Optimization over k-set Polytopes and Efficient k-set Enumeration

  • Artur Andrzejak
  • Komei Fukuda
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1663)

Abstract

We present two versions of an algorithm based on the reverse search technique for enumerating all k-sets of a point set in ℝ d . The key elements include the notion of a k-set polytope and the optimization of a linear function over a k-set polytope. In addition, we obtain several results related to the k-set polytopes. Among others, we show that the 1-skeleton of a k-set polytope restricted to vertices corresponding to the affine k-sets is not always connected.

Keywords

Voronoi Diagram Unique Path Total Time Complexity Reverse Search Linear Inequality System 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [ABFK92]
    Noga Alon, Imre Bárány, Zoltán Füredi, and Daniel J. Kleitman. Point selections and weak -nets for convex hulls. Combinatorics, Probability and Computing, 1992.Google Scholar
  2. [AF96]
    David Avis and Komei Fukuda. Reverse search for enumeration. Disc. Applied Math., 65:21–46, 1996.MathSciNetMATHCrossRefGoogle Scholar
  3. [AK95]
    E. Amaldi and V. Kann. The complexity and approximability of finding maximum feasible subsystems of linear relations. Theoretical Computer Science, 147(1-2):181–210, 1995.MathSciNetMATHCrossRefGoogle Scholar
  4. [AM95]
    Pankaj K. Agarwal and Jiří Matoušek. Dynamic half-space range reporting and its applications. Algorithmica, 13(4): 325–345, April 1995.MathSciNetMATHCrossRefGoogle Scholar
  5. [AW97a]
    Artur Andrzejak and Emo Welzl. k-sets and j-facets-A tour of discrete geometry. In preparation, 1997.Google Scholar
  6. [AW97b]
    Artur Andrzejak and Emo Welzl. Relations between numbers of k-sets and numbers of j-facets. In preparation, 1997.Google Scholar
  7. [Ede87]
    Herbert Edelsbrunner. Algorithms in Combinatorial Geometry. Springer-Verlag Berlin Heidelberg, New York, 1987.MATHGoogle Scholar
  8. [EVW97]
    Herbert Edelsbrunner, Pavel Valtr, and Emo Welzl. Cutting dense point sets in half. Discrete Comput. Geom., 17:243–255, 1997.MathSciNetMATHCrossRefGoogle Scholar
  9. [EW86]
    Herbert Edelsbrunner and Emo Welzl. Constructing belts in two-dimensional arrangements with applications. SIAM J. Comput., 15(1):271–284, 1986.MathSciNetMATHCrossRefGoogle Scholar
  10. [KM93]
    Nikolai M. Korneenko and Horst Martini. Hyperplane Approximations and Related Topics. Pach, János (ed.): New trends in discrete and computational geometry. Algorithms and Combinatorics v. 10, pages 135–161. Springer-Verlag, Berlin Heidelberg, New York, 1993.Google Scholar
  11. [Mul91]
    Ketan Mulmuley. On Levels in Arrangements and Voronoi Diagrams. Discrete Comput. Geom., 6:307–338, 1991.MathSciNetMATHCrossRefGoogle Scholar
  12. [Mul93]
    Ketan Mulmuley. Output sensitive and dynamic constructions of higher order Voronoi diagrams and levels in arrangements. Journal of Computer and System Sciences, 47(3):437–458, December 1993.MathSciNetMATHCrossRefGoogle Scholar
  13. [OSS95]
    Thomas Ottmann, Sven Schuierer, and Subbiah Soundaralakshmi. Enumerating Extreme Points in Higher Dimensions. Proceedings of 12th Annual Symposium on Theoretical Aspects of Computer Science (STACS 95), Munich, Germany, March 1995, LNCS 900, pages 562–570. Springer-Verlag Berlin Heidelberg, New York, 1995.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Artur Andrzejak
    • 1
  • Komei Fukuda
    • 2
  1. 1.Institute of Theoretical Computer ScienceETH ZürichZürichSwitzerland
  2. 2.Institute for Operations ResearchETH ZürichZürichSwitzerland

Personalised recommendations