Advertisement

Weighted orthogonal linear L-approximation and applications

  • Michael E. Houle
  • Hiroshi Imai
  • Keiko Imai
  • Jean-Marc Robert
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 382)

Abstract

Let S={s1, s2,...,s n } be a set of sites in E d , where every site s i has a positive real weight ω i . This paper gives an algorithm to find a weighted orthogonal L-approximation hyperplane for S. The algorithm is shown to require O(nlogn) time and O(n) space for d=2, and O(n[d/2+1]) time and O(n[(d+1)/2]) space for d>2. The L-approximation algorithm will be adapted to solve the problem of finding the width of a set of n points in E d , and the problem of finding a stabbing hyperplane for a set of n hyperspheres in E d with varying radii. The time and space complexities of the width and stabbing algorithms are seen to be the same as those of the L-approximation algorithm.

Keywords

Convex Hull Time Algorithm Feasibility Region Convex Polytope Supporting Hyperplane 
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. [1]
    Atallah, M. and Bajaj, C. Efficient Algorithms for Common Transversals, Inf. Proc. Letters 25 (1987), pp. 87–91.CrossRefGoogle Scholar
  2. [2]
    Avis, D. and Doskas, M. Algorithms for High Dimensional Stabbing Problems, Tech. Rept. SOCS-87.2, McGill Univ., January 1987.Google Scholar
  3. [3]
    Avis, D., Robert, J.-M., and Wenger, R. Lower Bounds for Line Stabbing, in preparation.Google Scholar
  4. [4]
    Bajaj, C. and Li, M. On the Duality of Intersection and Closest Points, in “Proc. 21st Ann. Allerton Conf. 1983”, pp. 459–460.Google Scholar
  5. [5]
    Borsuk, K. Multidimensional Analytic Geometry, Polish Scientific Publishers, Warsaw, 1969.Google Scholar
  6. [6]
    Brown, K. Q. Geometric Transforms for Fast Geometric Algorithms, Ph.D. Thesis, Rep. CMU-CS-80-101, Dept. Comp. Sci., Carnegie-Mellon Univ., Pittsburgh, PA, 1980.Google Scholar
  7. [7]
    Chvátal, V. Linear Programming, W. H. Freeman and Co., New York, 1983.Google Scholar
  8. [8]
    Edelsbrunner, H. Finding Transversals for Sets of Simple Geometric Figures, Theo. Comp. Sc. 35 (1985), pp. 55–69.CrossRefGoogle Scholar
  9. [9]
    Edelsbrunner, H., Guibas, L. J., and Sharir, M. The Upper Envelope of Piecewise Linear Functions: Algorithms and Applications, Tech. Rept. UIUCDCS-R-87-1390, Univ. of Illinois at Urbana-Champaign, November 1987.Google Scholar
  10. [10]
    Edelsbrunner, H., Maurer, H. A., Preparata, F. P., Rosenberg, A. L., Welzl, E., and Wood, D. Stabbing Line Segments, BIT 22 (1982), pp. 274–281.Google Scholar
  11. [11]
    Hershberger, J. private communication.Google Scholar
  12. [12]
    Houle, M. E. and Robert, J.-M. Orthogonal Weighted Linear Approximation and Applications, Tech. Rept. SOCS-88.13, McGill Univ., August 1988.Google Scholar
  13. [13]
    Houle, M. E. and Toussaint, G. T. Computing the Width of a Set, IEEE Trans. Pattern Anal. Machine Intell. 10 (1988), pp. 761–765.Google Scholar
  14. [14]
    Imai, H., Imai, K., and Yamamoto, P. Algorithms for Orthogonal L 1 Linear Approximation of Points in Two and Higher Dimensions, “13th Int. Symp. on Math. Prog. 1988”, Tokyo.Google Scholar
  15. [15]
    Kurozumi, Y. and Davis, W. A. Polygonal Approximation by the Minimax Method, Computer Graphics and Image Processing, 19 (1982), pp. 248–264.Google Scholar
  16. [16]
    Lee, D. T. and Wu, Y. F. Geometric Complexity of Some Location Problems, Algorithmica 1 (1986), pp. 193–212.Google Scholar
  17. [17]
    Morris, J. G. and Norback, J. P. Linear Facility Location — Solving Extensions of the Basic Problem, Eur. J. Oper. Res. 12 (1983), pp. 90–94.Google Scholar
  18. [18]
    Preparata, F. P. and Hong, S. J. Convex Hulls of Finite Sets of Points in Two and Three Dimensions, Comm. ACM 20 (1977), pp. 87–93.CrossRefGoogle Scholar
  19. [19]
    Seidel, R. A Convex Hull Algorithm Optimal for Points in Even Dimensions, Tech. Rept. 81-14, Univ. British Columbia, 1981.Google Scholar
  20. [20]
    Yamamoto, P., Kato, K., Imai, K. and Imai, H. Algorithms for Vertical and Orthogonal L 1 Linear Approximation of Points, in “Proc. 4th Ann. ACM Symp. Comp. Geom., 1988”, pp. 352–361.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Michael E. Houle
    • 1
  • Hiroshi Imai
    • 2
  • Keiko Imai
    • 3
  • Jean-Marc Robert
    • 1
  1. 1.School of Computer ScienceMcGill UniversityMontréalCanada
  2. 2.Department of Computer Science and Communication EngineeringKyushu UniversityFukuokaJapan
  3. 3.Information Science CenterKyushu Institute of Technology IizukaFukuokaJapan

Personalised recommendations