Computing Voronoi Adjacencies in High Dimensional Spaces by Using Linear Programming

  • Juan MendezEmail author
  • Javier Lorenzo
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 30)


Some algorithms in Pattern Recognition and Machine Learning as neighborhood-based classification and dataset condensation can be improved with the use of Voronoi tessellation. This paper shows the weakness of some existing algorithms of tessellation to deal with high-dimensional datasets. The use of linear programming can improve the tessellation procedures by focusing on Voronoi adjacency. It will be shown that the adjacency test based on linear programming is a version of the polytope search. However, the polytope search procedure provides more information than a simple Boolean test. This paper proposes a strategy to use the additional information contained in the basis of the linear programming algorithm to obtain other tests. The theoretical results are applied to tessellate several random datasets, and also for much-used datasets in Machine Learning repositories.


Voronoi adjacencies Nearest neighbors Machine learning Linear programming 


  1. 1.
    Agrell, E.: A Method for examining vector quantizer structures. In: Proceeding of IEEE International Symposium on Information Theory, pp. 394 (1993)Google Scholar
  2. 2.
    Frank, A. Asuncion, A. UCI Machine Learning Repository []. Irvine, CA: University of California, School of Information and Computer Science (2010)
  3. 3.
    Aupetit, M.: High-dimensional labeled data analysis with gabriel graphs. In: European Symposium on Artificial Neuron Networks, pp. 21–26 (April 2003)Google Scholar
  4. 4.
    Aupetit, M., Catz, T.: High-dimensional labeled data analysis with topology representating graphs. Neurocomputing 63, 139–169 (2005)CrossRefGoogle Scholar
  5. 5.
    Avis, D., Fukuda, K.: A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete Comput. Geom. 8(3), 295–313 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Barber, C.B., Dobkin, D.P., Huhdanpaa, H.: The quickhull algorithm for convex hulls. ACM Trans. Math. Software 22(4), 469–483 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bazaraa, M.S., Jarvis, J.J., Sherali, H.S.: Linnear Programming and Networks Flows. Wiley, New York (1990)Google Scholar
  8. 8.
    Bhattacharya, B., Poulsen, R., Toussaint, G.: Application of proximity graphs to editing nearest neighbor decision rules. Technical Report SOCS 92.19, School of Computer Science, McGill University (1992)Google Scholar
  9. 9.
    Bowyer, A.: Computing Dirichlet tessellations. Comput. J. 24(2), 162–166 (1981)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bremner, D., Fukuda, K., Marzetta, A.: Primal-dual methods for vertex and facet enumeration. In: SCG ’97: Proceedings of the Thirteenth Annual Symposium on Computational Geometry, pp. 49–56. ACM, New York (1997)Google Scholar
  11. 11.
    Chin, E., Garcia, E.K., Gupta, M.R.: Color management of printers by regression over enclosing neighborhoods. In: IEEE International Conference on Image Processing. ICIP, 2, 161–164, (Oct 2007)Google Scholar
  12. 12.
    Dantzig, G.B., Thapa, M.N.: Linear Programming 2: Theory and Extensions. Springer, Berlin (2003)Google Scholar
  13. 13.
    Devroye, L., Gyorfi, L., Lugosi, G.: A Probabilistic Theory of Pattern Recognition, Springer Verlag (1996)Google Scholar
  14. 14.
    Duda, R., Hart, P., Stork, D.: Pattern Classification. Wiley, New York (2001)zbMATHGoogle Scholar
  15. 15.
    Fukuda, K.: Frecuently asked questions in polyhedral computation. Technical report, Swiss Federal Institute of Technology, Lausanne, Switzerland (June 2004)Google Scholar
  16. 16.
    Fukuda, K., Liebling, T.M., Margot, F.: Analysis of backtrak algoritms for listing all vertices and all faces of convex polyhedron. Comput. Geom. 8, 1–12 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Gabriel, K.R., Sokal, R.R.: A new stattistical approach to geographic variation analysis. Systemat. Zool. 18, 259–270 (1969)CrossRefGoogle Scholar
  18. 18.
    Greeff, G.: The revised simplex algorithm on a GPU. Technical report, Department of Computer Science, University of Stellenbosch (February 2005)Google Scholar
  19. 19.
    Gupta, M.R., Garcia, E.K., Chin, E.: Adaptive local linear regression with application to printer color management. In: IEEE Transactions on Image Process (2008)Google Scholar
  20. 20.
    Kalai, G.: Linear programming, the simplex algorithm and simple polytopes. Math. Program. 79, 217–233 (1997)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Koivistoinen, H., Ruuska, M., Elomaa, T.: A voronoi diagram approach to autonomous clustering. Lecture Notes in Computer Science (4265), pp. 149–160 (2006)CrossRefGoogle Scholar
  22. 22.
    Mendez, J.: Cooperating Multi-core and Multi-GPU in the computation of the multidimensional voronoi adjacency in machine learning datasets. In: PDPTA, pp. 717–723 (2010)Google Scholar
  23. 23.
    Mendez, J., Lorenzo, J.: Efficient computation of voronoi neighbors based on polytope Search in pattern recognition. In: Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods, pp. 357–364. SciTePress (2012)Google Scholar
  24. 24.
    Navarro, G.: Searching in metric spaces by spatial approximation. VLDB J. 11, 28–46 (2002)CrossRefGoogle Scholar
  25. 25.
    Ramasubramanian, V., Paliwal, K.: Voronoi projection-based fast nearest-neighbor search algorithms: Box-search and mapping table-based search techniques. Digital Signal Process. 7, 260–277 (1997)CrossRefGoogle Scholar
  26. 26.
    Sibson, R.: A brief description of natural neighbour interpolation. In: Interpreting Multivariate Data, pp. 21–36. Wiley, New York (1981)Google Scholar
  27. 27.
    Watson, D.F.: Computing the n-dimensional tessellation with application to voronoi polytopes. Comput. J. 24(2), 167–172 (1981)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Web, A.: Statistical Pattern Recognition, 2nd edn. Wiley, New York (2002)CrossRefGoogle Scholar
  29. 29.
    Winston, W.L.: Operations Research Applications and Algorithms. Wadsworth, Belmont (1994)zbMATHGoogle Scholar
  30. 30.
    Wright, M.H.: The interior-point revolution in optimization: History, recent developments, and lasting consequences. Bull. AMS 42(1), 39–56 (2004)CrossRefGoogle Scholar
  31. 31.
    Yarmish, G., van Slyke, R.: RetroLP, an implementation of the standard Simplex method. Technical report, Department of Computer and Information Science, Brooklyn College (2001)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Departamento de Informática y SistemasUniversity Las Palmas de Gran CanariaLas Palmas de Gran CanariaSpain
  2. 2.Institute of Intelligent SystemsUniversity Las Palmas de Gran CanariaLas Palmas de Gran CanariaSpain

Personalised recommendations