Skip to main content
Log in

An Approximation Algorithm for a Problem of Partitioning a Sequence into Clusters with Constraints on Their Cardinalities

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

We consider the problem of partitioning a finite sequence of points in Euclidean space into a given number of clusters (subsequences) minimizing the sum over all clusters of intracluster sums of squared distances of elements of the clusters to their centers. It is assumed that the center of one of the desired clusters is the origin, while the centers of the other clusters are unknown and are defined as the mean values of cluster elements. Additionally, there are a few structural constraints on the elements of the sequence that enter the clusters with unknown centers: (1) the concatenation of indices of elements of these clusters is an increasing sequence, (2) the difference between two consequent indices is lower and upper bounded by prescribed constants, and (3) the total number of elements in these clusters is given as an input. It is shown that the problem is strongly NP-hard. A 2-approximation algorithm that is polynomial for a fixed number of clusters is proposed for this problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Tak-chung Fu, “A review on time series data mining,” Eng. Appl. Artificial Intell. 24 (1), 164–181 (2011).

    Article  Google Scholar 

  2. Remote Sensing Time Series: Revealing Land Surface Dynamics, Ed. by C. Kuenzer, S. Dech, and W. Wagner (Springer, New York, 2015), Ser. Remote Sensing and Digital Image Processing 22.

    Google Scholar 

  3. T. Warren Liao, “Clustering of time series data—a survey,” Pattern Recogn. 38 (11), 1857–1874 (2005).

    Article  MATH  Google Scholar 

  4. C. C. Aggarwal, Data Mining: The Textbook (Springer, New York, 2015).

    Book  MATH  Google Scholar 

  5. 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).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. V. Kel’manov and S. A. Khamidullin, “An approximating polynomial algorithm for a sequence partitioning problem,” J. Appl. Ind. Math. 8 (2), 53–66 (2014).

    MathSciNet  MATH  Google Scholar 

  7. A. V. Kel’manov and L. V. Mikhailova, “Joint detection of a given number of reference fragments in a quasiperiodic sequence and its partition into segments containing series of identical fragments,” Comp. Math. Math. Phys. 46 (1), 165–181 (2006).

    Article  MATH  Google Scholar 

  8. A. V. Kel’manov, S. A. Khamidullin, and V. I. Khandeev, “An exact pseudopolynomial algorithm for a sequence 2-cluster partitioning problem,” in Proceedings of the 15th All-Russia Conference on Mathematical Programming and Applications, Yekaterinburg, Russia, 2015 (IMM UrO RAN, Yekaterinburg, 2015), p. 139.

    Google Scholar 

  9. A. V. Kel’manov, S. A. Khamidullin, and V. I. Khandeev, “A fully polynomial-time approximation scheme for a sequence 2-cluster partitioning problem,” J. Appl. Ind. Math. 10 (2), 209–219 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  10. A. V. Kel’manov and S. M. Romanchenko, “An FPTAS for a vector subset search problem,” J. Appl. Ind. Math. 8 (3), 329–336 (2012).

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. V. Kel’manov.

Additional information

Original Russian Text © A.V. Kel’manov, L.V.Mikhailova, S.A.Khamidullin, V.I. Khandeev, 2016, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2016, Vol. 22, No. 3.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kel’manov, A.V., Mikhailova, L.V., Khamidullin, S.A. et al. An Approximation Algorithm for a Problem of Partitioning a Sequence into Clusters with Constraints on Their Cardinalities. Proc. Steklov Inst. Math. 299 (Suppl 1), 88–96 (2017). https://doi.org/10.1134/S0081543817090115

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543817090115

Keywords

Navigation