The 2-center problem in three dimensions
Pankaj K. Agarwal, Rinat Ben Avraham, Micha Sharir · 2010
Let P be a set of n points in ℜ3. The 2-center problem for P is to find two congruent balls of the 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 log8 n) expected time, and the second algorithm runs in O(n2 log8 n/(1--r*/r0)3) expected time, where r* is the radius of the 2-center 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 very close to r0, which is equivalent to the condition of the centers of the two balls in the 2-center of P not being very close to each other.