Faster Algorithms for Growing Prioritized Disks and Rectangles

Ahn, Hee-Kap; Bae, Sang Won; Choi, Jongmin; Korman, Matias; Mulzer, Wolfgang; Oh, Eunjin; Park, Ji-won; van Renssen, André; Vigneron, Antoine

doi:10.4230/LIPIcs.ISAAC.2017.3

File

LIPIcs.ISAAC.2017.3.pdf

Filesize: 0.64 MB
13 pages

Document Identifiers

DOI: 10.4230/LIPIcs.ISAAC.2017.3
URN: urn:nbn:de:0030-drops-82199

Author Details

Hee-Kap Ahn

Sang Won Bae

Jongmin Choi

Matias Korman

Wolfgang Mulzer

Eunjin Oh

Ji-won Park

André van Renssen

Antoine Vigneron

Cite AsGet BibTex

Hee-Kap Ahn, Sang Won Bae, Jongmin Choi, Matias Korman, Wolfgang Mulzer, Eunjin Oh, Ji-won Park, André van Renssen, and Antoine Vigneron. Faster Algorithms for Growing Prioritized Disks and Rectangles. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 3:1-3:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ISAAC.2017.3

Abstract

Motivated by map labeling, we study the problem in which we are given a collection of n disks in the plane that grow at possibly different speeds. Whenever two disks meet, the one with the higher index disappears. This problem was introduced by Funke, Krumpe, and Storandt[IWOCA 2016]. We provide the first general subquadratic algorithm for computing the times and the order of disappearance. Our algorithm also works for other shapes (such as rectangles) and in any fixed dimension. Using quadtrees, we provide an alternative algorithm that runs in near linear time, although this second algorithm has a logarithmic dependence on either the ratio of the fastest speed to the slowest speed of disks or the spread of the disk centers (the ratio of the maximum to the minimum distance between them). Our result improves the running times of previous algorithms by Funke, Krumpe, and Storandt [IWOCA 2016], Bahrdt et al. [ALENEX 2017], and Funke and Storandt [EWCG 2017]. Finally, we give an \Omega(n\log n) lower bound on the problem, showing that our quadtree algorithms are almost tight.

Keywords

map labeling
growing disks
elimination order

Metrics

Access Statistics
Total Accesses (updated on a weekly basis)

0

PDF Downloads

0

Metadata Views

References

Pankaj K. Agarwal, Boris Aronov, and Micha Sharir. Computing envelopes in four dimensions with applications. SIAM J. Comput., 26(6):1714-1732, 1997.
Daniel Bahrdt, Michael Becher, Stefan Funke, Filip Krumpe, André Nusser, Martin Seybold, and Sabine Storandt. Growing balls in ℝ^d. In Proc. 19th Workshop Algorithm Eng. Exp. (ALENEX), pages 247-258, 2017.
Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag, 2008.
Thom Castermans, Bettina Speckmann, Frank Staals, and Kevin Verbeek. Agglomerative clustering of growing squares. CoRR, abs/1706.10195, 2017. URL: http://arxiv.org/abs/1706.10195.
Bernard Chazelle, Herbert Edelsbrunner, Michelangelo Grigni, Leonidas J. Guibas, John Hershberger, Micha Sharir, and Jack Snoeyink. Ray shooting in polygons using geodesic triangulations. Algorithmica, 12(1):54-68, 1994.
Michael Formann. Weighted closest pairs. In Proc. 10th Sympos. Theoret. Aspects Comput. Sci. (STACS), pages 270-281, 1993.
Stefan Funke, Filip Krumpe, and Sabine Storandt. Crushing disks efficiently. In Proc. 27th Int. Workshop Comb. Alg. (IWOCA), pages 43-54, 2016.
Stefan Funke and Sabine Storandt. Parametrized runtimes for ball tournaments. In Proc. 33rd European Workshop Comput. Geom. (EWCG), pages 221-224, 2017.
Sariel Har-Peled. Geometric Approximation Algorithms. American Mathematical Society, Boston, MA, USA, 2011.
John Hershberger. Finding the upper envelope of n line segments in O(n log n) time. Inform. Process. Lett., 33(4):169-174, 1989.
Vladlen Koltun. Almost tight upper bounds for vertical decompositions in four dimensions. J. ACM, 51(5):699-730, 2004.
Jiří Matoušek. Lectures on Discrete Geometry. Springer-Verlag, 2002.