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Optimal slope selection

  • Richard Cole
  • Jeffrey Salowe
  • W. L. Steiger
  • Endre Szemerédi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 317)

Abstract

Given n points in the plane and an integer k, we consider the problem of selecting that pair which determines the line with the kth smallest or largest slope. In the restricted case where k is O(n), line sweeping gives an optimal, Θ(nlog n) algorithm. For general k we use the parametric search technique of Megiddo to describe an O(n (log n)2) algorithm. This is modified to produce a new, optimal Θ(nlog n) selection algorithm by incorporating an approximation idea.

Keywords

Linear Time Query Point Delete Line Sorting Network Approximate Rank 
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.
    Ajtai, M., Komlós, J., and Szemerédi, E. "Sorting in cLogn Parallel Steps". Combinatorica 3 (1983), 1–19.Google Scholar
  2. 2.
    Bentley, J.L. and Ottmann, Th. "Algorithms for Reporting and Counting Geometric Intersections". IEEE Trans. on Computers C 28 (1979), 643–647.Google Scholar
  3. 3.
    Blum, M., Floyd, R.W., Pratt, V., Rivest, R.L., and Tarjan, R.E. "Time Bounds for Selection". JCSS 7 (1973), 448–461.Google Scholar
  4. 4.
    Chazelle, B. "New Techniques for Computing Order Statistics in Euclidean Space". Proc. ACM Symposium on Comp. Geom. (1985), 125–134.Google Scholar
  5. 5.
    Cole, R. "Slowing Down Sorting Networks to Obtain Faster Sorting Algorithms". JACM 31 (1984), 200–208.Google Scholar
  6. 6.
    Knuth, Donald E.. The Art of Computer Programming. Volume 3:Sorting and Searching. Addison-Wesley, Reading, Mass., 1973.Google Scholar
  7. 7.
    Megiddo, N. "Applying Parallel Computational Algorithms in the Design of Serial Algorithms". JACM 30 (1983), 852–865.Google Scholar
  8. 8.
    Salowe, Jeffrey. Selection Problems in Computational Geometry. Ph.D. Th., Department of Computer Science, Rutgers University, New Brunswick, NJ, 1987.Google Scholar
  9. 9.
    Shamos, M.I. "Algorithms and Complexity: New Directions and Recent Results". Geometry and Statistics: Problems at the Interface (1976), 251–280. Academic Press, New York.Google Scholar
  10. 10.
    Theil, H. "A Rank-Invariant Method of Linear and Polynomial Regression Analysis I". Proc. Kon. Ned. Akad. v. Wetensch. A. 53 (1950), 386–392.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Richard Cole
    • 1
  • Jeffrey Salowe
    • 2
  • W. L. Steiger
    • 3
  • Endre Szemerédi
    • 4
    • 5
  1. 1.Courant InstituteNew York UniversityNew York
  2. 2.Computer ScienceUniversity of VirginiaCharlottesville
  3. 3.Computer ScienceRutgers UniversityNew Brunswick
  4. 4.Computer ScienceRutgers UniversityUSA
  5. 5.Mathematical InstituteHungarian Academy of SciencesHungary

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