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Covering Many or Few Points with Unit Disks

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Abstract

Let P be a set of n weighted points. We study approximation algorithms for the following two continuous facility-location problems.

In the first problem we want to place m unit disks, for a given constant m≥1, such that the total weight of the points from P inside the union of the disks is maximized. We present algorithms that compute, for any fixed ε>0, a (1−ε)-approximation to the optimal solution in O(nlog n) time.

In the second problem we want to place a single disk with center in a given constant-complexity region X such that the total weight of the points from P inside the disk is minimized. Here we present an algorithm that computes, for any fixed ε>0, in O(nlog 2 n) expected time a disk that is, with high probability, a (1+ε)-approximation to the optimal solution.

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Authors and Affiliations

  1. Department of Computer Science, TU Eindhoven, Eindhoven, The Netherlands

    Mark de Berg

  2. Department of Mathematics, FMF, University of Ljubljana, Ljubljana, Slovenia

    Sergio Cabello

  3. Department of Mathematics, IMFM, Ljubljana, Slovenia

    Sergio Cabello

  4. Department of Computer Science, University of Illinois, Illinois, USA

    Sariel Har-Peled

Authors
  1. Mark de Berg
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  2. Sergio Cabello
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  3. Sariel Har-Peled
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Corresponding author

Correspondence to Sergio Cabello.

Additional information

A preliminary version of this work has appeared in Approximation and Online Algorithms—WAOA 2006, LNCS, vol. 4368.

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de Berg, M., Cabello, S. & Har-Peled, S. Covering Many or Few Points with Unit Disks. Theory Comput Syst 45, 446–469 (2009). https://doi.org/10.1007/s00224-008-9135-9

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  • DOI: https://doi.org/10.1007/s00224-008-9135-9

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