Abstract
We consider a problem of partitioning a finite set of points in Euclidean space into clusters so as to minimize the sum over all clusters of the intracluster sums of the squared distances between clusters elements and their centers. The centers of one part of the clusters are given as an input, while the centers of the other part of the clusters are defined as centroids (geometrical centers). It is known that in the general case this problem is strongly NP-hard. We prove constructively that the one-dimensional case of this problem is solvable in polynomial time. This result is based, first, on the proved properties of the problem optimal solution and, second, on the justified dynamic programming scheme.
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Acknowledgments
The study was supported by the Russian Foundation for Basic Research, projects 19-01-00308 and 18-31-00398, by the Russian Academy of Science (the Program of basic research), project 0314-2019-0015, and by the Russian Ministry of Science and Education under the 5–100 Excellence Programme.
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Kel’manov, A., Khandeev, V. (2019). The Problem K-Means and Given J-Centers: Polynomial Solvability in One Dimension. In: Bykadorov, I., Strusevich, V., Tchemisova, T. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2019. Communications in Computer and Information Science, vol 1090. Springer, Cham. https://doi.org/10.1007/978-3-030-33394-2_16
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DOI: https://doi.org/10.1007/978-3-030-33394-2_16
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