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Computational Mathematics and Mathematical Physics

, Volume 58, Issue 12, pp 2078–2085 | Cite as

A Randomized Algorithm for a Sequence 2-Clustering Problem

  • A. V. Kel’manovEmail author
  • S. A. KhamidullinEmail author
  • V. I. KhandeevEmail author
Article
  • 7 Downloads

Abstract

We consider a strongly NP-hard problem of partitioning a finite Euclidean sequence into two clusters of given cardinalities minimizing the sum over both clusters of intracluster sums of squared distances from clusters elements to their centers. The center of one cluster is unknown and is defined as the mean value of all points in the cluster. The center of the other cluster is the origin. Additionally, the difference between the indices of two consequent points from the first cluster is bounded from below and above by some constants. A randomized algorithm that finds an approximation solution of the problem in polynomial time for given values of the relative error and failure probability and for an established parameter value is proposed. The conditions are established under which the algorithm is polynomial and asymptotically exact.

Keywords:

partitioning sequence Euclidean space minimum sum-of-squared distances NP-hardness randomized algorithm asymptotic accuracy 

Notes

ACKNOWLEDGMENTS

This work was supported by the Russian Science Foundation, project no. 16-11-10041.

REFERENCES

  1. 1.
    A. V. Kel’manov and S. A. Khamidullin, “Posterior detection of a given number of identical subsequences in a quasi-periodic sequence,” Comput. Math. Math. Phys. 41 (5), 762–774 (2001).MathSciNetzbMATHGoogle Scholar
  2. 2.
    A. V. Kel’manov and B. Jeon, “A posteriori joint detection and discrimination of pulses in a quasiperiodic pulse train,” IEEE Trans. Signal Process. 52 (3), 645–656 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J. A. Carter, E. Agol, et al., “Kepler-36: A pair of planets with neighboring orbits and dissimilar densities,” Science 337, 556–559 (2012).CrossRefGoogle Scholar
  4. 4.
    M. C. Bishop, Pattern Recognition and Machine Learning (Springer Science + Business Media, New York, 2006).Google Scholar
  5. 5.
    A. V. Kel’manov and S. A. Khamidullin, “An approximation polynomial-time algorithm for a sequence bi-clustering problem,” Comput. Math. Math. Phys. 55 (6), 1068–1076 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Tak-chung Fu, “A review on time series data mining,” Eng. Appl. Artificial Intelligence 24 (1), 164–181 (2011).CrossRefGoogle Scholar
  7. 7.
    E. Kh. Gimadi, A. V. Kel’manov, M. A. Kel’manova, and S. A. Khamidullin, “A posteriori detection of a quasiperiodic fragment with a given number of repetitions in a numerical sequence,” Sib. Zh. Ind. Mat. 9 (1), 55–74 (2006).MathSciNetzbMATHGoogle Scholar
  8. 8.
    E. Kh. Gimadi, A. V. Kel’manov, M. A. Kel’manova, and S. A. Khamidullin, “A posteriori detecting a quasiperiodic fragment in a numerical sequence,” Pattern Recogn. Image Anal. 18 (1), 30–42 (2008).CrossRefzbMATHGoogle Scholar
  9. 9.
    A. V. Kel’manov, “Off-line detection of a quasi-periodically recurring fragment in a numerical sequence,” Proc. Steklov Inst. Math. 263, Suppl. 2, 84–92 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    A. V. Kel’manov and A. V. Pyatkin, “Complexity of certain problems of searching for subsets of vectors and cluster analysis,” Comput. Math. Math. Phys. 49 (11), 1966–1971 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A. V. Kel’manov and A. V. Pyatkin, “On complexity of some problems of cluster analysis of vector sequences,” J. Appl. Ind. Math. 7 (3), 363–369 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. V. Kel’manov and S. A. Khamidullin, “An approximating polynomial algorithm for a sequence partitioning problem,” J. Appl. Ind. Math. 8 (2), 236–244 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    R. Motwani and P. Raghavan, Randomized Algorithms (Cambridge Univ. Press, New York, 1995).CrossRefzbMATHGoogle Scholar
  14. 14.
    A. V. Kel’manov, S. A. Khamidullin, and V. I. Khandeev, “Exact pseudopolynomial algorithm for a sequence bi-clustering problem,” Proceedings of 15th All-Russia Conference on Mathematical Programming and Applications (Yekaterinburg, 2015), pp. 139–140.Google Scholar
  15. 15.
    A. V. Kel’manov, S. A. Khamidullin, and V. I. Khandeev, “Fully polynomial-time approximation scheme for a sequence 2-clustering problem,” Diskret. Anal. Issled. Oper. 23 (2), 21–40 (2016).zbMATHGoogle Scholar
  16. 16.
    A. V. Kel’manov and V. I. Khandeev, “Fully polynomial-time approximation scheme for a special case of a quadratic Euclidean 2-clustering problem,” Comput. Math. Math. Phys. 56 (2), 334–341 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979).zbMATHGoogle Scholar
  18. 18.
    A. V. Kel’manov and V. I. Khandeev, “A randomized algorithm for two-cluster partition of a set of vectors,” Comput. Math. Math. Phys. 55 (2), 330–339 (2015).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of SciencesNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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