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 \(\varDelta =O(1)\).
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
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)