Skip to main content

Computing the Betti Numbers of Arrangements in Practice

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 3718)

Abstract

We describe an algorithm for computing the zero-th and the first Betti numbers of the union of n simply connected compact semi-algebraic sets in ℝk, where each such set is defined by a constant number of polynomials of constant degrees. The complexity of the algorithm is O(n 3). We also describe an implementation of this algorithm in the particular case of arrangements of ellipsoids in ℝ3 and describe some of our results.

Keywords

  • Spectral Sequence
  • Simplicial Complex
  • Betti Number
  • Algebraic Hypersurface
  • Input Formula

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arnon, D.S., Collins, G.E., McCallum, S.: Cylindrical algebraic decomposition. I. The basic algorithm. SIAM J. Comput. 13(4), 865–877 (1984)

    CrossRef  MathSciNet  Google Scholar 

  2. Arnon, D.S., Collins, G.E., McCallum, S.: Cylindrical algebraic decomposition. II. An adjacency algorithm for the plane. SIAM J. Comput. 13(4), 878–889 (1984)

    CrossRef  MathSciNet  Google Scholar 

  3. Arnon, D.S., Collins, G.E., McCallum, S.: An adjacency algorithm for cylindrical algebraic decompositions of three-dimensional space. J. Symbolic Comput. 5(1-2), 163–187 (1988)

    CrossRef  MATH  MathSciNet  Google Scholar 

  4. Basu, S.: Computing the Betti numbers of arrangements via spectral sequences. J. Comput. System Sci. 67(2), 244–262 (2003); Special issue on STOC 2002 (Montreal, QC)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. Basu, S.: Computing the first few Betti numbers of semi-algebraic sets in single exponential time (2005) (preprint)

    Google Scholar 

  6. Basu, S.: Computing the top Betti numbers of semi-algebraic sets defined by quadratic inequalities in polynomial time. In: Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing (2005)

    Google Scholar 

  7. Basu, S., Pollack, R., Roy, M.-F.: Algorithms in real algebraic geometry. Algorithms and Computation in Mathematics, vol. 10. Springer, Berlin (2003)

    MATH  Google Scholar 

  8. Basu, S., Pollack, R., Roy, M.-F.: Computing the first Betti number and the connected components of semi-algebraic sets. In: Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing (2005)

    Google Scholar 

  9. Brown, C.W.: An overview of QEPCAD B: a tool for real quantifier elimination and formula simplification. Journal of Japan Society for Symbolic and Algebraic Computation 10(1), 13–22 (2003)

    Google Scholar 

  10. Carlsson, G., Zomorodian, A., Collins, A., Guibas, L.: Persistence barcodes for shapes. In: SGP 2004: Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometric Processing, pp. 124–135. ACM Press, New York (2004)

    CrossRef  Google Scholar 

  11. Chazelle, B., Edelsbrunner, H., Guibas, L.J., Sharir, M.: A singly-exponential stratification scheme for real semi-algebraic varieties and its applications. Theoretical Computer Science 84, 77–105 (1991)

    CrossRef  MATH  Google Scholar 

  12. Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)

    Google Scholar 

  13. Edelsbrunner, H.: The union of balls and its dual shape. Discrete Comput. Geom. 13(3-4), 415–440 (1995)

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. Fortuna, E., Gianni, P., Parenti, P., Traverso, C.: Algorithms to compute the topology of orientable real algebraic surfaces. J. Symbolic Comput. 36(3-4), 343–364 (2003)

    CrossRef  MATH  MathSciNet  Google Scholar 

  15. Halperin, D.: Arrangements. In: Handbook of discrete and computational geometry, CRC Press Ser. Discrete Math. Appl., pp. 389–412. CRC, Boca Raton (1997)

    Google Scholar 

  16. Koltun, V.: Sharp bounds for vertical decompositions of linear arrangements in four dimensions. Discrete Comput. Geom. 31(3), 435–460 (2004)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Basu, S., Kettner, M. (2005). Computing the Betti Numbers of Arrangements in Practice. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2005. Lecture Notes in Computer Science, vol 3718. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11555964_2

Download citation

  • DOI: https://doi.org/10.1007/11555964_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28966-1

  • Online ISBN: 978-3-540-32070-8

  • eBook Packages: Computer ScienceComputer Science (R0)