A Near Linear Algorithm for Testing Linear Separability in Two Dimensions

  • Sylvain Contassot-Vivier
  • David Elizondo
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 311)


We present a near linear algorithm for determining the linear separability of two sets of points in a two-dimensional space. That algorithm does not only detects the linear separability but also computes separation information. When the sets are linearly separable, the algorithm provides a description of a separation hyperplane. For non linearly separable cases, the algorithm indicates a negative answer and provides a hyperplane of partial separation that could be useful in the building of some classification systems.


Classification linear separability 2D geometry 


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  1. 1.
    Bazaraa, M.S., Jarvis, J.J.: Linear Programming and Network Flow. John Wiley and Sons, London (1977)Google Scholar
  2. 2.
    Boser, B.E., Guyon, I.M., Vapnik, V.N.: A training algorithm for optimal margin classifiers. In: Proceedings of the Fifth Annual Workshop on Computational Learning Theory, COLT 1992, pp. 144–152. ACM, New York (1992), CrossRefGoogle Scholar
  3. 3.
    Cortes, C., Vapnik, V.: Support-vector network. Machine Learning 20, 273–297 (1995)zbMATHGoogle Scholar
  4. 4.
    Elizondo, D.A.: The linear separability problem: some testing methods. IEEE Transactions on Neural Networks 17(2), 330–344 (2006), CrossRefGoogle Scholar
  5. 5.
    Elizondo, D.A.: Artificial neural networks, theory and applications (2008), French HDRGoogle Scholar
  6. 6.
    Elizondo, D.A., Ortiz-de-Lazcano-Lobato, J.M., Birkenhead, R.: A Novel and Efficient Method for Testing Non Linear Separability. In: de Sá, J.M., Alexandre, L.A., Duch, W., Mandic, D.P. (eds.) ICANN 2007, Part I. LNCS, vol. 4668, pp. 737–746. Springer, Heidelberg (2007), CrossRefGoogle Scholar
  7. 7.
    Ferreira, L., Kaszkurewicz, E., Bhaya, A.: Solving systems of linear equations via gradient systems with discontinuous righthand sides: application to ls-svm. IEEE Transactions on Neural Networks 16, 501–505 (2005)CrossRefGoogle Scholar
  8. 8.
    Fisher, R.A.: The use of multiple measurements in taxonomic problems. Annual Eugenics 7(II), 179–188 (1936)Google Scholar
  9. 9.
    Fourier, J.B.J.: Solution d’une question pariculière du calcul des inégalités. In: Oeuvres II, pp. 317–328 (1826)Google Scholar
  10. 10.
    McCulloch, W., Pitts, W.: A logical calculus of the ideas imminent in nervous activity. Bulletin of Mathematical Biophysics 5, 115–133 (1943)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Novikoff, A.: On convergence proofs on perceptrons. In: Symposium on the Mathematical Theory of Automata, vol. XII, pp. 615–622 (1962)Google Scholar
  12. 12.
    Pang, S., Kim, D., Bang, S.Y.: Membership authentication using svm classification tree generated by membership-based lle data partition. IEEE Transactions on Neural Networks 16, 436–446 (2005)CrossRefGoogle Scholar
  13. 13.
    Rosenblatt, F.: The perceptron: A probabilistic model for information storage in the brain. Psychological Review 65, 386–408 (1958)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rosenblatt, F.: Principles of Neurodynamics. Spartan, Washington, D.C. (1962)Google Scholar
  15. 15.
    Tajine, M., Elizondo, D.: New methods for testing linear separability. Neurocomputing 47(1-4), 295–322 (2002)CrossRefGoogle Scholar
  16. 16.
    Tarski, A.: A decision method for elementary algebra and geometry. Tech. rep., University of California Press, Berkeley and Los Angeles (1951)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sylvain Contassot-Vivier
    • 1
  • David Elizondo
    • 2
  1. 1.Loria, UMR 7503Université de LorraineNancyFrance
  2. 2.Centre for Computational IntelligenceDe Montfort UniversityLeicesterUnited Kingdom

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