Discrete & Computational Geometry

, Volume 4, Issue 2, pp 101–115 | Cite as

AnO(n logn) algorithm for the all-nearest-neighbors Problem

  • Pravin M. Vaidya
Article

Abstract

Given a setV ofn points ink-dimensional space, and anL q -metric (Minkowski metric), the all-nearest-neighbors problem is defined as follows: for each pointp inV, find all those points inV−{p} that are closest top under the distance metricL q . We give anO(n logn) algorithm for the all-nearest-neighbors problem, for fixed dimensionk and fixed metricL q . Since there is an Θ(n logn) lower bound, in the algebraic decision-tree model of computation, on the time complexity of any algorithm that solves the all-nearest-neighbors problem (fork=1), the running time of our algorithm is optimal up to a constant factor.

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References

  1. 1.
    M. Ben-Or, Lower bounds for algebraic computation trees,Proc. 15th Annual ACM Symp. Theory Comput., 1983, pp. 80–86.Google Scholar
  2. 2.
    J. L. Bentley, Multidimensional divide-and-conquer,Comm. ACM23 (1980), 214–229.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    J. L. Bentley, D. F. Stanat, and E. H. Williams, Jr., The complexity of finding fixed radius nearest neighbors,Inform. Process. Lett. 6 (1977), 209–212.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    J. L. Bentley, B. Weide, and A. Y. Yao, Optimal expected-time algorithms for closest-point problems,ACM Tran. Math. Software 6 (1982), 563–579.MathSciNetCrossRefGoogle Scholar
  5. 5.
    K. Clarkson, Fast algorithms for the all-nearest-neighbors problem,Proc. 24th Annual Symp. Found. Comput. Sci., 1983, pp. 226–232.Google Scholar
  6. 6.
    H. N. Gabow, J. L. Bentley, and R. E. Tarjan, Scaling and related techniques for geometry problems,Proc. 16th Annual ACM Symp. Theory Comput., 1984, pp. 135–143.Google Scholar
  7. 7.
    F. P. Preparata and M. I. Shamos,Computational Geometry: An Introduction, Springer-Verlag, New York, 1985.CrossRefGoogle Scholar
  8. 8.
    E. M. Reingold, J. Nievergelt, and N. Deo,Combinatorial Algorithms: Theory and Practice, Prentice Hall, Englewood Cliffs, NJ, 1977.Google Scholar
  9. 9.
    M. I. Shamos, Computational Geometry, Ph.D. dissertation, Yale University, New Haven, CT, 1978.Google Scholar
  10. 10.
    M. O. Rabin, Probabilistic algorithms, inAlgorithms and Complexity (J. F. Traub, ed.), Academic Press, New York, 1976, pp. 21–30.Google Scholar
  11. 11.
    F. Yuval, Finding nearest neighbors,Inform. Process. Lett. 3 (1975), 113–114.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • Pravin M. Vaidya
    • 1
  1. 1.Department of Computer ScienceUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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