# 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

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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

- Computer Science
- GIScience
- 2018

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 two… Expand

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

- Mathematics
- 2011

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. These… Expand

Finding the Upper Envelope of n Line Segments in O(n log n) Time

- Mathematics, Computer Science
- Inf. Process. Lett.
- 1989

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

- Computer Science, Mathematics
- Graduate texts in mathematics
- 2002

This book is primarily a textbook introduction to various areas of discrete geometry, in which several key results and methods are explained, in an accessible and concrete manner, in each area. Expand

Almost tight upper bounds for vertical decompositions in four dimensions

- Mathematics
- 2004

We show that the complexity of the vertical decomposition of an arrangement of n fixed-degree algebraic surfaces or surface patches in four dimensions is O(n4+e), for any e > 0. This improves the… Expand

Ray shooting in polygons using geodesic triangulations

- Mathematics, Computer Science
- Algorithmica
- 2005

A simple decomposition scheme that partitions the interior of P intoO(n) so-called geodesic triangles, so that any line segment interior toP crosses at most 2 logn of these triangles can be used to preprocessP in a very simple manner, so any ray-shooting query can be answered in timeO(logn). Expand

Preprocessing Imprecise Points for Delaunay Triangulation: Simplified and Extended

- Mathematics, Computer Science
- Algorithmica
- 2010

This work shows how to leverage the knowledge of ℛ for faster Delaunay computation and optimally handles a wide variety of inputs, e.g., overlapping disks of different sizes and fat regions. Expand