An Efficient Algorithm for the Proximity Connected Two Center Problem

Abstract

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 $$\delta \ge 0$$ and we have the additional constraint that the distance between the centers of the disks should be at most $$\delta $$ . Note that when $$\delta =0$$ , the PCTC problem is reduced to the 1-center(minimum enclosing disk) problem and when $$\delta $$ 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.