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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Edelsbrunner, H.,Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, 1987.
Edelsbrunner, H., Guibas, L. J., and Sharir, M., The upper envelope of piecewise linear functions: algorithms and applications,Discrete Comput. Geom., to appear.
Greenberg, M. J.,Lectures on Algebraic Topology, Benjamin, Reading, MA, 1967.
Grünbaum, B.,Convex Polytopes, Wiley, Chichester, 1967.
Hart, S. and Sharir, M., Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes,Combinatorica 6 (1986), 151–177.
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.
Wiernik, A. and Sharir, M., Planar realization of nonlinear Davenport-Schinzel sequences by segments,Discrete Comput. Geom. 3 (1988), 15–47.
Author information
Authors and Affiliations
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
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02187734