TY - JOUR

T1 - Faster algorithms for growing prioritized disks and rectangles

AU - Ahn, Hee Kap

AU - Bae, Sang Won

AU - Choi, Jongmin

AU - Korman, Matias

AU - Mulzer, Wolfgang

AU - Oh, Eunjin

AU - Park, Ji won

AU - van Renssen, André

AU - Vigneron, Antoine

N1 - Publisher Copyright:
© 2019 Elsevier B.V.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019/7

Y1 - 2019/7

N2 - Motivated by map labeling, Funke, Krumpe, and Storandt [IWOCA 2016] introduced the following problem: we are given a sequence of n disks in the plane. Initially, all disks have radius 0, and they grow at constant, but possibly different, speeds. Whenever two disks touch, the one with the higher index disappears. The goal is to determine the elimination order, i.e., the order in which the disks disappear. We provide the first general subquadratic algorithm for this problem. Our solution extends to other shapes (e.g., rectangles), and it works in any fixed dimension. We also describe an alternative algorithm that is based on quadtrees. Its running time is O(n(logn+min{logΔ,logΦ})), where Δ is the ratio of the fastest and the slowest growth rate and Φ is the ratio of the largest and the smallest distance between two disk centers. This improves the running times of previous algorithms by Funke, Krumpe, and Storandt [IWOCA 2016], Bahrdt et al. [ALENEX 2017], and Funke and Storandt [EuroCG 2017]. Finally, we give an Ω(nlogn) lower bound, showing that our quadtree algorithms are almost tight.

AB - Motivated by map labeling, Funke, Krumpe, and Storandt [IWOCA 2016] introduced the following problem: we are given a sequence of n disks in the plane. Initially, all disks have radius 0, and they grow at constant, but possibly different, speeds. Whenever two disks touch, the one with the higher index disappears. The goal is to determine the elimination order, i.e., the order in which the disks disappear. We provide the first general subquadratic algorithm for this problem. Our solution extends to other shapes (e.g., rectangles), and it works in any fixed dimension. We also describe an alternative algorithm that is based on quadtrees. Its running time is O(n(logn+min{logΔ,logΦ})), where Δ is the ratio of the fastest and the slowest growth rate and Φ is the ratio of the largest and the smallest distance between two disk centers. This improves the running times of previous algorithms by Funke, Krumpe, and Storandt [IWOCA 2016], Bahrdt et al. [ALENEX 2017], and Funke and Storandt [EuroCG 2017]. Finally, we give an Ω(nlogn) lower bound, showing that our quadtree algorithms are almost tight.

KW - Condition number

KW - Data structures

KW - Elimination order

KW - Lower bound

KW - Quadtree

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U2 - 10.1016/j.comgeo.2019.02.001

DO - 10.1016/j.comgeo.2019.02.001

M3 - Article

AN - SCOPUS:85062414647

VL - 80

SP - 23

EP - 39

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

ER -