BIT Numerical Mathematics

, Volume 32, Issue 2, pp 249–267 | Cite as

Applications of a semi-dynamic convex hull algorithm

  • John Hershberger
  • Subhash Suri
Algorithm Theory


We obtain new results for manipulating and searching semi-dynamic planar convex hulls (subject to deletions only), and apply them to derive improved bounds for two problems in geometry and scheduling. The new convex hull results are logarithmic time bounds for set splitting and for finding a tangent when the two convex hulls are not linearly separated. Using these results, we solve the following two problems optimally inO(n logn) time: (1) [matching] givenn red points andn blue points in the plane, find a matching of red and blue points (by line segments) in which no two edges cross, and (2) [scheduling] givenn jobs with due dates, linear penalties for late completion, and a single machine on which to process them, find a schedule of jobs that minimizes the maximum penalty.

CR categories

F.2.2 I.3.5 E..1 

Additional keywords

Convex hull semi-dynamic algorithm geometric matching scheduling 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Akiyama and N. Alon.Disjoint simplices and geometric hypergraphs. Annals New York Academy of Science, pages 1–3, 1989.Google Scholar
  2. 2.
    M. Atallah.A matching problem in the plane. Journal of Computer and System Sciences, 31: 63–70, 1985.Google Scholar
  3. 3.
    M. Ben-Or.Lower bounds for algebraic computation trees. In Proceedings of the 15th ACM Symposium on Theory of Computing, pages 80–86, 1983.Google Scholar
  4. 4.
    K. Q. Brown.Geometric Transforms for Fast Geometric Algorithms. PhD thesis, Carnegie-Mellon University, 1980.Google Scholar
  5. 5.
    B. Chazelle.On the convex layers of a planar set. IEEE Transactions on Information Theory, IT-31 (4): 509–517, July 1985.Google Scholar
  6. 6.
    H. Edelsbrunner.Algorithms in Combinatorial Geometry, volume 10 ofEATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1987.Google Scholar
  7. 7.
    M. Fields and G. Frederickson.A faster algorithm for the maximum weighted tardiness problem. Manuscript, 1989.Google Scholar
  8. 8.
    J. Friedman, J. Hershberger and J. Snoeyink.Compliant motion in a simple polygon. In Proceedings of the 5th ACM Symposium on Computational Geometry, pages 175–186, 1989.Google Scholar
  9. 9.
    R. Graham.An efficient algorithm for determining the convex hull of a finite planar set. Information Processing Letters, 1: 132–133, 1972.Google Scholar
  10. 10.
    L. Guibas, J. Hershberger and J. Snoeyink.Compact interval trees: A data structure for convex hulls. International Journal of Computational Geometry & Applications, 1 (1): 1–22, 1991.Google Scholar
  11. 11.
    J. Hershberger and S. Suri.Finding tailored partitions. Journal of Algorithms, 12 (3): 431–463, September 1991.Google Scholar
  12. 12.
    D. Hochbaum and R. Shamir.An O log 2 n)algorithm for the maximum weighted tardiness problem. Information Processing Letters, 31: 215–219, 1989.Google Scholar
  13. 13.
    D. Kirkpatrick and R. Seidel.The ultimate planar convex hull algorithm? SIAM Journal on Computing, 15: 287–299, 1986.Google Scholar
  14. 14.
    L. C. Larson.Problem-Solving Through Problems. Springer-Verlag, New York, 1983.Google Scholar
  15. 15.
    E. L. Lawler,Optimal sequencing of a single machine subject to precedence constraints. Management Science, 19: 544–546, 1973.Google Scholar
  16. 16.
    M. Overmars and J. van Leeuwen.Maintenance of configurations in the plane. Journal of Computer and System Sciences, 23: 166–204, 1981.Google Scholar
  17. 17.
    F. P. Preparata.An optimal real time algorithm for planar convex hulls. Communications of the ACM, 22: 402–405, 1979.Google Scholar
  18. 18.
    F. P. Preparata and M. I. Shamos.Computational Geometry. Springer-Verlag, New York, 1985.Google Scholar

Copyright information

© BIT Foundations 1992

Authors and Affiliations

  • John Hershberger
    • 1
    • 2
  • Subhash Suri
    • 1
    • 2
  1. 1.DEC Systems Research CenterPalo AltoUSA
  2. 2.Bell Communications ResearchMorristownUSA

Personalised recommendations