A Randomized Algorithm for 2-Partition of a Sequence

  • Alexander Kel’manovEmail author
  • Sergey Khamidullin
  • Vladimir KhandeevEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10716)


In the paper we consider one strongly NP-hard problem of partitioning a finite Euclidean sequence into two clusters minimizing the sum over both clusters of intracluster sum of squared distances from clusters elements to their centers. The cardinalities of clusters are assumed to be given. The center of the first cluster is unknown and is defined as the mean value of all points in the cluster. The center of the second one is the origin. Additionally, the difference between the indexes of two consequent points from the first cluster is bounded from below and above by some constants. A randomized algorithm for the problem is proposed. For an established parameter value, given a relative error \(\varepsilon > 0\) and fixed \(\gamma \in (0, 1)\), this algorithm allows to find a \((1 + \varepsilon )\)-approximate solution of the problem with a probability of at least \(1 - \gamma \) in polynomial time. The conditions are established under which the algorithm is polynomial and asymptotically exact.


Partitioning Sequence Euclidean space Minimum sum-of-squared distances NP-hardness Randomized algorithm Asymptotic accuracy 



This work was supported by the Russian Foundation for Basic Research, project nos. 15-01-00462, 16-31-00186, and 16-07-00168.


  1. 1.
    Fu, T.: A review on time series data mining. Eng. Appl. Artif. Intell. 24(1), 164–181 (2011)CrossRefGoogle Scholar
  2. 2.
    Kuenzer, C., Dech, S., Wagner, W.: Remote Sensing Time Series. Remote Sensing and Digital Image Processing, vol. 22. Springer, Switzerland (2015). Google Scholar
  3. 3.
    Liao, T.W.: Clustering of time series data – a survey. Pattern Recogn. 38(11), 1857–1874 (2005)CrossRefzbMATHGoogle Scholar
  4. 4.
    Aggarwal, C.C.: Data Mining: The Textbook. Springer, Switzerland (2015). CrossRefzbMATHGoogle Scholar
  5. 5.
    Carter, J.A., Agol, E., et al.: Kepler-36: a pair of planets with neighboring orbits and dissimilar densities. Science 337(6094), 556–559 (2012)CrossRefGoogle Scholar
  6. 6.
    Kel’manov, A.V., Jeon, B.: A posteriori joint detection and discrimination of pulses in a quasiperiodic pulse train. IEEE Trans. Sig. Process. 52(3), 645–656 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Rajeev, M., Prabhakar, R.: Randomized Algorithms. Cambridge University Press, New York (1995)zbMATHGoogle Scholar
  8. 8.
    Kel’manov, A.V., Pyatkin, A.V.: On complexity of some problems of cluster analysis of vector sequences. J. Appl. Ind. Math. 7(3), 363–369 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kel’manov, A.V., Khamidullin, S.A.: An approximating polynomial algorithm for a sequence partitioning problem. J. Appl. Ind. Math. 8(2), 236–244 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kel’manov, A.V., Khamidullin, S.A., Khandeev, V.I.: Exact pseudopolynomial algorithm for one sequence partitioning problem. Autom. Remote Control 78(1), 67–74 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kel’manov, A.V., Khamidullin, S.A., Khandeev, V.I.: A fully polynomial-time approximation scheme for a sequence 2-cluster partitioning problem. J. Appl. Ind. Math. 10(2), 209–219 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gimadi, E.K., Kel’manov, A.V., Kel’manova, M.A., Khamidullin, S.A.: A posteriori detection of a quasiperiodic fragment with a given number of repetitions in a numerical sequence (in Russian). Sibirsk. Zh. Ind. Mat. 9(1), 55–74 (2006)zbMATHGoogle Scholar
  13. 13.
    Gimadi, E.K., Kel’manov, A.V., Kel’manova, M.A., Khamidullin, S.A.: A posteriori detecting a quasiperiodic fragment in a numerical sequence. Pattern Recogn. Image Anal. 18(1), 30–42 (2008)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kel’manov, A.V.: Off-line detection of a quasi-periodically recurring fragment in a numerical sequence. Proc. Steklov Inst. Math. 263(S2), 84–92 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kel’manov, A.V., Pyatkin, A.V.: Complexity of certain problems of searching for subsets of vectors and cluster analysis. Comput. Math. Math. Phys. 49(11), 1966–1971 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kel’manov, A.V., Khandeev, V.I.: 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.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  18. 18.
    Kel’manov, A.V., Khandeev, V.I.: A randomized algorithm for two-cluster partition of a set of vectors. Comput. Math. Math. Phys. 55(2), 330–339 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kel’manov, A.V., Khamidullin, S.A.: 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

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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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