Covering Many or Few Points with Unit Disks

  • Mark de Berg
  • Sergio Cabello
  • Sariel Har-Peled
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4368)

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 a deterministic algorithm that can compute, for any ε>0, a (1−ε)-approximation to the optimal solution in O(n logn + ε\(^{{\rm -4}{\it m}}\)log\(^{\rm 2{\it m}}\) (1/ε)) 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 can compute, for any ε>0, with high probability a (1+ε)-approximation to the optimal solution in O(n (log3n + ε− 4 log2n )) expected time.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Mark de Berg
    • 1
  • Sergio Cabello
    • 2
  • Sariel Har-Peled
    • 3
  1. 1.Department of Computer ScienceTU EindhovenThe Netherlands
  2. 2.Department of Mathematics, FMFUniversity of Ljubljana, and Department of Mathematics, IMFMSlovenia
  3. 3.Department of Computer ScienceUniversity of IllinoisUSA

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