Output-sensitive cell enumeration in hyperplane arrangements

  • Nora Sleumer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1432)

Abstract

We present a simple and practical algorithm for enumerating the set of cells C of an arrangement of m hyperplanes. For fixed dimension its time complexity is O(m ίddot ¦ C ¦). This is an improvement by a factor of m over the reverse search algorithm by Avis and Fukuda. The algorithm needs little space, is output-sensitive, straightforward to parallelize and the implementation is simple for all dimensions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AF96]
    D. Avis and K. Fukuda. Reverse search for enumeration. Discrete Applied Mathematics, 65:21–46, 1996.CrossRefMathSciNetGoogle Scholar
  2. [BFM97]
    D. Bremner, K. Fukuda, and A. Marzetta. Primal-dual methods for vertex and facet enumeration. In Proceedings of the 13th Annual ACM Symposium on Computational Geometry (SoCG), pages 49–56, 1997.Google Scholar
  3. [BMFN96]
    A. Brüngger, A. Marzetta, K. Fukuda, and J. Nievergelt. The Parallel Search Bench ZRAM and its Applications. To appear in: Annals of Operations Research. PS file available from ftp://ftp.ifor.ethz.ch/pub/fukuda/reports, 1996.Google Scholar
  4. [BVS+93]
    A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White, and G. Ziegler. Oriented Matroids, volume 46. Cambridge University Press, 1993.Google Scholar
  5. [Ede87]
    H. Edelsbrunner. Algorithms in Combinatorial Geometry. Springer-Verlag, Heidelberg, 1987.Google Scholar
  6. [EG86]
    H. Edelsbrunner and L. Guibas. Topologically sweeping an arrangement. In Proceedings of the 18th ACM Symposium on the Theory Computing (STOC), pages 389–403, 1986.Google Scholar
  7. [EOS86]
    H. Edelsbrunner, J. O'Rourke, and R. Seidel. Constructing arrangements of lines and hyperplanes with applications. SIAM Journal on Computing, 15(2):341–363, 1986.CrossRefMathSciNetGoogle Scholar
  8. [Fuk95]
    K. Fukuda. cdd+ Reference Manual. PS file available from ftp://ftp.ifor.ethz.ch/pub/fukuda/cdd, 1995.Google Scholar
  9. [McK87]
    M. McKenna. Worst-case optimal hidden-surface removal. ACM Transactions on Graphics, 6:19–28, 1987.CrossRefGoogle Scholar
  10. [Meg84]
    N. Megiddo. Linear programming in linear time when dimension is fixed. Journal of the ACM, 31:114–127, 1984.MATHCrossRefMathSciNetGoogle Scholar
  11. [O'R96]
    J. O'Rourke. Computational geometry column 28. International Journal of Computational Geometry and Applications, 6(2):243–244, 1996.CrossRefGoogle Scholar
  12. [OSS95]
    T. Ottmann, S. Schuierer, and S. Soundaralakshmi. Enumerating extreme points in higher dimensions. In Proceedings of the 12th Symposium on the Theoretical Aspects of Computer Science (STACS), volume 900 of LNCS, pages 562–570. Springer-Verlag, 1995.Google Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Nora Sleumer
    • 1
  1. 1.Institut für Theoretische InformatikETH ZentrumZürichSwitzerland

Personalised recommendations