Discrete & Computational Geometry

, Volume 17, Issue 3, pp 263–282 | Cite as

On Nearest-Neighbor Graphs

  • D. EppsteinEmail author
  • M. S. PatersonEmail author
  • F. F. YaoEmail author


The “nearest-neighbor” relation, or more generally the “k-nearest-neighbors” relation, defined for a set of points in a metric space, has found many uses in computational geometry and clustering analysis, yet surprisingly little is known about some of its basic properties. In this paper we consider some natural questions that are motivated by geometric embedding problems. We derive bounds on the relationship between size and depth for the components of a nearest-neighbor graph and prove some probabilistic properties of the k-nearest-neighbors graph for a random set of points.


Minimum Span Tree Computational Geometry Related Edge Outer Vertex Rectilinear Steiner Tree 
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.


  1. 1.
    N. Alon and J. H. Spencer. The Probabilistic Method. Wiley-Interscience, New York, 1992.zbMATHGoogle Scholar
  2. 2.
    M. Bern. Two probabilistic results on rectilinear Steiner trees. Algorithmica 3 (1988), 191–204.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J. Boris. A vectorized “near neighbors” algorithm of order N using a monotonic logical grid. Journal of Computational Physics 66 (1986), 1–20.zbMATHCrossRefGoogle Scholar
  4. 4.
    P. B. Callahan. Optimal parallel all-nearest-neighbors using the well-separated pair decomposition. Proc. 34 th IEEE Symp. on Foundations of Computer Science (1993), pp. 332–340.Google Scholar
  5. 5.
    P. B. Callahan and S. R. Kosaraju. A decomposition of multi-dimensional point-sets with applications to k-nearest-neighbors and n-body potential fields. Proc. 24th ACM Symp. on Theory of Computing (1992), pp. 546–556.Google Scholar
  6. 6.
    K. L. Clarkson. Fast algorithms for the all-nearest-neighbors problem. Proc. 24th IEEE Symp. on Foundations of Computer Science (1983), pp. 226–232.Google Scholar
  7. 7.
    J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices and Groups. Springer-Verlag, New York, 1988.zbMATHGoogle Scholar
  8. 8.
    L. Devroye. The expected size of some graphs in computational geometry. Computers & Mathematics with Applications 15 (1988), 53–64.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    M. T. Dickerson and D. Eppstein. Algorithms for proximity problems in higher dimensions. Computational Geometry, Theory & Applications 5 (1996), 277–291.zbMATHMathSciNetGoogle Scholar
  10. 10.
    P. Eades and S. Whitesides. The realization problem for Euclidean minimum spanning trees is NP-hard. Proc. 10th ACM Symp. on Computational Geometry (1994), pp. 49–56.Google Scholar
  11. 11.
    H. Edelsbrunner. Algorithms in Combinatorial Geometry. Springer-Verlag, New York, 1987.zbMATHGoogle Scholar
  12. 12.
    D. Eppstein and J. Erickson. Iterated nearest neighbors and finding minimal polytopes. Discrete & Computational Geometry 11 (1994), 321–350.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    C. Monma and S. Suri. Transitions in geometric spanning trees. Proc. 7th ACM Symp. on Computational Geometry (1991), pp. 239–249.Google Scholar
  14. 14.
    M. S. Paterson and F. F. Yao. On nearest-neighbor graphs. Proc. 19th Internat. Coll. on Automata, Languages and Programming. LNCS, 623. Springer-Verlag, Berlin, 1992, pp. 416–426.Google Scholar
  15. 15.
    S.-H. Teng and F. F. Yao. Percolation and k-nearest neighbor clustering. Manuscript, 1993.Google Scholar
  16. 16.
    P. Vaidya. An O(n log n) algorithm for the all-nearest-neighbors problem. Discrete & Computational Geometry 4 (1989), 101–115.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  1. 1.Department of Information and Computer ScienceUniversity of CaliforniaIrvineUSA
  2. 2.Department of Computer ScienceUniversity of WarwickCoventryEngland
  3. 3.Xerox Palo Alto Research CenterPalo AltoUSA

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