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 - Funding Information:
A preliminary version appeared as H.-K. Ahn, S.W. Bae, J. Choi, M. Korman, W. Mulzer, E. Oh, J.-W. Park, A. van Renssen, A. Vigneron. Faster Algorithms for Growing Prioritized Disks and Rectangles. Proc. 28th ISAAC, pp. 3:1–3:13. The work by H.-K. Ahn, J. Choi, E. Oh was supported by the MSIT (Ministry of Science and ICT), Korea, under the SW Starlab support program (IITP–2017–0–00905) supervised by the IITP (Institute for Information & communications Technology Promotion). S.W. Bae was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2015R1D1A1A01057220 and 2018R1D1A1B07042755). M. Korman was supported in part by National Science Foundation awards CCF-1422311 and KAKENHI No. 17K12635, Japan. W. Mulzer was supported in part by Deutsche Forschungsgemeinschaft Grants MU 3501/1 and MU 3501/2 and by European Research Council StG 757609. J.-W. Park was supported by the MSIT (Ministry of Science and ICT), Korea, under the Next-Generation Information Computing Development Program supervised by the National Research Foundation of Korea (NRF-2017M3C4A7066317). A. van Renssen was supported by JST ERATO Grant Number JPMJER1201, Japan. A. Vigneron was supported by the 2016 Research Fund (1.160054.01) of UNIST (Ulsan National Institute of Science and Technology). This work was initiated during the 20th Korean Workshop on Computational Geometry. The authors would like to thank the other participants for motivating and insightful discussions. We would also like to thank the anonymous reviewers for their close reading of the paper and for many helpful comments that significantly improved the presentation of the paper.
Funding Information:
A preliminary version appeared as H.-K. Ahn, S.W. Bae, J. Choi, M. Korman, W. Mulzer, E. Oh, J.-W. Park, A. van Renssen, A. Vigneron. Faster Algorithms for Growing Prioritized Disks and Rectangles. Proc. 28th ISAAC, pp. 3:1–3:13. The work by H.-K. Ahn, J. Choi, E. Oh was supported by the MSIT (Ministry of Science and ICT), Korea, under the SW Starlab support program (IITP–2017–0–00905) supervised by the IITP (Institute for Information & communications Technology Promotion). S.W. Bae was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2015R1D1A1A01057220 and 2018R1D1A1B07042755). M. Korman was supported in part by National Science Foundation awards CCF-1422311 and KAKENHI No. 17K12635, Japan. W. Mulzer was supported in part by Deutsche Forschungsgemeinschaft Grants MU 3501/1 and MU 3501/2 and by European Research Council StG 757609. J.-W. Park was supported by the MSIT (Ministry of Science and ICT), Korea, under the Next-Generation Information Computing Development Program supervised by the National Research Foundation of Korea (NRF-2017M3C4A7066317). A. van Renssen was supported by JST ERATO Grant Number JPMJER1201, Japan. A. Vigneron was supported by the 2016 Research Fund (1.160054.01) of UNIST (Ulsan National Institute of Science and Technology).
Publisher Copyright:
© 2019 Elsevier B.V.

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 -