Skip to main content

The upper envelope of piecewise linear functions: Tight bounds on the number of faces

Abstract

This note proves that the maximum number of faces (of any dimension) of the upper envelope of a set ofn possibly intersectingd-simplices ind+1 dimensions is Θ(n dα(n)). This is an extension of a result of Pach and Sharir [PS] who prove the same bound for the number ofd-dimensional faces of the upper envelope.

References

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

    Book  MATH  Google Scholar 

  2. Edelsbrunner, H., Guibas, L. J., and Sharir, M., The upper envelope of piecewise linear functions: algorithms and applications,Discrete Comput. Geom., to appear.

  3. Greenberg, M. J.,Lectures on Algebraic Topology, Benjamin, Reading, MA, 1967.

    MATH  Google Scholar 

  4. Grünbaum, B.,Convex Polytopes, Wiley, Chichester, 1967.

    MATH  Google Scholar 

  5. Hart, S. and Sharir, M., Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes,Combinatorica 6 (1986), 151–177.

    MathSciNet  Article  MATH  Google Scholar 

  6. Pach, J. and Sharir, M., The upper envelope of piecewise linear functions and the boundary of a region enclosed by convex plates: combinatorial analysis,Discrete Comput. Geom., to appear.

  7. Wiernik, A. and Sharir, M., Planar realization of nonlinear Davenport-Schinzel sequences by segments,Discrete Comput. Geom. 3 (1988), 15–47.

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

This work was supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and by the National Science Foundation under Grant CCR-8714565. Research on the presented result was partially carried out while the author worked for the IBM T. J. Watson Research Center at Yorktown Height, New York, USA.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Edelsbrunner, H. The upper envelope of piecewise linear functions: Tight bounds on the number of faces. Discrete Comput Geom 4, 337–343 (1989). https://doi.org/10.1007/BF02187734

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02187734

Keywords

  • Recurrence Relation
  • Euler Characteristic
  • Discrete Comput Geom
  • Cell Complex
  • Piecewise Linear Function