Locating an Obnoxious Line among Planar Objects

  • Danny Z. Chen
  • Haitao Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5878)

Abstract

Given a set P of n points in the plane such that each point has a positive weight, we study the problem of finding an obnoxious line that intersects the convex hull of P and maximizes the minimum weighted Euclidean distance to all points of P. We also consider a variant of this problem whose input is a set of m polygons with totally n vertices in the plane such that each polygon has a positive weight and whose goal is to locate an obnoxious line with respect to the weighted polygons. We improve the previous results for both problems. Our algorithms are based on new geometric observations and interesting algorithmic techniques.

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References

  1. 1.
    Bereg, S., Díaz-Báñez, J.M., Seara, C., Ventura, I.: On finding widest empty curved corridors. Computational Geometry: Theory and Applications 38(3), 154–169 (2007)MATHMathSciNetGoogle Scholar
  2. 2.
    Chattopadhyay, S., Das, P.: The k-dense corridor problems. Pattern Recognition Letters 11(7), 463–469 (1990)MATHCrossRefGoogle Scholar
  3. 3.
    Chen, D.Z., Wang, H.: Locating an obnoxious line among planar objects (2009) (manuscript)Google Scholar
  4. 4.
    Cheng, S.: Widest empty L-shaped corridor. Information Processing Letters 58(6), 277–283 (1996)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cole, R.: Slowing down sorting networks to obtain faster sorting algorithms. Journal of the ACM 34(1), 200–208 (1987)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)MATHGoogle Scholar
  7. 7.
    Díaz-Báñez, J.M., Ramos, P.A., Sabariego, P.: The maximin line problem with regional demand. European Journal of Operational Research 181(1), 20–29 (2007)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Drezner, Z., Wesolowsky, G.O.: Location of an obnoxious route. Journal of Operational Research Society 40(11), 1011–1018 (1989)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Houle, M., Maciel, A.: Finding the widest empty corridor through a set of points. In: Tousssaint, G.T. (ed.) Snapshots of Computational and Discrete Geometry. TR SOCS–88.11, Dept. of Computer Science, McGill University, Montreal, Canada (1988)Google Scholar
  10. 10.
    Houle, M.E., Toussaint, G.T.: Computing the width of a set. IEEE Trans. Pattern Anal. Mach. Intell. 10(5), 761–765 (1988)MATHCrossRefGoogle Scholar
  11. 11.
    Janardan, R., Preparata, F.P.: Widest-corridor problems. Nordic Journal of Computing 1(2), 231–245 (1994)MathSciNetGoogle Scholar
  12. 12.
    Lee, D.T., Wu, Y.F.: Geometric complexity of some location problems. Algorithmica 1(1-4), 193–211 (1986)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Nielsen, F., Yvinec, M.: Output-sensitive convex hull algorithms of planar convex objects. International Journal of Computational Geometry and Applications 8(1), 39–66 (1998)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Rappaport, D.: A convex hull algorithm for discs, and applications. Computational Geometry: Theory and Applications 1(3), 171–187 (1992)MATHMathSciNetGoogle Scholar
  15. 15.
    Shin, C., Shin, S.Y., Chwa, K.: The widest k-dense corridor problems. Information Processing Letters 68(1), 25–31 (1998)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Danny Z. Chen
    • 1
  • Haitao Wang
    • 1
  1. 1.Department of Computer Science and EngineeringUniversity of Notre DameNotre DameUSA

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