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.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
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)
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)
Agarwal, P.K., Aronov, B., Koltun, V.: Efficient algorithms for bichromatic separability. ACM Transactions on Algorithms (2005) (to appear)
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)
Agarwal, P.K., Aronov, B., Sharir, M.: Exact and approximation algorithms for minimum-width cylindrical shells. Discrete Comput. Geom. 26(3), 307–320 (2001)
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)
Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Approximating extent measures of points. J. Assoc. Comput. Mach. 51, 606–635 (2004)
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)
Aronov, B., Har-Peled, S.: On approximating the depth and related problems. In: Proc. 16th ACM-SIAM Sympos. Discrete Algorithms (2005)
Brodal, G.S., Jacob, R.: Dynamic planar convex hull. In: Proc. 43rd Annu. IEEE Sympos. Found. Comput. Sci., pp. 617–626 (2002)
Chan, T.M.: Output-sensitive results on convex hulls, extreme points, and related problems. Discrete Comput. Geom. 16, 369–387 (1996)
Chan, T.M.: Approximating the diameter, width, smallest enclosing cylinder and minimum-width annulus. Internat. J. Comput. Geom. Appl. 12(2), 67–85 (2002)
Chan, T.M.: Low-dimensional linear programming with violations. In: Proc. 43rd Annu. IEEE Sympos. Found. Comput. Sci., pp. 570–579 (2002)
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)
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)
Chazelle, B.: An optimal convex hull algorithm in any fixed dimension. Discrete Comput. Geom. 10, 377–409 (1993)
Cristianini, N., Shaw-Taylor, J.: Support Vector Machines. Cambridge University Press, Cambridge (2000)
Dudley, R.M.: Metric entropy of some classes of sets with differentiable boundaries. J. Approx. Theory 10, 227–236 (1974)
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)
Edelsbrunner, H.: Algorithms in Combinatorial Geometry. EATCS Monographs on Theoretical Computer Science, vol. 10. Springer, Heidelberg (1987)
Edelsbrunner, H., O’Rourke, J., Seidel, R.: Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput. 15, 341–363 (1986)
Edelsbrunner, H., Preparata, F.P.: Minimum polygonal separation. Inform. Comput. 77, 218–232 (1988)
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)
Fekete, S.P.: On the complexity of min-link red-blue separation. Manuscript, Department of Applied Mathematics, SUNY Stony Brook, NY (1992)
Har-Peled, S., Varadarajan, K.: High-dimensional shape fitting in linear time. In: Proc. 19th Annu. ACM Sympos. Comput. Geom., pp. 39–47 (2003)
Har-Peled, S., Wang, Y.: Shape fitting with outliers. SIAM J. Comput. 33(2), 269–285 (2004)
Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning. Springer, Berlin (2001)
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)
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)
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)
Matoušek, J.: On geometric optimization with few violated constraints. Discrete Comput. Geom. 14, 365–384 (1995)
Megiddo, N.: Linear programming in linear time when the dimension is fixed. J. Assoc. Comput. Mach. 31, 114–127 (1984)
Mitchell, J.S.B.: Approximation algorithms for geometric separation problems. Technical report, Department of Applied Mathematics, SUNY Stony Brook, NY (July 1993)
O’Rourke, J., Kosaraju, S.R., Megiddo, N.: Computing circular separability. Discrete Comput. Geom. 1, 105–113 (1986)
Sharir, M., Smorodinsky, S., Tardos, G.: An improved bound for k-sets in three dimensions. Discrete Comput. Geom. 26, 195–204 (2001)
Vapnik, V.: The Nature of Statistical Learning Theory. Springer, New York (1996)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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)