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Polynomial Time Algorithms for Bichromatic Problems

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Algorithms and Discrete Applied Mathematics (CALDAM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10156))

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Abstract

In this article, we consider a collection of geometric problems involving points colored by two colors (red and blue), referred to as bichromatic problems. The motivation behind studying these problems is two fold; (i) these problems appear naturally and frequently in the fields like Machine learning, Data mining, and so on, and (ii) we are interested in extending the algorithms and techniques for single point set (monochromatic) problems to bichromatic case. For all the problems considered in this paper, we design low polynomial time exact algorithms. These algorithms are based on novel techniques which might be of independent interest.

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Notes

  1. 1.

    The red rectangle may contain blue points on its boundary.

References

  1. Aggarwal, A., Suri, S.: Fast algorithms for computing the largest empty rectangle. In: SoCG, Waterloo, Canada, pp. 278–290 (1987)

    Google Scholar 

  2. Arkin, E.M., Banik, A., Carmi, P., Citovsky, G., Katz, M.J., Mitchell, J.S.B., Simakov, M.: Conflict-free covering. In: CCCG, Kingston, Ontario, Canada, 10–12 August 2015

    Google Scholar 

  3. Arkin, E.M., Banik, A., Carmi, P., Citovsky, G., Katz, M.J., Mitchell, J.S.B., Simakov, M.: Choice is hard. In: Elbassioni, K., Makino, K. (eds.) ISAAC 2015. LNCS, vol. 9472, pp. 318–328. Springer, Heidelberg (2015). doi:10.1007/978-3-662-48971-0_28

    Chapter  Google Scholar 

  4. Arkin, E.M., Díaz-Báñez, J.M., Hurtado, F., Kumar, P., Mitchell, J.S.B., Palop, B., Pérez-Lantero, P., Saumell, M., Silveira, R.I.: Bichromatic 2-center of pairs of points. Comput. Geom. 48(2), 94–107 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Armaselu, B., Daescu, O.: Maximum area rectangle separating red and blue points. In: CCCG 2016, British Columbia, Canada, 3–5 August 2016, pp. 244–251 (2016)

    Google Scholar 

  6. Aronov, B., Har-Peled, S.: On approximating the depth and related problems. SIAM J. Comput. 38(3), 899–921 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Backer, J., Keil, J.M.: The mono- and bichromatic empty rectangle and square problems in all dimensions. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 14–25. Springer, Heidelberg (2010). doi:10.1007/978-3-642-12200-2_3

    Chapter  Google Scholar 

  8. Backer, J., Mark Keil, J.: The bichromatic square and rectangle problems. Technical report 2009–01, University of Saskatchewan (2009)

    Google Scholar 

  9. Bar-Noy, A., Cheilaris, P., Smorodinsky, S.: Deterministic conflict-free coloring for intervals: from offline to online. ACM Trans. Algorithms 4(4), 44 (2008)

    Article  MathSciNet  Google Scholar 

  10. Biniaz, A., Bose, P., Maheshwari, A., Smid, M.: Plane bichromatic trees of low degree. In: Mäkinen, V., Puglisi, S.J., Salmela, L. (eds.) IWOCA 2016. LNCS, vol. 9843, pp. 68–80. Springer, Heidelberg (2016). doi:10.1007/978-3-319-44543-4_6

    Chapter  Google Scholar 

  11. Biniaz, A., Maheshwari, A., Nandy, S.C., Smid, M.: An optimal algorithm for plane matchings in multipartite geometric graphs. In: Dehne, F., Sack, J.-R., Stege, U. (eds.) WADS 2015. LNCS, vol. 9214, pp. 66–78. Springer, Heidelberg (2015). doi:10.1007/978-3-319-21840-3_6

    Chapter  Google Scholar 

  12. Bitner, S., Cheung, Y.K., Daescu, O.: Minimum separating circle for bichromatic points in the plane. In: ISVD 2010, Quebec, Canada, June 28–30, 2010, pp. 50–55 (2010)

    Google Scholar 

  13. Chan, T.M.: Output-sensitive results on convex hulls, extreme points, and related problems. In: SOCG, Vancouver, B.C., Canada, 5–12 June 1995, pp. 10–19 (1995)

    Google Scholar 

  14. Chaudhuri, J., Nandy, S.C., Das, S.: Largest empty rectangle among a point set. J. Algorithms 46(1), 54–78 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chazelle, B., (Scot) Drysdale III, R.L., Lee, D.T.: Computing the largest empty rectangle. SIAM J. Comput. 15(1), 300–315 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cheilaris, P., Gargano, L., Rescigno, A.A., Smorodinsky, S.: Strong conflict-free coloring for intervals. Algorithmica 70(4), 732–749 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chen, K., Fiat, A., Kaplan, H., Levy, M., Matousek, J., Mossel, E., Pach, J., Sharir, M., Smorodinsky, S., Wagner, U., Welzl, E.: Online conflict-free coloring for intervals. SIAM J. Comput. 36(5), 1342–1359 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cortés, C., Díaz-Báñez, J.M., Pérez-Lantero, P., Seara, C., Urrutia, J., Ventura, I.: Bichromatic separability with two boxes: a general approach. J. Algorithms 64(2–3), 79–88 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cristianini, N., Shawe-Taylor, J.: An Introduction to Support Vector Machines: And Other Kernel-based Learning Methods. Cambridge University Press, New York (2000)

    Book  MATH  Google Scholar 

  20. Dey, T.K.: Improved bounds on planar k-sets and k-levels. In: FOCS, Miami Beach, Florida, USA, 19–22 October 1997, pp. 156–161 (1997)

    Google Scholar 

  21. Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification. Wiley, New York (2000)

    MATH  Google Scholar 

  22. Eckstein, J., Hammer, P.L., Liu, Y., Nediak, M., Simeone, B.: The maximum box problem and its application to data analysis. Comp. Opt. Appl. 23(3), 285–298 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Even, G., Lotker, Z., Ron, D., Smorodinsky, S.: Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks. SIAM J. Comput. 33(1), 94–136 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Goswami, P.P., Das, S., Nandy, S.C.: Triangular range counting query in 2d and its application in finding k nearest neighbors of a line segment. Comput. Geom. 29(3), 163–175 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kaneko, A., Kano, M.: Discrete geometry on red and blue points in the plane a survey. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete and Computational Geometry, Algorithms and Combinatorics, vol. 25, pp. 551–570. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  26. Katz, M.J., Lev-Tov, N., Morgenstern, G.: Conflict-free coloring of points on a line with respect to a set of intervals. Comput. Geom. 45(9), 508–514 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu, Y., Nediak, M.: Planar case of the maximum box and related problems. In: CCCG 2003, Halifax, Canada, 11–13 August 2003, pp. 14–18 (2003)

    Google Scholar 

  28. Naamad, A., Lee, D.T., Hsu, W.-L.: On the maximum empty rectangle problem. Discret. Appl. Math. 8(3), 267–277 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  29. Orlowski, M.: A new algorithm for the largest empty rectangle problem. Algorithmica 5(1), 65–73 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank an anonymous reviewer of an earlier version of this paper for suggestions that has helped us improve the running time of the algorithm for MaxCol.

Author information

Authors and Affiliations

  1. Computer Science, University of Iowa, Iowa City, USA

    Sayan Bandyapadhyay

  2. Department of Computer Science and Engineering, Indian Institute of Technology, Jodhpur, India

    Aritra Banik

Authors
  1. Sayan Bandyapadhyay
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  2. Aritra Banik
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Corresponding author

Correspondence to Aritra Banik .

Editor information

Editors and Affiliations

  1. University of Lethbridge, Lethbridge, Alberta, Canada

    Daya Gaur

  2. Indian Institute of Technology Madras, Chennai, India

    N.S. Narayanaswamy

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Bandyapadhyay, S., Banik, A. (2017). Polynomial Time Algorithms for Bichromatic Problems. In: Gaur, D., Narayanaswamy, N. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2017. Lecture Notes in Computer Science(), vol 10156. Springer, Cham. https://doi.org/10.1007/978-3-319-53007-9_2

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  • DOI: https://doi.org/10.1007/978-3-319-53007-9_2

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  • Online ISBN: 978-3-319-53007-9

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