An Efficient Algorithm for the Proximity Connected Two Center Problem

Given a set P of n points in the plane, the k -center problem is to find k congruent disks of minimum possible radius such that their union covers all the points in P . The 2-center problem is a special case of the k -center problem that has been extensively studied in the recent past [ 7 , 20 , 22 ]. In this paper, we consider a generalized version of the 2-center problem called proximity connected 2-center (PCTC) problem. In this problem, we are also given a parameter δ ≥ 0 and we have the additional constraint that the distance between the centers of the disks should be at most δ . Note that when δ = 0 , the PCTC problem is reduced to the 1-center(minimum enclosing disk) problem and when δ tends to infinity, it is reduced to the 2-center problem. The PCTC problem first appeared in the context of wireless networks in 1992 [ 12 ], but obtaining a nontrivial deterministic algorithm for the problem remained open. In this paper, we resolve this open problem by providing a deterministic O ( n 2 log n ) time algorithm for the problem.