research-article

Kinetic 2-centers in the black-box model

Authors:

Marcel Roeloffzen,

Bettina SpeckmannAuthors Info & Claims

SoCG '13: Proceedings of the twenty-ninth annual symposium on Computational geometry

Pages 145 - 154

https://doi.org/10.1145/2462356.2462393

Published: 17 June 2013 Publication History

Abstract

We study two versions of the 2-center problem for moving points in the plane. Given a set P of n points, the Euclidean 2-center problem asks for two congruent disks of minimum size that together cover P; the rectilinear 2-center problem correspondingly asks for two congruent axis-aligned squares of minimum size that together cover P. Our methods work in the black-box KDS model, where we receive the locations of the points at regular time steps and we know an upper bound d_{max} on the maximum displacement of any point within one time step. We show how to maintain the rectilinear 2-center in amortized sub-linear time per time step, under certain assumptions on the distribution of the point set P. For the Euclidean 2-center we give a similar result: we can maintain in amortized sub-linear time (again under certain assumptions on the distribution) a (1+ε)-approximation of the optimal 2-center. In many cases---namely when the distance between the centers of the disks is relatively large or relatively small---the solution we maintain is actually optimal.

We also present results for the case where the maximum speed of the centers is bounded. We describe a simple scheme to maintain a 2-approximation of the rectilinear 2-center, and we provide a scheme which gives a better approximation factor depending on several parameters of the point set and the maximum allowed displacement of the centers. The latter result can be used to obtain a 2.29-approximation for the Euclidean 2-center; this is an improvement over the previously best known bound of 8/π approx 2.55. These algorithms run in amortized sub-linear time per time step, as before under certain assumptions on the distribution.

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Cited By

Evans WKirkpatrick D(2024)Minimizing Query Frequency to Bound Congestion Potential for Moving Entities at a Fixed Target TimeAlgorithms10.3390/a1706024617:6(246)Online publication date: 6-Jun-2024
https://doi.org/10.3390/a17060246
Evans WKirkpatrick D(2023)A Frequency-Competitive Query Strategy for Maintaining Low Collision Potential Among Moving EntitiesApproximation and Online Algorithms10.1007/978-3-031-49815-2_2(14-28)Online publication date: 22-Dec-2023
https://doi.org/10.1007/978-3-031-49815-2_2
Evans WKirkpatrick D(2023)Minimizing Query Frequency to Bound Congestion Potential for Moving Entities at a Fixed Target TimeFundamentals of Computation Theory10.1007/978-3-031-43587-4_12(162-175)Online publication date: 21-Sep-2023
https://doi.org/10.1007/978-3-031-43587-4_12
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Index Terms

Kinetic 2-centers in the black-box model
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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cover image ACM Conferences

SoCG '13: Proceedings of the twenty-ninth annual symposium on Computational geometry

June 2013

472 pages

ISBN:9781450320313

DOI:10.1145/2462356

General Chairs:
Guilherme D. da Fonseca
UNIRIO
,
Thomas Lewiner
PUC-Rio
,
Luis Peñaranda
IMPA
,
Program Chairs:
Timothy Chan
University of Waterloo
,
Rolf Klein
University of Bonn

Copyright © 2013 ACM.

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Published: 17 June 2013

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SoCG '13

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SoCG '13: Symposium on Computational Geometry 2013

June 17 - 20, 2013

Rio de Janeiro, Brazil

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SoCG '13 Paper Acceptance Rate 48 of 137 submissions, 35%;

Overall Acceptance Rate 625 of 1,685 submissions, 37%

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Cited By

Evans WKirkpatrick D(2024)Minimizing Query Frequency to Bound Congestion Potential for Moving Entities at a Fixed Target TimeAlgorithms10.3390/a1706024617:6(246)Online publication date: 6-Jun-2024
https://doi.org/10.3390/a17060246
Evans WKirkpatrick D(2023)A Frequency-Competitive Query Strategy for Maintaining Low Collision Potential Among Moving EntitiesApproximation and Online Algorithms10.1007/978-3-031-49815-2_2(14-28)Online publication date: 22-Dec-2023
https://doi.org/10.1007/978-3-031-49815-2_2
Evans WKirkpatrick D(2023)Minimizing Query Frequency to Bound Congestion Potential for Moving Entities at a Fixed Target TimeFundamentals of Computation Theory10.1007/978-3-031-43587-4_12(162-175)Online publication date: 21-Sep-2023
https://doi.org/10.1007/978-3-031-43587-4_12
Hoog v.d. Ivan Kreveld MMeulemans WVerbeek KWulms J(2021)Topological Stability of Kinetic k-CentersTheoretical Computer Science10.1016/j.tcs.2021.03.026Online publication date: Mar-2021
https://doi.org/10.1016/j.tcs.2021.03.026
Wang HZhang J(2019)Covering Uncertain Points in a TreeAlgorithmica10.1007/s00453-018-00537-681:6(2346-2376)Online publication date: 17-May-2019
https://dl.acm.org/doi/10.1007/s00453-018-00537-6
Meulemans WSpeckmann BVerbeek KWulms J(2018)A Framework for Algorithm Stability and Its Application to Kinetic Euclidean MSTsLATIN 2018: Theoretical Informatics10.1007/978-3-319-77404-6_58(805-819)Online publication date: 13-Mar-2018
https://doi.org/10.1007/978-3-319-77404-6_58
Hoog v.d. Ivan Kreveld MMeulemans WVerbeek KWulms J(2018)Topological Stability of Kinetic k-centersWALCOM: Algorithms and Computation10.1007/978-3-030-10564-8_4(43-55)Online publication date: 21-Dec-2018
https://doi.org/10.1007/978-3-030-10564-8_4
Wang HZhang J(2017)Computing the Center of Uncertain Points on Tree NetworksAlgorithmica10.1007/s00453-016-0158-378:1(232-254)Online publication date: 1-May-2017
https://dl.acm.org/doi/10.1007/s00453-016-0158-3
Wang HZhang J(2017)Covering Uncertain Points in a TreeAlgorithms and Data Structures10.1007/978-3-319-62127-2_47(557-568)Online publication date: 5-Jul-2017
https://doi.org/10.1007/978-3-319-62127-2_47
Wang HZhang J(2015)Computing the Center of Uncertain Points on Tree NetworksAlgorithms and Data Structures10.1007/978-3-319-21840-3_50(606-618)Online publication date: 28-Jul-2015
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