Advertisement

Algorithms for polytope covering and approximation

  • Kenneth L. Clarkson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 709)

Abstract

This paper gives an algorithm for polytope covering: let L and U be sets of points in Rd, comprising n points altogether. A cover for L from U is a set C⊂U with L a subset of the convex hull of C. Suppose c is the size of a smallest such cover, if it exists. The randomized algorithm given here finds a cover of size no more than c(5dln c), for c large enough. The algorithm requires O(c2n1+δ) expected time. More exactly, the time bound is
$$O(cn^{1 + \delta } + c(nc)^{1/(1 + \gamma /(1 + \delta ))} )$$
, where γγ1/[d/2]. The previous best bounds were cO(log n) cover size in O(nd) time.[MS92b] A variant algorithm is applied to the problem of approximating the boundary of a polytope with the boundary of a simpler polytope. For an appropriate measure, an approximation with error ε requires c=O(d/ε)d−1 vertices, and the algorithm gives an approximation with c(5d3 ln(1/ε)) vertices. The algorithms apply ideas previously used for small-dimensional linear programming.

Keywords

Convex Hull Range Query Query Time Successful Iteration Expected Time 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Cla87]
    K. L. Clarkson. New applications of random sampling in computational geometry. Discrete and Computational Geometry, 2:195–222, 1987.Google Scholar
  2. [Cla88]
    K. L. Clarkson. A Las Vegas algorithm for linear programming when the dimension is small. In Proc. 29ih IEEE Symp. on Foundations of Computer Science, pages 452–456, 1988. Revised version: Las Vegas algorithms for linear and integer programming when the dimension is small (preprint).Google Scholar
  3. [DJ90]
    G. Das and D. Joseph. The complexity of minimum nested polyhedra. In Canadian Conference on Computational Geometry, 1990.Google Scholar
  4. [HW87]
    D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete and Computational Geometry, 2:127–151, 1987.Google Scholar
  5. [Lit87]
    N. Littlestone. Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. In Proc. 28th IEEE Symp. on Foundations of Computer Science, pages 68–77, 1987.Google Scholar
  6. [Mat92]
    J. Matoušek. Reporting points in halfspaces. Computational Geometry: Theory and Applications, pages 169–186, 1992.Google Scholar
  7. [MS92a]
    J. Matoušek and O. Schwartzkopf. Linear optimization queries. In Proc. Eighth ACM Symp. on Comp. Geometry, pages 16–25, 1992.Google Scholar
  8. [MS92b]
    J. Mitchell and S. Suri. Separation and approximation of polyhedral objects. In Proc. 3rd ACM Symp. on Discrete Algorithms, pages 296–306, 1992.Google Scholar
  9. [Wel88]
    E. Welzl. Partition trees for triangle counting and other range searching problems. In Proc. Fourth ACM Symp. on Comp. Geometry, pages 23–33, 1988.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Kenneth L. Clarkson
    • 1
  1. 1.AT&T Bell LaboratoriesMurray Hill

Personalised recommendations