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An Exact Algorithm of Searching for the Largest Cluster in an Integer-Valued Problem of 2-Partitioning a Sequence

  • Mathematical Method in Pattern Recognition
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Abstract

We analyze mathematical aspects of one of the fundamental data analysis problems consisting in the search (selection) for the subset with the largest number of similar elements among a collection of objects. In particular, the problem appears in connection with the analysis of data in the form of time series (discrete signals). One of the problems in modeling this challenge is considered, namely, the problem of finding the cluster of the largest size (cardinality) in a 2-partition of a finite sequence of points in Euclidean space into two clusters (subsequences) under two constraints. The first constraint is on the choice of the indices of elements included in the clusters. This constraint simulates the set of time-admissible configurations of similar elements in the observed discrete signal. The second constraint is imposed on the value of the quadratic clustering function. This constraint simulates the level of intracluster proximity of objects. The clustering function under the second constraint is the sum (over both clusters) of the intracluster sums of squared distances between the cluster elements and its center. The center of one of the clusters is unknown and defined as the centroid (the arithmetic mean over all elements of this cluster). The center of the other cluster is the origin. Under the first constraint, the difference between any two subsequent indices of elements contained in a cluster with an unknown center is bounded above and below by some constants. It is established in the paper that the optimization problem under consideration, which models one of the simplest significant problems of data analysis, is strongly NP-hard. We propose an exact algorithm for the case of a problem with integer coordinates of its input points. If the dimension of the space is bounded by a constant, then the algorithm is pseudopolynomial.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, pr. Akad. Koptyuga 4, Novosibirsk, Russia

    A. V. Kel’manov, S. A. Khamidullin, V. I. Khandeev & A. V. Pyatkin

  2. Novosibirsk State University, ul. Pirogova 2, Novosibirsk, Russia

    A. V. Kel’manov, V. I. Khandeev & A. V. Pyatkin

Authors
  1. A. V. Kel’manov
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  2. S. A. Khamidullin
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  3. V. I. Khandeev
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  4. A. V. Pyatkin
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Correspondence to A. V. Kel’manov.

Additional information

Sergei Asgadullovich Khamidullin. Born 1952. Graduated from Novosibirsk State University in 1974 with specialty in Physics and Applied Mathematics. Received Candidate’s Degree in Engineering Cybernetics and Information Theory in 1997. Currently with the Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Senior research scientist in Data Analysis Laboratory. Scientific interests: mathematical methods and computer technologies for data analysis and pattern recognition, discrete optimization. Author of more than 70 publications.

Aleksandr Vasil’evich Kel’manov. Born 1952. Graduated from Izhevsk State Technical University in 1974 with specialty in Applied Mathematics. Received Candidate’s Degree in Engineering Cybernetics and Information Theory in 1980 and Doctor of Sciences degree in Physics and Mathematics in 1994. Currently with the Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Head of Data Analysis Laboratory. Scientific interests: data analysis, data mining, pattern recognition, clusterization, discrete optimization, NP-hard problems, efficient algorithms with performance guarantees. Author of more than 200 publications.

Vladimir Il’ich Khandeev. Born 1991. Graduated from Novosibirsk State University in 2014 with specialty in Applied Mathematics and Computer Science. Received Candidate’s Degree in Physics and Mathematics in 2017. Currently with the Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Research scientist in Data Analysis Laboratory. Scientific interests: data mining, pattern recognition, clusterization, discrete optimization, NP-hard problems, efficient algorithms with performance guarantees. Author of 12 publications.

Artem Valer’evich Pyatkin. Born 1973. Graduated from Novosibirsk State University in 1996 with specialty in Mathematics. Received Candidate’s Degree in Physics and Mathematics in 1999 and Doctor of Sciences degree in Physics and Mathematics in 2009. Currently with the Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, head of Discrete Optimization in Operations Research Laboratory. Scientific interests: graph Theory, data analysis, pattern recognition, clustering, discrete optimization, NP-hard problems. Author of more than 60 publications.

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Kel’manov, A.V., Khamidullin, S.A., Khandeev, V.I. et al. An Exact Algorithm of Searching for the Largest Cluster in an Integer-Valued Problem of 2-Partitioning a Sequence. Pattern Recognit. Image Anal. 28, 703–711 (2018). https://doi.org/10.1134/S105466181804017X

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