Farthest neighbors, maximum spanning trees and related problems in higher dimensions

  • Pankaj K. Agarwal
  • Jiří Matoušek
  • Subhash Suri
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 519)


We present a randomized algorithm of expected time complexity O(m2/3n2/3log4/3m+mlog2m+nlog2n) for computing bi-chromatic farthest neighbors between n red points and m blue points in ɛ3. The algorithm can also be used to compute all farthest neighbors or external farthest neighbors of n points in ɛ3 in O(n4/3log4/3n) expected time. Using these procedures as building blocks, we can compute a Euclidean maximum spanning tree or a minimum-diameter two-partition of n points in ɛ3 in O(n4/3log7/3n) expected time. The previous best bound for these problems was O(n3/2log1/2n). Our algorithms can be extended to higher dimensions.

We also propose fast and simple approximation algorithms for these problems. These approximation algorithms produce solutions that approximate the true value with a relative accuracy ɛ and run in time O(nɛ(1−k)/2logn) or O(nɛ(1−k)/2log2n) in k-dimensional space.


Span Tree Minimum Span Tree Voronoi Diagram Computational Geometry Blue Point 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    P. Agarwal, H. Edelsbrunner, O. Schwarzkopf, and E. Welzl. Euclidean minimum spanning trees and bichromatic closest pairs. Proc. of 6th ACM Symp. on Computational Geometry, 1990, pp. 203–210.Google Scholar
  2. [2]
    F. Aurenhammer. Improved algorithms for discs and balls using power diagrams. Journal of Algorithms, 9 (1988), 151–161.Google Scholar
  3. [3]
    B. Chazelle. How to search in history. Information and Control, 64 (1985), 77–99.Google Scholar
  4. [4]
    K. Clarkson. Fast expected-time and approximate algorithms for geometric minimum spanning tree. Proc. 16th Annual Symposium on Theory of Computing, 1984, pp. 342–348.Google Scholar
  5. [5]
    K. Clarkson. A randomized algorithm for closest-point queries. SIAM J. Computing, 17 (1988), 830–847.Google Scholar
  6. [6]
    K. Clarkson and P. Shor. Applications of random sampling in computational geometry, II. Discrete and Computational Geometry. 4 (1989), pp. 387–421.Google Scholar
  7. [7]
    H. Edelsbrunner. An acyclicity theorem for cell complexes in d dimensions. Proc. of 5th ACM Symp. on Computational Geometry, 1989, pp. 145–151.Google Scholar
  8. [8]
    H. Edelsbrunner. Algorithms in Combinatorial Geometry, Springer-Verlag, 1987.Google Scholar
  9. [9]
    H. Edelsbrunner and R. Seidel. Voronoi diagrams and arrangements. Discrete and Computational Geometry, 1 (1985), 25–44.Google Scholar
  10. [10]
    H. Edelsbrunner and M. Sharir, A hyperplane incidence problem with applications to counting distances. Proc. of International Symposium on Algorithms, LNCS 450, Springer-Verlag, 1990.Google Scholar
  11. [11]
    O. Egecioglu and B. Kalantari. Approximating the diameter of a set of points in the Euclidean space. Technical report, Rutgers University, 1989.Google Scholar
  12. [12]
    H. N. Gabow and R. E. Tarjan. A linear-time algorithm for a special case of disjoint set union. Journal of Computer and System Sciences, 30 (1985), 209–221.Google Scholar
  13. [13]
    R. L. Graham and P. Hell. On the history of minimum spanning tree problem. Annals of History of Computing, 7 (1985), 43–57.Google Scholar
  14. [14]
    D. Haussler and E. Welzl. ∈-nets and simplex range queries, Discrete and Computational Geometry. 2 (1987), 127–151.Google Scholar
  15. [15]
    C. Monma, M. Paterson, S. Suri and F. F. Yao. Computing Euclidean maximum spanning trees. Algorithmica, 5 (1990), 407–419.Google Scholar
  16. [16]
    C. Monma and S. Suri. Partitioning points and graphs to minimize the maximum or the sum of diameters. Proceedings of 6th International Conference on the Theory and Applications of Graphs, John Wiley and Sons, 1989Google Scholar
  17. [17]
    F. P. Preparata and M. I. Shamos. Computational Geometry. Springer Verlag, New York, 1985.Google Scholar
  18. [18]
    F. Preparata and R. Tamassia. Efficient spatial point location. Workshop on Algorithms and Data Structures, LNCS 382, Springer-Verlag, (1989), 3–11.Google Scholar
  19. [19]
    R. Seidel. A convex hull algorithm optimal for point sets in even dimensions. University of British Columbia, Vancouver, 1981.Google Scholar
  20. [20]
    P. Vaidya. Minimum spanning trees in k-dimensional space. SIAM J. of Computing, 17 (1988), 572–582.Google Scholar
  21. [21]
    P. Vaidya. An O(n log n) algorithm for the all-nearest-neighbor problem. Discrete and Computational Geometry, 4 (1989), 101–115.Google Scholar
  22. [22]
    A. Yao. On constructing minimum spanning trees in k-dimensional spaces and related problems, SIAM J. Computing, 11 (1982), 721–736.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Pankaj K. Agarwal
    • 1
  • Jiří Matoušek
    • 2
  • Subhash Suri
    • 3
  1. 1.Duke UniversityUSA
  2. 2.Charles University and Georgia TechUSA
  3. 3.BellcoreUSA

Personalised recommendations