Output-Sensitive Algorithms for Computing Nearest-Neighbour Decision Boundaries

  • David Bremner
  • Erik Demaine
  • Jeff Erickson
  • John Iacono
  • Stefan Langerman
  • Pat Morin
  • Godfried Toussaint
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2748)

Abstract

Given a set R of red points and a set B of blue points, the nearest-neighbour decision rule classifies a new point q as red (respectively, blue) if the closest point to q in R ∪ B comes from R (respectively, B). This rule implicitly partitions space into a red set and a blue set that are separated by a red-blue decision boundary. In this paper we develop output-sensitive algorithms for computing this decision boundary for point sets on the line and in ℝ2. Both algorithms run in time O(n log k), where k is the number of points that contribute to the decision boundary. This running time is the best possible when parameterizing with respect to n and k.

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References

  1. 1.
    Ben-Or, M.: Lower bounds for algebraic computation trees (preliminary report). In: Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pp. 80–86 (1983)Google Scholar
  2. 2.
    Bhattacharya, B.K., Sen, S.: On a simple, practical, optimal, output-sensitive randomized planar convex hull algorithm. Journal of Algorithms 25(1), 177–193 (1997)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Blum, M., Floyd, R.W., Pratt, V., Rivest, R.L., Tarjan, R.E.: Time bounds for selection. Journal of Computing and Systems Science 7, 448–461 (1973)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chan, T.M.: Optimal output-sensitive convex hull algorithms in two and three dimensions. Discrete & Computational Geometry 16, 361–368 (1996)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chan, T.M., Snoeyink, J., Yap, C.K.: Primal dividing and dual pruning: Output-sensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams. Discrete & Computational Geometry 18, 433–454 (1997)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cover, T.M., Hart, P.E.: Nearest neighbour pattern classification. IEEE Transactions on Information Theory 13, 21–27 (1967)MATHCrossRefGoogle Scholar
  7. 7.
    Dasarathy, B., White, L.J.: A characterization of nearest-neighbour rule decision surfaces and a new approach to generate them. Pattern Recognition 10, 41–46 (1978)MATHCrossRefGoogle Scholar
  8. 8.
    Devroye, L.: On the inequality of Cover and Hart. IEEE Transactions on Pattern Analysis and Machine Intelligence 3, 75–78 (1981)MATHCrossRefGoogle Scholar
  9. 9.
    Dobkin, D.P., Kirkpatrick, D.G.: Fast detection of poyhedral intersection. Theoretical Computer Science 27, 241–253 (1983)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dobkin, D.P., Kirkpatrick, D.G.: A linear algorithm for determining the separation of convex polyhedra. Journal of Algorithms 6, 381–392 (1985)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hoare, C.A.R.: ACM Algorithm 64: Quicksort. Communications of the ACM 4(7), 321 (1961)CrossRefGoogle Scholar
  12. 12.
    Kirkpatrick, D.G.: Optimal search in planar subdivisions. SIAM Journal on Computing 12(1), 28–35 (1983)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kirkpatrick, D.G., Seidel, R.: The ultimate planar convex hull algorithm? SIAM Journal on Computing 15(1), 287–299 (1986)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Preparata, F.P., Shamos, M.I.: Computational Geometry. Springer, Heidelberg (1985)Google Scholar
  15. 15.
    Shamos, M.I.: Geometric complexity. In: Proceedings of the 7th ACM Symposium on the Theory of Computing (STOC 1975), pp. 224–253 (1975)Google Scholar
  16. 16.
    Stone, C.: Consistent nonparametric regression. Annals of Statistics 8, 1348–1360 (1977)CrossRefGoogle Scholar
  17. 17.
    Toussaint, G.T.: Proximity graphs for instance-based learning. (2003) (manuscript) Google Scholar
  18. 18.
    Toussaint, G.T., Bhattacharya, B.K., Poulsen, R.S.: The application of Voronoi diagrams to non-parametric decision rules. In: Proceedings of Computer Science and Statistics: 16th Symposium of the Interface (1984)Google Scholar
  19. 19.
    Wenger, R.: Randomized quick hull. Algorithmica 17, 322–329 (1997)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • David Bremner
    • 1
  • Erik Demaine
    • 2
  • Jeff Erickson
    • 3
  • John Iacono
    • 4
  • Stefan Langerman
    • 5
  • Pat Morin
    • 6
  • Godfried Toussaint
    • 7
  1. 1.Faculty of Computer ScienceUniversity of New Brunswick 
  2. 2.MIT Laboratory for Computer Science 
  3. 3.Computer Science DepartmentUniversity of Illinois 
  4. 4.Polytechnic University 
  5. 5.Chargé de recherches du FNRSUniversité Libre de Bruxelles 
  6. 6.School of Computer ScienceCarleton University 
  7. 7.School of Computer ScienceMcGill University 

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