Robust Nonparametric Data Approximation of Point Sets via Data Reduction

  • Stephane Durocher
  • Alexandre Leblanc
  • Jason Morrison
  • Matthew Skala
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7676)


In this paper we present a novel nonparametric method for simplifying piecewise linear curves and we apply this method as a statistical approximation of structure within sequential data in the plane. We consider the problem of minimizing the average length of sequences of consecutive input points that lie on any one side of the simplified curve. Specifically, given a sequence P of n points in the plane that determine a simple polygonal chain consisting of n − 1 segments, we describe algorithms for selecting a subsequence Q ⊂ P (including the first and last points of P) that determines a second polygonal chain to approximate P, such that the number of crossings between the two polygonal chains is maximized, and the cardinality of Q is minimized among all such maximizing subsets of P. Our algorithms have respective running times \(O(n^2\sqrt{\log n})\) when P is monotonic and O(n 2log4/3 n) when P is any simple polyline.


Blue Point Consecutive Segment Polygonal Chain Chebyshev Distance Angular Order 
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  1. 1.
    Agarwal, P.K., Har-Peled, S., Mustafa, N.H., Wang, Y.: Near-Linear Time Approximation Algorithms for Curve Simplification. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 29–202. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Agarwal, P.K., Varadarajan, K.R.: Efficient algorithms for approximating polygonal chains. Discrete and Computational Geometry 23, 273–291 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Alt, H., Guibas, L.J.: Discrete geometric shapes: Matching, interpolation, and approximation. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 121–153. Elsevier (2000)Google Scholar
  4. 4.
    Chan, T.M., Pǎtraşcu, M.: Counting inversions, offline orthogonal range counting and related problems. In: SODA, pp. 161–173 (2010)Google Scholar
  5. 5.
    Chan, W.S., Chin, F.: Approximation of Polygonal Curves with Minimum Number of Line Segments. In: Ibaraki, T., Iwama, K., Yamashita, M., Inagaki, Y., Nishizeki, T. (eds.) ISAAC 1992. LNCS, vol. 650, pp. 378–387. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  6. 6.
    Douglas, D., Peucker, T.: Algorithms for the reduction of points required to represent a digitised line or its caricature. The Canadian Cartographer 10, 112–122 (1973)CrossRefGoogle Scholar
  7. 7.
    Han, Y.: Deterministic sorting in o(nloglogn) time and linear space. J. of Algorithms 50, 96–105 (2004)zbMATHCrossRefGoogle Scholar
  8. 8.
    Han, Y., Thorup, M.: Integer sorting in \(o(n\sqrt{\log \log n})\) expected time and linear space. In: FOCS, pp. 135–144 (2002)Google Scholar
  9. 9.
    Hershberger, J., Snoeyink, J.: Cartographic line simplification and polygon csg formulae in o(nlog* n) time. Computational Geometry: Theory and Applications 11(3-4), 175–185 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Imai, H., Iri, M.: Polygonal approximation of curve-formulations and algorithms. In: Toussaint, G.T. (ed.) Computational Morphology, pp. 71–86. North-Holland (1988)Google Scholar
  11. 11.
    Melkman, A., O’Rourke, J.: On polygonal chain approximation. In: Toussaint, G.T. (ed.) Computational Morphology, pp. 87–95. North-Holland (1988)Google Scholar
  12. 12.
    Skiena, S.S.: The Algorithm Design Manual, 2nd edn. Springer (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Stephane Durocher
    • 1
  • Alexandre Leblanc
    • 1
  • Jason Morrison
    • 1
  • Matthew Skala
    • 1
  1. 1.University of ManitobaWinnipegCanada

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