Exact algorithms for size constrained 2-clustering in the plane

We study the problem of determining an optimal bipartition { A , B } of a set X of n points in R 2 , under the size constraints | A | = k and | B | = n - k , that minimizes the dispersion of points around their centroid in A and B, both in the cases of Euclidean and Manhattan norms. Under the Euclidean norm, we show that the problem can be solved in O ( n k 3 log 2 ź n ) time by using known properties on k-sets and convex hulls; moreover, the solutions for all k = 1 , 2 , ź , ź n / 2 ź can be computed in O ( n 2 log ź n ) time. In the case of Manhattan norm, we present an algorithm working in O ( n 2 log ź n ) time, which uses an extended version of red-black trees to maintain a bipartition of a planar point set. Also in this case we provide a full version of the algorithm yielding the solutions for all size constraints k. All these procedures work in O ( n ) space and rely on separation results of the clusters of optimal solutions.