Abstract
Given a set of prioritized disks with fixed centers in \(\mathbb {R}^2\) whose radii grow linearly over time, we are interested in computing an elimination order of these disks assuming that when two disks touch, the one with lower priority is ‘crushed’. A straightforward algorithm has running time \(O(n^2\log n)\) which we improve to expected \(O(n(\log ^6 n+\varDelta ^2 \log ^2 n + \varDelta ^4\log n))\) where \(\varDelta \) is the ratio between largest and smallest radii amongst the disks. For a very natural application of this problem in the map rendering domain, we have \(\varDelta =O(1)\).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Been, K., Daiches, E., Yap, C.: Dynamic map labeling. IEEE Trans. Visual Comput. Graphics 12(5), 773–780 (2006)
Been, K., Nöllenburg, M., Poon, S.-H., Wolff, A.: Optimizing active ranges for consistent dynamic map labeling. Comput. Geom. 43(3), 312–328 (2010)
Chan, T.M.: Closest-point problems simplified on the RAM. In: Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2002, pp. 472–473. Society for Industrial and Applied Mathematics, Philadelphia (2002)
Chan, T.M.: A dynamic data structure for 3-D convex hulls, 2-D nearest neighbor queries. J. ACM 57(3), 16:1–16:15 (2010)
Edelsbrunner, H., Mücke, E.P.: Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans. Graph. 9(1), 66–104 (1990)
Eppstein, D., Erickson, J.: Raising roofs, crashing cycles, and playing pool: applications of a data structure for finding pairwise interactions. Discrete Comput. Geometry 22(4), 569–592 (1999)
Funke, S., Klein, C., Mehlhorn, K., Schmitt, S.: Controlled perturbation for delaunay triangulations. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, pp. 1047–1056. Society for Industrial and Applied Mathematics, Philadelphia (2005)
Halperin, D., Shelton, C.R.: A perturbation scheme for spherical arrangements with application to molecular modeling. Comput. Geometry 10(4), 273–287 (1998). Special Issue on Applied Computational Geometry
Mount, D.M., Park, E.: A dynamic data structure for approximate range searching. In: Proceedings of the Twenty-sixth Annual Symposium on Computational Geometry, SoCG 2010, pp. 247–256. ACM, New York (2010)
Schwartges, N., Allerkamp, D., Haunert, J., Wolff, A.: Optimizing active ranges for point selection in dynamic maps. In: Proceeding 16th ICA Generalisation Workshop (ICAGW 2013), 10 pages (2013)
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Funke, S., Krumpe, F., Storandt, S. (2016). Crushing Disks Efficiently. In: Mäkinen, V., Puglisi, S., Salmela, L. (eds) Combinatorial Algorithms. IWOCA 2016. Lecture Notes in Computer Science(), vol 9843. Springer, Cham. https://doi.org/10.1007/978-3-319-44543-4_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-44543-4_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-44542-7
Online ISBN: 978-3-319-44543-4
eBook Packages: Computer ScienceComputer Science (R0)