Finding Tailored Partitions

Abstract We consider the following problem: given a planar set of points S , a measure μ acting on S , and a pair of values μ 1 and μ 2 , does there exist a bipartition S = S 1 ∪ S 2 satisfying μ ( S i ) ≤ μ i for i = 1,2? We present algorithms for several natural measures, including the diameter ( set measure ), the area, perimeter, or diagonal of the smallest enclosing axes-parallel rectangle ( rectangular measure ), the side length of the smallest enclosing axes-parallel square ( square measure ), and the radius of the smallest enclosing circle ( circular measure ). The algorithms run in time O ( n log n ) for the set, rectangle, and square measures, and in time O ( n 2 log n ) for the circular measure. The problem of partitioning S into an arbitrary number k of subsets is known to be NP-complete for many of these measures.