The 2-center problem in three dimensions

Let P be a set of n points in R^3. The 2-center problem for P is to find two congruent balls of minimum radius whose union covers P. We present a randomized algorithm for computing a 2-center of P that runs in O(@b(r^@?)n^2log^4nloglogn) expected time; here @b(r)=1/(1-r/r"0)^3, r^@? is the radius of the 2-center balls of P, and r"0 is the radius of the smallest enclosing ball of P. The algorithm is near quadratic as long as r^@? is not too close to r"0, which is equivalent to the condition that the centers of the two covering balls be not too close to each other. This improves an earlier slightly super-cubic algorithm of Agarwal, Efrat, and Sharir (2000) [2] (at the cost of making the algorithm performance depend on the center separation of the covering balls).