Skip to main content
SpringerLink
Log in

Separability with Outliers

  • Conference paper
Algorithms and Computation (ISAAC 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3827))

Included in the following conference series:

Abstract

We develop exact and approximate algorithms for computing optimal separators and measuring the extent to which two point sets in d-dimensional space are separated, with respect to different classes of separators and various extent measures. This class of geometric problems generalizes two widely studied problem families, namely separability and the computation of statistical estimators.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Agarwal, P.K., Aronov, B., Chan, T.M., Sharir, M.: On levels in arrangements of lines, segments, planes, and triangles. Discrete Comput. Geom. 19, 315–331 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Agarwal, P.K., Aronov, B., Har-Peled, S., Sharir, M.: Approximation and exact algorithms for minimum-width annuli and shells. Discrete Comput. Geom. 24(4), 687–705 (2000)

    MATH  MathSciNet  Google Scholar 

  3. Agarwal, P.K., Aronov, B., Koltun, V.: Efficient algorithms for bichromatic separability. ACM Transactions on Algorithms (2005) (to appear)

    Google Scholar 

  4. Agarwal, P.K., Aronov, B., Sharir, M.: Line traversals of balls and smallest enclosing cylinders in three dimensions. Discrete Comput. Geom. 21, 373–388 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Agarwal, P.K., Aronov, B., Sharir, M.: Exact and approximation algorithms for minimum-width cylindrical shells. Discrete Comput. Geom. 26(3), 307–320 (2001)

    MATH  MathSciNet  Google Scholar 

  6. Agarwal, P.K., Guibas, L.J., Har-Peled, S., Rabinovitch, A., Sharir, M.: Penetration depth of two convex polytopes in 3D. Nordic J. Comput. 7(3), 227–240 (2000)

    MATH  MathSciNet  Google Scholar 

  7. Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Approximating extent measures of points. J. Assoc. Comput. Mach. 51, 606–635 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Arkin, E., Hurtado, F., Mitchell, J., Seara, C., Skiena, S.: Some lower bounds on geometric separability problems. In: 11th Fall Workshop on Computational Geometry (2001)

    Google Scholar 

  9. Aronov, B., Har-Peled, S.: On approximating the depth and related problems. In: Proc. 16th ACM-SIAM Sympos. Discrete Algorithms (2005)

    Google Scholar 

  10. Brodal, G.S., Jacob, R.: Dynamic planar convex hull. In: Proc. 43rd Annu. IEEE Sympos. Found. Comput. Sci., pp. 617–626 (2002)

    Google Scholar 

  11. Chan, T.M.: Output-sensitive results on convex hulls, extreme points, and related problems. Discrete Comput. Geom. 16, 369–387 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Chan, T.M.: Approximating the diameter, width, smallest enclosing cylinder and minimum-width annulus. Internat. J. Comput. Geom. Appl. 12(2), 67–85 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chan, T.M.: Low-dimensional linear programming with violations. In: Proc. 43rd Annu. IEEE Sympos. Found. Comput. Sci., pp. 570–579 (2002)

    Google Scholar 

  14. Chan, T.M.: Faster core-set constructions and data stream algorithms in fixed dimensions. In: Proc. 20th Annu. ACM Sympos. Comput. Geom., pp. 152–159 (2004)

    Google Scholar 

  15. Charikar, M., Khuller, S., Mount, D.M., Narasimhan, G.: Algorithms for facility location problems with outliers. In: Proc. 12th ACM-SIAM Sympos. Discrete Algorithms, pp. 642–651 (2001)

    Google Scholar 

  16. Chazelle, B.: An optimal convex hull algorithm in any fixed dimension. Discrete Comput. Geom. 10, 377–409 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  17. Cristianini, N., Shaw-Taylor, J.: Support Vector Machines. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  18. Dudley, R.M.: Metric entropy of some classes of sets with differentiable boundaries. J. Approx. Theory 10, 227–236 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  19. Duncan, C.A., Goodrich, M.T., Ramos, E.A.: Efficient approximation and optimization algorithms for computational metrology. In: Proc. 8th ACM-SIAM Sympos. Discrete Algorithms, pp. 121–130 (1997)

    Google Scholar 

  20. Edelsbrunner, H.: Algorithms in Combinatorial Geometry. EATCS Monographs on Theoretical Computer Science, vol. 10. Springer, Heidelberg (1987)

    MATH  Google Scholar 

  21. Edelsbrunner, H., O’Rourke, J., Seidel, R.: Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput. 15, 341–363 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  22. Edelsbrunner, H., Preparata, F.P.: Minimum polygonal separation. Inform. Comput. 77, 218–232 (1988)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  24. Fekete, S.P.: On the complexity of min-link red-blue separation. Manuscript, Department of Applied Mathematics, SUNY Stony Brook, NY (1992)

    Google Scholar 

  25. Har-Peled, S., Varadarajan, K.: High-dimensional shape fitting in linear time. In: Proc. 19th Annu. ACM Sympos. Comput. Geom., pp. 39–47 (2003)

    Google Scholar 

  26. Har-Peled, S., Wang, Y.: Shape fitting with outliers. SIAM J. Comput. 33(2), 269–285 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  27. Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning. Springer, Berlin (2001)

    MATH  Google Scholar 

  28. Hurtado, F., Mora, M., Ramos, P.A., Seara, C.: Two problems on separability with lines and polygonals. In: Proc. 15th European Workshop on Computational Geometry, pp. 33–35 (1999)

    Google Scholar 

  29. Hurtado, F., Noy, M., Ramos, P.A., Seara, C.: Separating objects in the plane with wedges and strips. Discrete Appl. Math. 109, 109–138 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  30. Hurtado, F., Seara, C., Sethia, S.: Red-blue separability problems in 3d. In: Proc. 3rd Int. Conf. Comput. Sci. and Its Appl., pp. 766–775 (2003)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  32. Megiddo, N.: Linear programming in linear time when the dimension is fixed. J. Assoc. Comput. Mach. 31, 114–127 (1984)

    MATH  MathSciNet  Google Scholar 

  33. Mitchell, J.S.B.: Approximation algorithms for geometric separation problems. Technical report, Department of Applied Mathematics, SUNY Stony Brook, NY (July 1993)

    Google Scholar 

  34. O’Rourke, J., Kosaraju, S.R., Megiddo, N.: Computing circular separability. Discrete Comput. Geom. 1, 105–113 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  35. Sharir, M., Smorodinsky, S., Tardos, G.: An improved bound for k-sets in three dimensions. Discrete Comput. Geom. 26, 195–204 (2001)

    MATH  MathSciNet  Google Scholar 

  36. Vapnik, V.: The Nature of Statistical Learning Theory. Springer, New York (1996)

    Google Scholar 

  37. Yamamoto, P., Kato, K., Imai, K., Imai, H.: Algorithms for vertical and orthogonal L 1 linear approximation of points. In: Proc. 4th Annu. ACM Sympos. Comput. Geom., pp. 352–361. ACM Press, New York (1988)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Illinois, 201 N. Goodwin Avenue, Urbana, IL, 61801, USA

    Sariel Har-Peled

  2. Computer Science Department, Stanford University, 353 Serra Mall, Gates 464, Stanford, CA, 94305, USA

    Vladlen Koltun

Authors
  1. Sariel Har-Peled
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vladlen Koltun
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, City University of Hong Kong, Tat Chee Avenue, Kowloon , Hong Kong

    Xiaotie Deng

  2. Department of Computer Science, University of Texas at Dallas, EE/CS Building, 75083, Richardson, TX, USA

    Ding-Zhu Du

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Har-Peled, S., Koltun, V. (2005). Separability with Outliers. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_5

Download citation

  • DOI: https://doi.org/10.1007/11602613_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-30935-2

  • Online ISBN: 978-3-540-32426-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation