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clustering; data mining; $k$-means; Voronoi diagram
The paper gives a new interpretation and a possible optimization of the well-known $k$-means algorithm for searching for a locally optimal partition of the set $\mathcal {A}= \{a_i\in \mathbb {R}^n\colon i=1,\dots ,m\}$ which consists of $k$ disjoint nonempty subsets $\pi _1,\dots ,\pi _k$, $1\leq k\leq m$. For this purpose, a new divided $k$-means algorithm was constructed as a limit case of the known smoothed $k$-means algorithm. It is shown that the algorithm constructed in this way coincides with the $k$-means algorithm if during the iterative procedure no data points appear in the Voronoi diagram. If in the partition obtained by applying the divided $k$-means algorithm there are data points lying in the Voronoi diagram, it is shown that the obtained result can be improved further.
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