Finding Optimal Bipartitions of Points and Polygons

We give eecient algorithms to compute an optimal bipartition of a set of points or a set of simple polygons in the plane. In particular, we give an O(n 2) algorithm for partitioning a set of n points into two subsets in order to minimize the sum of the perimeters of the convex hulls. We also examine the problems of minimizing the maximum of the perimeters, or the sum/maximum of the areas, for the case in which the partitioning is a line partitioning, induced by some line, and we examine related problems for the bipartitioning of polygons. 1 Introduction In the \bipartition problem", we are interested in partitioning a set S of n points into two subsets (S 1 and S 2) in such a way as to optimize some function of the \sizes" ((S i)) of the two subsets. Avis ((2]) gave an O(n 2 log n) time algorithm to nd a bipartition that minimizes the maximum of the diameters of the sets S 1 and S 2. Asano, Bhattacharya, Keil and Yao ((1]) improved the bound on the time complexity of this problem, obtaining an optimal O(n log n) algorithm. Monma and Suri ((13]) gave an O(n 2) time algorithm for minimizing the sum of diameters. Recently, Hershberger and Suri ((8]) have considered the problem in which the measure of \size" (S i) is (a). the diameter, (b). the area, perimeter, or diagonal of the smallest enclosing axes-parallel rectangle, or (c). the side length of the smallest enclosing axes-parallel square. They provide O(n logn) time algorithms to nd a bipartition that satisses (S i) i (i = 1; 2) for two given numbers 1 and 2. Here, we consider the version of the bipartition problem in which (S i) is the perimeter or area of the convex hull, conv(S i), and we desire a partition that minimizes the sum (S 1) + (S 2) or the maximum maxf(S 1); (S 2)g. In the case of minimizing the sum of the perimeters, we obtain an O(n 2) time algorithm to nd an optimal bipartition. For the other three versions of the problem (minimizing the maximum of the perimeters, or minimizing the sum/maximum of the areas), our algorithm nds an optimal line partitioning (a bipartition that is induced by a line). We also consider the generalization in which S is a set of disjoint polygons. Table 1 summarizes the problems …