Theory of Computing Systems

, 45:446 | Cite as

Covering Many or Few Points with Unit Disks

Article

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 2n) expected time a disk that is, with high probability, a (1+ε)-approximation to the optimal solution.

Keywords

Facility location Geometric optimization Random sample Weighted points 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Mark de Berg
    • 1
  • Sergio Cabello
    • 2
    • 3
  • Sariel Har-Peled
    • 4
  1. 1.Department of Computer ScienceTU EindhovenEindhovenThe Netherlands
  2. 2.Department of Mathematics, FMFUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Department of MathematicsIMFMLjubljanaSlovenia
  4. 4.Department of Computer ScienceUniversity of IllinoisIllinoisUSA

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