# On separable and rectangular clusterings

• H. Heusinger
• H. Noltemeier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 333)

## 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 Si such that Si is separable from S/Si by a line and |Si|=ki, where ki are given numbers. For a function f which maps C subsets Si of S into ℝ we present an algorithm which finds an, with respect to f, optimal C-s-clustering in O(Cn3/2log2n+PC(Cn3/2Uf(n)+Pf(n))) steps. (where Pf(n) resp. Uf(n) are the time to calculate resp. to update f, if the arguments are slightly changed and pC is the number of, for the algorithm distinct, orderings of the ki. These orderings are also a part of the input of the algorithm.) Then an O(nC−1logn) solution for the C-r-clustering problem of finding all sets of C (C≤3) axis-parallel rectangles Ri such that |Ri∩S|=ki and Ri ∩ Rj ∩ S=ø for i ≠ j is given. If we assume in addition that Ri∩Rj=ø for all i ≠ j we give an O(nC-2) algorithm for C>2 and an O(n) algorithm for C=2.

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