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Algorithmica

, Volume 55, Issue 1, pp 14–28 | Cite as

A Separation Bound for Real Algebraic Expressions

  • Christoph Burnikel
  • Stefan Funke
  • Kurt Mehlhorn
  • Stefan Schirra
  • Susanne Schmitt
Open Access
Article

Abstract

Real algebraic expressions are expressions whose leaves are integers and whose internal nodes are additions, subtractions, multiplications, divisions, k-th root operations for integral k, and taking roots of polynomials whose coefficients are given by the values of subexpressions. We consider the sign computation of real algebraic expressions, a task vital for the implementation of geometric algorithms. We prove a new separation bound for real algebraic expressions and compare it analytically and experimentally with previous bounds. The bound is used in the sign test of the number type leda::real.

Keywords

Exact geometric computation Separation bound Zero testing 

References

  1. 1.
    Burnikel, C., Fleischer, R., Mehlhorn, K., Schirra, S.: Exact efficient computational geometry made easy. In: Proceedings of the 15th Annual Symposium on Computational Geometry (SCG’99), pp. 341–350 (1999) Google Scholar
  2. 2.
    Burnikel, C., Fleischer, R., Mehlhorn, K., Schirra, S.: A strong and easily computable separation bound for arithmetic expressions involving radicals. Algorithmica 27, 87–99 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Burnikel, C., Mehlhorn, K., Schirra, S.: How to compute the Voronoi diagram of line segments: Theoretical and experimental results. In: Proceedings of the 2nd Annual European Symposium on Algorithms—ESA’94. Lecture Notes in Computer Science, vol. 855, pp. 227–239. Springer, Berlin (1994) Google Scholar
  4. 4.
    Burnikel, C., Mehlhorn, K., Schirra, S.: The LEDA class real number. Technical report MPI-I-96-1-001, Max-Planck-Institut für Informatik, Saarbrücken (1996) Google Scholar
  5. 5.
    Canny, J.F.: The Complexity of Robot Motion Planning. MIT Press, Cambridge (1987) Google Scholar
  6. 6.
    Hecke, E.: Vorlesungen über die Theorie der Algebraischen Zahlen. Chelsea, New York (1970) zbMATHGoogle Scholar
  7. 7.
    Karamcheti, V., Li, C., Pechtchanski, I., Yap, C.: A core library for robust numeric and geometric computation. In: Proceedings of the 15th Annual ACM Symposium on Computational Geometry, pp. 351–359. Miami, FL, 1999 Google Scholar
  8. 8.
    LEDA (Library of Efficient Data Types and Algorithms): www.mpi-sb.mpg.de/LEDA/leda.html
  9. 9.
    Li, C., Yap, C.: A new constructive root bound for algebraic expressions. In: Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’01), pp. 496–505 (2001) Google Scholar
  10. 10.
    Loos, R.: Computing in algebraic extensions. In: Buchberger, B., Collins, G.E., Loos, R. (eds.) Computer Algebra. Symbolic and Algebraic Computation. Computing Supplementum, vol. 4, pp. 173–188. Springer, Vienna (1982) Google Scholar
  11. 11.
    Mehlhorn, K., Näher, S.: The LEDA Platform for Combinatorial and Geometric Computing. Cambridge University Press, Cambridge (1999) Google Scholar
  12. 12.
    Mehlhorn, K., Schirra, S.: Exact computation with leda_real - theory and geometric applications. In: Alefeld, G., Rohn, J., Rumpf, S., Yamamoto, T. (eds.) Symbolic Algebraic Methods and Verification Methods. Springer, Vienna (2001) Google Scholar
  13. 13.
    Mignotte, M.: Identification of algebraic numbers. J. Algorithms 3(3), 197–204 (1982) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Mignotte, M.: Mathematics for Computer Algebra. Springer, Berlin (1992) zbMATHGoogle Scholar
  15. 15.
    Neukirch, J.: Algebraische Zahlentheorie. Springer, Berlin (1990) Google Scholar
  16. 16.
    Scheinerman, E.R.: When close enough is close enough. Am. Math. Mon. 107, 489–499 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Yap, C.K.: Towards exact geometric computation. Comput. Geom. Theory Appl. 7 (1997) Google Scholar
  18. 18.
    Yap, C.K.: Fundamental Problems in Algorithmic Algebra. Oxford University Press, Oxford (1999) Google Scholar
  19. 19.
    Yap, C.K., Dube, T.: The exact computation paradigm. In: Computing in Euclidean Geometry II. World Scientific, Singapore (1995) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Christoph Burnikel
    • 1
  • Stefan Funke
    • 2
  • Kurt Mehlhorn
    • 3
  • Stefan Schirra
    • 4
  • Susanne Schmitt
    • 3
  1. 1.ENCOM GmbHSaarlouisGermany
  2. 2.Institut für Mathematik und InformatikErnst-Moritz-Arndt Universität GreifswaldGreifswaldGermany
  3. 3.MPI für InformatikSaarbrückenGermany
  4. 4.Fakultät für InformatikOtto-von-Guericke Universität MagdeburgMagdeburgGermany

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