Skip to main content
SpringerLink
Log in

Exact Volume Computation for Polytopes: A Practical Study

  • Chapter
Book cover Polytopes — Combinatorics and Computation

Part of the book series: DMV Seminar ((OWS,volume 29))

Abstract

We study several known volume computation algorithms for convex d-polytopes by classifying them into two classes, triangulation methods and signed-decomposition methods. By incorporating the detection of simplicial faces and a storing/reusing scheme for face volumes we propose practical and theoretical improvements for two of the algorithms. Finally we present a hybrid method combining advantages from the two algorithmic classes. The behaviour of the algorithms is theoretically analysed for hypercubes and practically tested on a wide range of polytopes, where the new hybrid method proves to be superior.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Avis, D. Bremner, and R. Seidel. How good are convex hull algorithms.Computational Geometry: Theory and Applications7:265–302, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Bremner. Incremental convex hull algorithms are not output sensitive.Discrete Comput. Geom.21 (1999), 57–68.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Bremner, K. Fukuda, and A. Marzetta. Primal-dual methods for vertex and facet enumeration. InProc. 13th Annu. ACM Sympos. Comput. Geom.pages 49–56, 1997. Full paper inDiscrete Comput. Geom. 20 (1998), 333–357.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Cohen and T. Hickey. Two algorithms for determining volumes of convex polyhedra.Journal of the ACM26(3):401–414, July 1979.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. E. Dyer and A. M. Frieze. The complexity of computing the volume of a polyhedron.SIAM J. Comput.17:967–974, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  6. M.E. Dyer. The complexity of vertex enumeration methods.Math. Oper. Res.8:381402, 1983.

    MathSciNet  Google Scholar 

  7. H. Edelsbrunner. Algorithms in Combinatorial Geometry. Springer-Verlag, 1987.

    Google Scholar 

  8. P. Filliman. The volume of duals and sections of polytopes.Mathematika39:67–80, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  9. K. Fukuda, T. M. Liebling, and F. Margot. Analysis of backtrack algorithms for listing all vertices and all faces of a convex polyhedron.Computational Geometry8:1–12, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  10. ] K. Fukuda and A. Prodon. Double description method revisited. In M. Deza, R. Euler, and I. Manoussakis, editorsCombinatorics and Computer Sciencevolume 1120 of Lecture Notes in Computer Sciencepages 91–111. Springer-Verlag, 1996.

    Google Scholar 

  11. P. Gritzmann and V. Klee. On the complexity of some basic problems in computational convexity: II. Volume and mixed volumes. In T. Bisztriczky, P. McMullen, R. Schneider, and A.I. Weiss, editorsPolytopes: Abstract convex and computational (Scarborough ON 1993)NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 440, pages 373–466. Kluwer Acad. Publ., Dordrecht, 1994.

    Google Scholar 

  12. R. Kannan, L. Lovász, and M. Simonovits. Random walks and an O*(n5) volume algorithm for convex bodies.Random Struct. Algorithms11(1):1–50, 1997.

    Article  MATH  Google Scholar 

  13. J. B. Lasserre. An analytical expression and an algorithm for the volume of a convex polyhedron in IV.J. of Optimization Theory and Applications39(3):363–377, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. B. Lasserre. Integration on a convex polytope. Technical Report 96173, Centre National de la Recherche Scientifique, Laboratoire d’Analyse et d’Architectures des Systems, Toulouse, 1996.

    Google Scholar 

  15. J. Lawrence. Polytope volume computation.Mathematics of Computation57(195):259–271, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  16. R. Seidel. Small-dimensional linear programming and convex hulls made easy.Discrete Comput. Geom.6:423–434, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Verschelde, K. Gatermann, and R. Cools. Mixed-volume computation by dynamic lifting applied to polynomial system solving.Discrete Comput. Geom.16:69–112, 1996.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Operations Research Swiss Federal Institute of Technology, CH-8092, Zurich, Switzerland

    Benno Büeler

  2. Lehrstuhl für Diskrete Mathematik Optimierung und Operations Research Institut für Mathematik, Universität Augsburg, Deutschland

    Andreas Enge

  3. Department of Mathematics, Swiss Federal Institute of Technology, Lausanne, CH-1015, Switzerland

    Komei Fukuda

Authors
  1. Benno Büeler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andreas Enge
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Komei Fukuda
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Mathematics and Computer Science, The Hebrew University, Givat Ram, 91904, Jerusalem, Israel

    Gil Kalai

  2. Fachbereich Mathematik MA 7-1, Technische Universität Berlin, Strasse des 17. Juni 135, D-10623, Berlin, Germany

    Günter M. Ziegler

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Basel AG

About this chapter

Cite this chapter

Büeler, B., Enge, A., Fukuda, K. (2000). Exact Volume Computation for Polytopes: A Practical Study. In: Kalai, G., Ziegler, G.M. (eds) Polytopes — Combinatorics and Computation. DMV Seminar, vol 29. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8438-9_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8438-9_6

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-7643-6351-2

  • Online ISBN: 978-3-0348-8438-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation