Abstract
We consider a strongly NP-hard problem of partitioning a finite sequence of vectors in a Euclidean space into two clusters of given size with the criterion of minimizing the total sum of square distances from cluster elements to their centers. The center of the first cluster is subject to optimization, defined by the mean value of all vectors in this cluster. The center of the second cluster is fixed at the origin. The partition is subject to the following condition: the difference between indices of two subsequent vectors included in the first cluster is bounded from above and below by given constants. We propose an exact pseudopolynomial algorithm for the case of a problem where the dimension of the space is fixed, and components of input vectors are integer-valued.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Jain, A.K., Data Clustering: 50 Years Beyond k-Means, Patt. Recognit. Lett., 2010, vol. 31, no. 8, pp. 651–666.
MacQueen, J.B., Some Methods for Classification and Analysis of Multivariate Observations, Proc. 5 Berkeley Symp. Math. Statist. Probab., Berkeley: Univ. of California Press, 1967, vol. 1, pp. 281–297.
Rao, M., Cluster Analysis and Mathematical Programming, J. Am. Statist. Assoc., 1971, vol. 66, pp. 622–626.
Hastie, T., Tibshirani, R., and Friedman, J., The Elements of Statistical Learning: Data Mining, Inference, and Prediction, New York: Springer-Verlag, 2001.
Bishop, C.M., Pattern Recognition and Machine Learning, New York: Springer-Verlag, 2006.
Aloise, D., Deshpande, A., Hansen, P., and Popat, P., NP-hardness of Euclidean Sum-of-Squares Clustering, Machine Learning, 2009, vol. 75, no. 2, pp. 245–248.
Kel’manov, A.V., Off-line Detection of a Quasi-periodically Recurring Fragment in a Numerical Sequence, Proc. Steklov Inst. Math., 2008, no. 2, pp. S84–S92.
Kel’manov, A.V. and Pyatkin, A.V., On the Complexity of a Search for a Subset of “Similar” Vectors, Dokl. Math., 2008, vol. 78, no. 1, pp. 574–575.
Kel’manov, A.V. and Pyatkin, A.V., Complexity of Certain Problems of Searching for Subsets of Vectors and Cluster Analysis, Comput. Math. Math. Phys., 2009, vol. 49, no. 11, pp. 1966–1971.
Kel’manov, A.V. and Pyatkin, A.V., On Complexity of Some Problems of Cluster Analysis of Vector Sequences, J. Appl. Indust. Math., 2013, vol. 7, no. 3, pp. 363–369.
Kel’manov, A.V. and Jeon, B., A Posteriori Joint Detection and Discrimination of Pulses in a Quasiperiodic Pulse Train, IEEE Trans. Signal Proc., 2004, vol. 52, no. 3, pp. 645–656.
Kel’manov, A.V. and Khamidullin, S.A., Posterior Detection of a Given Number of Identical Subsequences in a Quasi-periodic Sequence, Comput. Math. Math. Phys., 2001, vol. 41, no. 5, pp. 762–774.
Carter, J.A., Agol, E., et al., Kepler-36: A Pair of Planets with Neighboring Orbits and Dissimilar Densities, Science, 2012, vol. 337, no. 6094, pp. 556–559.
Carter, J.A. and Agol, E., The Quasiperiodic Automated Transit Search Algorithm, Astrophysic. J., 2013, vol. 765, no. 2, doi:10.1088/0004-637X/765/2/132.
Kel’manov, A.V. and Khamidullin, S.A., An Approximating Polynomial Algorithm for a Sequence Partitioning Problem, J. Appl. Indust. Math., 2014, vol. 8, no. 2, pp. 236–244.
Garey, M.R., and Johnson, D.S., Computers and Intractability: A Guide to the Theory of NP-Completeness, San Francisco: Freeman, 1979.
Dolgushev, A.V. and Kel’manov, A.V., An Approximation Algorithm for Solving a Problem of Cluster Analysis, J. Appl. Indust. Math., 2011, vol. 5, no. 4, pp. 551–558.
Dolgushev, A.V., Kel’manov, A.V., and Shenmaier, V.V., A Polynomial Approximation Scheme for One Problem of Partitioning a Finite Set into Two Clusters, Tr. Inst. Mat. Mekh. UrO RAN, 2015, vol. 21, no. 3, pp. 100–109.
Kel’manov, A.V. and Khandeev, V.I., Randomized Algorithm for Two-Cluster Partition of a Set of Vectors, Comput. Math. Math. Phys., 2015, vol. 55, no. 2, pp. 330–339.
Kel’manov, A.V. and Khandeev, V.I., An Exact Pseudopolynomial Algorithm for a Problem of the Two-Cluster Partitioning of a Set of Vectors, J. Appl. Indust. Math., 2015, vol. 9, no. 4, pp. 497–502.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.V. Kel’manov, S.A. Khamidullin, V.I. Khandeev, 2017, published in Avtomatika i Telemekhanika, 2017, No. 1, pp. 80–90.
This paper was recommended for publication by A.A. Lazarev, a member of the Editorial Board
Rights and permissions
About this article
Cite this article
Kel’manov, A.V., Khamidullin, S.A. & Khandeev, V.I. Exact pseudopolynomial algorithm for one sequence partitioning problem. Autom Remote Control 78, 67–74 (2017). https://doi.org/10.1134/S0005117917010052
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117917010052