Faster Algorithms for Growing Prioritized Disks and Rectangles

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



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

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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.

Subject Classification

Keywords
  • map labeling
  • growing disks
  • elimination order

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References

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