Finding tailored partitions

We consider the following problem: given a planar set of points <italic>S</italic>, a measure <italic>μ</italic> acting on <italic>S</italic>, and a pair of values <italic>μ</italic>1 and <italic>μ</italic>2, does there exist a bipartition <italic>S</italic> = <italic>S</italic><subscrpt>1</subscrpt> U <italic>S</italic><subscrpt>2</subscrpt> satisfying <italic>μ</italic>(<italic>S<subscrpt>i</subscrpt></italic>) ≤ <italic>μ<subscrpt>i</subscrpt></italic> for <italic>i</italic> = 1,2? We present algorithms of complexity <italic>&Ogr;</italic>(<italic>n</italic> log <italic>n</italic>) for several natural measures, including the diameter (<italic>set measure</italic>), the area, perimeter or diagonal of the smallest enclosing axes-parallel rectangle (<italic>rectangular measure</italic>), and the side length of the smallest enclosing axes-parallel square (<italic>square measure</italic>). The problem of partitioning <italic>S</italic> into <italic>k</italic> subsets, where <italic>k</italic> ≥ 3, is known to be <italic>NP</italic>-complete for many of these measures.

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