Proximity on a grid

Preliminary version
  • Rolf G. Karlsson
  • J. Ian Munro
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 182)


Given n keys taken from {1,...,m}×{1,...,m}, this paper presents an O(loglogm) time nearest neighbor query algorithm using O(n) space. An O(logm) time, O(nlogm) space semidynamic algorithm and an O(log3/2m) time, O(nlogm) space dynamic algorithm are also given. All the results apply under L1 and L metrics.


Voronoi Diagram Dynamic Algorithm Cluster Label Voronoi Region Trie Level 
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.


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  1. [1]
    J.L. Bentley and J.B. Saxe, Decomposable Searching Problems I. Static-to-Dynamic Transformations, J. Algorithms 1, 4 (Dec. 1980), 301–358.Google Scholar
  2. [2]
    D.P. Dobkin and R.J. Lipton, Multidimensional Searching Problems, SIAM J. Comput. 5, 2 (June 1976), 181–186.Google Scholar
  3. [3]
    L.J. Guibas and J. Stolfi, On Computing All North-East Nearest Neighbors in the L 1 Metric, Information Processing Lett. 17 (Nov. 1983), 219–223.Google Scholar
  4. [4]
    R.G. Karlsson, Algorithms on Bounded Domains, Ph.D. dissertation in preparation (1984).Google Scholar
  5. [5]
    J.M. Keil and D.G. Kirkpatrick, Computational Geometry on Integer Grids, Allerton Conference 1981, 41–50.Google Scholar
  6. [6]
    D.G. Kirkpatrick, Optimal Search in Planar Subdivisions, SIAM J.Comput. 12 (Feb. 1983), 28–35.Google Scholar
  7. [7]
    D.E. Knuth, The Art of Computer Programming, Vol. 3, Sorting and Searching, Addison-Wesley, Reading, Mass., 1973.Google Scholar
  8. [8]
    D.T. Lee and C.K. Wong, Voronoi Diagrams in L 1 (L ) metrics with 2-dimensional storage applications, SIAM J.Comput. 9 (Feb. 1980), 200–211.Google Scholar
  9. [9]
    R.J. Lipton and R.E. Tarjan, Applications of a Planar Separator Theorem, Proc. 18th Annual IEEE Symposium on Foundations of Computer Science (1978), 28–34.Google Scholar
  10. [10]
    M.H. Overmars and J. van Leeuwen, Worst-case Optimal Insertion and Deletion Methods for Decomposable Searching Problems, Information Processing Lett. 12, 4 (Aug. 1981), 168–173.Google Scholar
  11. [11]
    M.I. Shamos, Geometric Complexity, Proc. 7th Annual ACM Symposium on Theory of Computing (1975), 224–233.Google Scholar
  12. [12]
    D.E. Willard, Two Very Fast Trie Data Structures, Allerton Conference 1981, 355–363.Google Scholar
  13. [13]
    D.E. Willard, Log-logarithmic Worst-case Range Queries Are Possible in Space Θ(n), Information Processing Lett. 17 (Aug. 1983), 81–84.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Rolf G. Karlsson
    • 1
  • J. Ian Munro
    • 1
  1. 1.Data Structuring Group Department of Computer ScienceUniversity of WaterlooWaterlooCanada

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