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Smallest Square Covering k Points for Large Value of k

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Information Systems Design and Intelligent Applications

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 339))

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Abstract

Given a set of n planar points, a positive integer \( k\left( {1 \le k \le n} \right) \) and a geometric object, the objective of the k-cover problem is to find a smallest object such that it covers at least k input points. A deterministic algorithm is proposed to solve the k-cover problem when the object is an axis-parallel square and \( k > \frac{n}{2} \). The time and space complexities of the algorithm are \( O(n + (n - k) \log ^{2} (n - k)) \) and \( O(n) \) respectively.

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References

  1. Megiddo, N.: Linear-time algorithms for linear programming in ℝ3 and related problems. SIAM J. Comput. 12, 759–776 (1995)

    Article  MathSciNet  Google Scholar 

  2. Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, Berlin (1990)

    Google Scholar 

  3. Toussaint, G.T.: Solving geometric problems with the rotating calipers. In: Proceedings of IEEE MELECON (1983)

    Google Scholar 

  4. Katz, M.J., Kedem, K., Segal, M.: Discrete rectilinear 2-center problems. Comput. Geom. 15, 203–214 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. O’Rourke, J., Aggarwal, A., Maddila, S., Baldwin, M.: An optimal algorithm for finding minimal enclosing triangles. J. Algorithms 7, 258–269 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  6. Agarwal, P.K., Sharir, M., Toledo, S.: Applications of parametric searching in geometric optimization. J. Algorithms 17, 292–318 (1994)

    Article  MathSciNet  Google Scholar 

  7. Mukherjee, J., Mahapatra, P.R.S., Karmakar, A., Das, S.: Minimum-width rectangular annulus. Theoret. Comput. Sci. 508, 74–80 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gluchshenko, O., Hamacher, H.W., Tamir, A.: An optimal O(n log n) algorithm for finding an enclosing planar rectilinear annulus of minimum width. Oper. Res. Lett. 37, 168–170 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Datta, A., Lenhof, H.E., Schwarz, C., Smid, M.: Static and dynamic algorithms for k-point clustering problems. J. Algorithms 19, 474–503 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Smid, M.: Finding k points with a smallest enclosing square. Report MPI-92-152, Max-Plank-Institute fur Informatik, Saarbriucken (1992)

    Google Scholar 

  11. Aggarwal, A., Imai, H., Katoh, N., Suri, S.: Finding k points with minimum diameter and related problems. J. Algorithms 12, 38–56 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ahn, H.K., Bae, S.W., Demaine, E.D., Demaine, M.L., Kim, S.S., Korman, M., Reinbacher, I., Son, W.: Covering points by disjoint boxes with outliers. Comput. Geom. Theory Appl. 44, 178–190 (2011)

    Google Scholar 

  13. Segal, M., Kedem, K.: Enclosing k points in the smallest axis parallel rectangle. Inf. Process. Lett. 65, 95–99 (1998)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Chan, T.M.: Geometric applications of a randomized optimization technique. Discrete Comput. Geom. 22, 547–567 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. Das, S., Goswami, P.P., Nandy, S.C.: Smallest k-point enclosing rectangle and square of arbitrary orientation. Inf. Process. Lett. 94, 259–266 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Mahapatra, P.R.S., Karmakar, A., Das, S., Goswami, P.P.: k-enclosing axis-parallel square. LNCS 6784, pp. 84–93. Springer, New York (2011)

    Google Scholar 

  18. Mahapatra, P.R.S., Goswami, P.P., Das, S.: Covering points by isothetic unit squares. In: Proceeding of the 19th Canadian Conference on Computational Geometry (CCCG 2007), pp. 169–172 (2007)

    Google Scholar 

  19. Mahapatra, P.R.S., Goswami, P.P., Das, S.: Maximal covering by two isothetic unit squares. In: Proceeding of the 20th Canadian Conference on Computational Geometry (CCCG 2008), pp. 103–106 (2008)

    Google Scholar 

  20. Atta, S., Mahapatra, P.R.S.: Genetic Algorithm Based Approach for Serving Maximum Number of Customers Using Limited Resources, vol. 10, pp. 492–497. Elsevier, Amsterdam (2013)

    Google Scholar 

  21. Atta, S., Mahapatra, P.R.S.: Genetic Algorithm Based Approaches to Install Different Types of Facilities, vol. 248, pp. 195–203, AISC (Springer) (2014)

    Google Scholar 

  22. Andrews, H.C.: Introduction to Mathematical Techniques in Pattern Recognition. Wiley-Intersciences, New York (1972)

    MATH  Google Scholar 

  23. Asano, T., Bhattacharya, B., Keil, M., Yao, F.: Clustering algorithms based on maximum and minimum spanning trees. In: Proceeding of the 4th Annual Symposium on Computational Geometry, pp. 252–257 (1988)

    Google Scholar 

  24. Hartigan, J.A.: Clustering Algorithms. Wiley, New York (1975)

    MATH  Google Scholar 

  25. Cook, S.: The complexity of theorem proving procedures. In: Proceeding of the 3rd Annual ACM Symposium on Theory of Computing, pp. 151–158 (1971)

    Google Scholar 

  26. Frederickson, G.N., Johnson, D.B.: Generalized selection and ranking: sorted matrices. SIAM J. Comput. 13, 14–30 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  27. Drezner, Z., Hamacher, H.: Facility Location: Applications and Theory. Springer, Berlin (2002)

    Book  Google Scholar 

  28. Glozman, A., Kedem, K., Shpitalnik, G.: On some geometric selection and optimization problems via sorted matrices. Comput. Geom. Theory Appl. 11, 17–28 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, University of Kalyani, Kalyani, India

    Priya Ranjan Sinha Mahapatra

Authors
  1. Priya Ranjan Sinha Mahapatra
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Correspondence to Priya Ranjan Sinha Mahapatra .

Editor information

Editors and Affiliations

  1. University of Kalyani, Kalyani, West Bengal, India

    J. K. Mandal

  2. Department of Computer Science and Engineering, Anil Neerukonda Institute of Technology and Sciences, Vishakapatnam, India

    Suresh Chandra Satapathy

  3. Dean, Faculty of Engineering, Technology, University of Kalyani, Kalyani, West Bengal, India

    Manas Kumar Sanyal

  4. Engineering and Technological Studies, University of Kalyani, Kalyani, West Bengal, India

    Partha Pratim Sarkar

  5. Department Computer Science & Engineering, University of Kalyani, Kalyani, India

    Anirban Mukhopadhyay

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Mahapatra, P.R.S. (2015). Smallest Square Covering k Points for Large Value of k . In: Mandal, J., Satapathy, S., Kumar Sanyal, M., Sarkar, P., Mukhopadhyay, A. (eds) Information Systems Design and Intelligent Applications. Advances in Intelligent Systems and Computing, vol 339. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2250-7_33

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  • DOI: https://doi.org/10.1007/978-81-322-2250-7_33

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  • Publisher Name: Springer, New Delhi

  • Print ISBN: 978-81-322-2249-1

  • Online ISBN: 978-81-322-2250-7

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