The 2-Center Problem in Three Dimensions
Abstract
Let P be a set of n points in R3. 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*/r0)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.
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