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 two randomized algorithms for computing a 2-center of P. The first algorithm runs in O(n³ log⁵ n) expected time, and the second algorithm runs in O((n² log⁵ n) /(1-r/r_0)³⁾ expected time, where r is the radius of the 2-center balls of P and r0 is the radius of the smallest enclosing ball of P. The second algorithm is faster than the first one as long as r* is not too close to r0, which is equivalent to the condition that the centers of the two covering balls be not too close to each other.