computing in Euclidean Geometry. Lecture Notes Series on Computing, vol. 4, pp. 47–123. World Scientific (1995)
Bose, P., Demaine, E.D., Hurtado, F., Iacono, J., Langerman, S., Morin, P.: Geodesic ham-sandwich cuts. In: SoCG, pp. 1–9 (2004)
Demaine, E.D., Erickson, J., Hurtado, F., Iacono, J., Langerman, S., Meijer, H., Overmars, M.H., Whitesides, S.: Separating point sets in polygonal environments. International Journal of Computational Geometry and Applications 15(4), 403–420 (2005)
Douglas, D.H., Peucker, T.K.: Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. The Canadian Cartographer 10(2), 112–122 (1973)
Ghosh, S.: Visibility Algorithms in the Plane. Cambridge University Press, New York (2007)
Guibas, L.J., Hershberger, J.: Optimal shortest path queries in a simple polygon. Journal of Computer and System Sciences 39(2), 126–152 (1989)
Guibas, L.J., Hershberger, J., Leven, D., Sharir, M., Tarjan, R.E.: Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica 2, 209–233 (1987)
Guibas, L.J., Hershberger, J., Mitchell, J.S.B., Snoeyink, J.: Approximating polygons and subdivisions with minimum link paths. Journal of Computer and System Sciences 3(4), 383–415 (1993)
Gupta, H., Wenger, R.: Constructing pairwise disjoint paths with few links. ACM Transactions on Algorithms 3(3) (2007)
Hershberger, J., Suri, S.: An optimal algorithm for euclidean shortest paths in the plane. SIAM Journal of Computing 28(6), 2215–2256 (1999)
Hershberger, J., Snoeyink, J.: Computing minimum length paths of a given homotopy class. Computational Geometry: Theory and Applications 4, 63–97 (1994)
Hertel, S., Mehlhorn, K.: Fast triangulation of simple polygons. In: Karpinski, M. (ed.) FCT 1983. LNCS, vol. 158, pp. 207–218. Springer, Heidelberg (1983)
Melkman, A.A.: On-line construction of the convex hull of a simple polyline. Information Processing Letters 25(1), 11–12 (1987)
Mitchell, J.S.B.: L 1 shortest paths among polygonal obstacles in the plane. Algorithmica 8, 55–88 (1992)
Mitchell, J.S.B.: Shortest paths and networks. In: Handbook of Discrete and Computational Geometry, 2nd edn., pp. 607–642. Chapman & Hall/CRC (2004)
Mitchell, J.S.B., Polishchuk, V., Sysikaski, M.: Minimum-link paths revisited. ArXiv e-prints (2013), arXiv:1302.3091
Papadopoulou, E., Lee, D.T.: A new approach for the geodesic Voronoi diagram of points in a simple polygon and other restricted polygonal domains. Algorithmica 20(4), 319–352 (1998)
Rote, G., Santos, F., Streinu, I.: Pseudo-triangulations—a survey. In: Surveys on Discrete and Computational Geometry—Twenty Years Later. Contemporary Mathematics, pp. 343–410 (2008)
Speckmann, B., Tóth, C.D.: Allocating vertex π-guards in simple polygons via pseudo-triangulations. Discrete and Computational Geometry 33(2), 345–364 (2005)
Suri, S.: A linear time algorithm for minimum link paths inside a simple polygon. Computer Vision, Graphics, and Image Processing 35(1), 99–110 (1986)
Toussaint, G.T.: Computing geodesic properties inside a simple polygon. Revue D’Intelligence Artificielle 3(2), 9–42 (1989)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Aichholzer, O., Hackl, T., Korman, M., Pilz, A., Vogtenhuber, B. (2013). Geodesic-Preserving Polygon Simplification. In: Cai, L., Cheng, SW., Lam, TW. (eds) Algorithms and Computation. ISAAC 2013. Lecture Notes in Computer Science, vol 8283. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45030-3_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-45030-3_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-45029-7
Online ISBN: 978-3-642-45030-3
eBook Packages: Computer ScienceComputer Science (R0)