# Faster algorithms for growing prioritized disks and rectangles

@article{Ahn2019FasterAF,
title={Faster algorithms for growing prioritized disks and rectangles},
author={Hee-Kap Ahn and Sang Won Bae and Jong Min Choi and Matias Korman and Wolfgang Mulzer and Eunjin Oh and Ji-won Park and Andr{\'e} van Renssen and Antoine Vigneron},
journal={Comput. Geom.},
year={2019},
volume={80},
pages={23-39}
}
Motivated by map labeling, we study the problem in which we are given a collection of $n$ disks $D_1, \dots, D_n$ in the plane that grow at possibly different speeds. Whenever two disks meet, the one with the lower 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. This algorithm also works for other shapes (such as rectangles) and in any fixed… Expand
2 Citations

#### Tables and Topics from this paper

Parametrized Runtimes for Label Tournaments
• Mathematics, Computer Science
• COCOA
• 2019
A new parameter C which denotes the number of different disk sizes in the input is introduced which is upper bounded by n and designed an algorithm which runs in time $$\mathcal {O}(n C \log ^{\mathcal{O} (1)} n)$$. Expand
Labeling Points of Interest in Dynamic Maps using Disk Labels
A map labeling scheme, which allows to label maps at an interactive speed, and some extensions to the labeling that could be used for more sophisticated labeling features such as area labels turning into point labels at smaller map scales. Expand

#### References

SHOWING 1-10 OF 24 REFERENCES
Crushing Disks Efficiently
• Mathematics, Computer Science
• IWOCA
• 2016
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 twoExpand
Computing Envelopes in Four Dimensions with Applications
• Mathematics, Computer Science
• SIAM J. Comput.
• 1997
Improved algorithmic solutions to several problems in computational geometry are obtained, including computing the width of a point set in 3-space, computing the "biggest stick" in a simple polygon in the plane, and computing the smallest-width annulus covering a planar point set. Expand
Efficient Point Location in a Convex Spatial Cell-Complex
• Mathematics, Computer Science
• SIAM J. Comput.
• 1992
A new approach to point location in a three-dimensional cell complex {\em P}, which may be viewed as a nontrivial generalization of a corresponding two-dimensional technique due to Sarnak and Tarjan is proposed. Expand
Davenport-Schinzel sequences and their geometric applications
• Mathematics, Computer Science
• 1995
A close to linear bound on the maximum length of Davenport--Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems. Expand
Geometric Approximation Algorithms
Exact algorithms for dealing with geometric objects are complicated, hard to implement in practice, and slow. Over the last 20 years a theory of geometric approximation algorithms has emerged. TheseExpand
Finding the Upper Envelope of n Line Segments in O(n log n) Time
The method can be used to compute the upper envelope of “segments” that intersect pairwise at most k times and computes theupper envelope in O(λk + 1(n)log n) time. Expand
Lectures on discrete geometry
• J. Matousek
• Computer Science, Mathematics