# On separable and rectangular clusterings

## Abstract

For a finite set of points S in the Euclidean plane we introduce the so called C-s-clustering problem which can be stated as: Partition S into C subsets S_{i} such that S_{i} is separable from S/S_{i} by a line and |S_{i}|=k_{i}, where k_{i} are given numbers. For a function f which maps C subsets S_{i} of S into ℝ we present an algorithm which finds an, with respect to f, optimal C-s-clustering in O(Cn^{3/2}log2n+P_{C}(Cn^{3/2}U_{f}(n)+P_{f}(n))) steps. (where ^{P}f^{(n)} resp. U_{f}(n) are the time to calculate resp. to update f, if the arguments are slightly changed and p_{C} is the number of, for the algorithm distinct, orderings of the k_{i}. These orderings are also a part of the input of the algorithm.) Then an O(n^{C−1}logn) solution for the C-r-clustering problem of finding all sets of C (C≤3) axis-parallel rectangles R_{i} such that |R_{i}∩S|=k_{i} and R_{i} ∩ R_{j} ∩ S=ø for i ≠ j is given. If we assume in addition that R_{i}∩R_{j}=ø for all i ≠ j we give an O(n^{C-2}) algorithm for C>2 and an O(n) algorithm for C=2.

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