eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-12-07
3:1
3:13
10.4230/LIPIcs.ISAAC.2017.3
article
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
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol092-isaac2017/LIPIcs.ISAAC.2017.3/LIPIcs.ISAAC.2017.3.pdf
map labeling
growing disks
elimination order