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Outlier Respecting Points Approximation

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Book cover Algorithms and Computation (ISAAC 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7074))

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Abstract

In this paper, we consider a generalized problem formulation of computing a functional curve to approximate a point set in the plane with outliers. The goal is to seek a solution that not only optimizes its original objectives, but also somehow accommodates the impact of the outliers. Based on a new model of accommodating outliers, we present efficient geometric algorithms for various versions of this problem (e.g., the approximating functions are step functions or piecewise linear functions, the points are unweighted or weighted, etc). All our results are first known. Our new model and techniques for handling outliers may be useful to other applications as well.

This research was supported in part by NSF under Grant CCF-0916606.

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References

  1. Arning, A., Agrawal, R., Raghavan, P.: A linear method for deviation detection in large databases. In: Proc. of the Second International Conference on Knowledge Discovery and Data Mining, pp. 164–169 (1996)

    Google Scholar 

  2. Barnett, V., Lewis, T.: Outliers in Statistical Data. John Wiley and Sons, New York (1994)

    MATH  Google Scholar 

  3. Brodal, G., Jacob, R.: Dynamic planar convex hull. In: Proc. of the 43rd Symposium on Foundations of Computer Science (FOCS), pp. 617–626 (2002)

    Google Scholar 

  4. Chan, T.: Output-sensitive results on convex hulls and extreme points, and related problems. Discrete & Computational Geometry 16(3), 369–387 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chan, T.: Low-dimensional linear programming with violations. In: Proc. of 43rd Symposium on Foundations of Computer Science (FOCS), pp. 570–579 (2002)

    Google Scholar 

  6. Chen, D.Z., Wang, H.: Approximating Points by a Piecewise Linear Function: I. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 224–233. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  7. Chen, D.Z., Wang, H.: Approximating Points by a Piecewise Linear Function: II. Dealing with Outliers. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 234–243. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  8. Díaz-Bánez, J., Mesa, J.: Fitting rectilinear polygonal curves to a set of points in the plane. European Journal of Operational Research 130(1), 214–222 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dobkin, D., Kirkpatrick, D.: A linear algorithm for determining the separation of convex polyhedra. Journal of Algorithms 6(3), 381–392 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dyer, M.: Linear time algorithms for two- and three-variable linear programs. SIAM J. Comp. 13(1), 31–45 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Everett, H., Robert, J.M., van Kreveld, M.: An optimal algorithm for the (≤ k)-levels and with applications to separation and transversal problems. Int. J. Comput. Geometry Appl. 6(3), 247–261 (1996)

    Article  MATH  Google Scholar 

  12. Fournier, H., Vigneron, A.: Fitting a Step Function to a Point Set. In: Halperin, D., Mehlhorn, K. (eds.) Esa 2008. LNCS, vol. 5193, pp. 442–453. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  13. Guha, S., Koudas, N., Shim, K.: Data streams and histograms. In: Proc. of the 33rd Annual Symposium on Theory of Computing (STOC), pp. 471–475 (2001)

    Google Scholar 

  14. Guha, S., Shim, K.: A note on linear time algorithms for maximum error histograms. IEEE Trans. Knowl. Data Eng. 19(7), 993–997 (2007)

    Article  Google Scholar 

  15. Guha, S., Shim, K., Woo, J.: Rehist: Relative error histogram construction algorithms. In: Proc. of the 30th International Conference on Very Large Data Bases (VLDB), pp. 300–311 (2004)

    Google Scholar 

  16. Karras, P., Sacharidis, D., Mamoulis, N.: Exploiting duality in summarization with deterministic guarantees. In: Proc. of the 13th International Conference on Knowledge Discovery and Data Mining, pp. 380–389 (2007)

    Google Scholar 

  17. Knorr, E., Ng, R.: Algorithms for mining distance-based outliers in large datasets. In: Proc. of the 24th International Conference on Very Large Data Bases, pp. 382–403 (1998)

    Google Scholar 

  18. Matoušek, J.: On geometric optimization with few violated constraints. Discrete Comput. Geom. 14(1), 365–384 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Megiddo, N.: Linear programming in linear time when the dimension is fixed. Journal of the ACM 31(1), 114–127 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  20. Muller, D., Preparata, F.: Finding the intersection of two convex polyhedra. Theor. Compu. Sci. 7, 217–236 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  21. O’Rourke, J.: An on-line algorithm for fitting straight lines between data ranges. Commun. of ACM 24, 574–578 (1981)

    Article  MATH  Google Scholar 

  22. Preparata, F., Hong, S.: Convex hulls of finite sets of points in two and three dimensions. Communications of the ACM 20(2), 87–93 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ramaswamy, S., Rastogi, R., Shim, K.: Efficient algorithms for mining outliers from large data sets. ACM SIGMOD Record 29(2), 427–438 (2000)

    Article  Google Scholar 

  24. Roos, T., Widmayer, P.: k-violation linear programming. Information Processing Letters 52(2), 109–114 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sharir, M., Smorodinsky, S., Tardos, G.: An improved bound for k-sets in three dimensions. Discrete & Computational Geometry 26, 195–204 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN, 46556, USA

    Danny Z. Chen & Haitao Wang

Authors
  1. Danny Z. Chen
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  2. Haitao Wang
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Editor information

Editors and Affiliations

  1. Chuo University, Kasuga, Bunkyo-ku, 112-8551, Tokyo, Japan

    Takao Asano

  2. Gunma University, 1-5-1 Tenjin-Cho, 376-8515, Kiryu-Shi, Japan

    Shin-ichi Nakano

  3. Japan Advanced Institute of Science and Technology, 1-1 Asahidai, 923-1292, Nomi, Ishikawa, Japan

    Yoshio Okamoto

  4. Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, 152-8552, Tokyo, Japan

    Osamu Watanabe

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© 2011 Springer-Verlag Berlin Heidelberg

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Chen, D.Z., Wang, H. (2011). Outlier Respecting Points Approximation. In: Asano, T., Nakano, Si., Okamoto, Y., Watanabe, O. (eds) Algorithms and Computation. ISAAC 2011. Lecture Notes in Computer Science, vol 7074. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25591-5_61

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  • DOI: https://doi.org/10.1007/978-3-642-25591-5_61

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-25590-8

  • Online ISBN: 978-3-642-25591-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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