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Smallest Maximum-Weight Circle for Weighted Points in the Plane

  • Sergey Bereg
  • Ovidiu DaescuEmail author
  • Marko Zivanic
  • Timothy Rozario
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9156)

Abstract

Let P be a weighted set of points in the plane. In this paper we study the problem of computing a circle of smallest radius such that the total weight of the points covered by the circle is maximized. We present an algorithm with polynomial time depending on the number of points with positive and negative weight. We also consider a restricted version of the problem where the center of the circle should be on a given line and give an algorithm that runs in \(O(n(m+n) \log (m+n))\) time using \(O(m+n)\) space. The algorithm can report all k smallest maximal weight circles with an additional O(k) space. Moreover, for this version, if all positively weighted points are required to be included within the circle then we prove a number of interesting properties and provide an algorithm that runs in \(O((n+m) \log (n+m))\) time.

Keywords

Voronoi Diagram Binary Search Blue Point Event Point Bisector Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Barbay, J., Chan, T.M., Navarro, G., Pérez-Lantero, P.: Maximum-weight planar boxes in \(O(n^2)\) time (and better). Inf. Process. Lett. 114(8), 437–445 (2014)zbMATHCrossRefGoogle Scholar
  2. 2.
    Carr, R.D., Doddi, S., Konjevod, G., Marathe, M.: On the red-blue set cover problem. In: Proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 345–353 (2000)Google Scholar
  3. 3.
    Chan, T.M., Hu, N.: Geometric red-blue set cover for unit squares and related problems. Comput. Geom. 48(5), 380–385 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bitner, S., Cheung, Y.K., Daescu, O.: Minimum separating circle for bichromatic points in the plane. In: Proceedings of the 7th International Symposium on Voronoi Diagrams in Science and Engineering, pp. 50–55 (2010)Google Scholar
  5. 5.
    Cheung, Y.K., Daescu, O., Zivanic, M.: Kinetic red-blue minimum separating circle. In: Wang, W., Zhu, X., Du, D.-Z. (eds.) COCOA 2011. LNCS, vol. 6831, pp. 448–463. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  6. 6.
    Dobkin, D.P., Eppstein, D., Mitchell, D.P.: Computing the discrepancy with applications to supersampling patterns. ACM Transactions on Graphics 15, 354–376 (1996)CrossRefGoogle Scholar
  7. 7.
    Har-Peled, S., Lee, M.: Weighted geometric set cover problems revisited. Journal of Computational Geometry 3(1), 65–85 (2012)MathSciNetGoogle Scholar
  8. 8.
    Har-Peled, S., Mazumdar, S.: Fast algorithms for computing the smallest k-enclosing circle. Algorithmica 41(3), 147–157 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    O’Rourke, J., Kosaraju, S., Megiddo, N.: Computing circular separability. Discrete. Computational Geometry 1, 105–113 (1986)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Sergey Bereg
    • 1
  • Ovidiu Daescu
    • 1
    Email author
  • Marko Zivanic
    • 1
  • Timothy Rozario
    • 1
  1. 1.The University of Texas at DallasRichardsonUSA

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