The 2-center problem in three dimensions

Authors:
Pankaj K. Agarwal;Rinat Ben Avraham;Micha Sharir
Affiliations:
Duke University, Durham, NC, USA;Tel Aviv University, Tel Aviv, Israel;Tel Aviv University, Tel Aviv, Israel
Venue:
Proceedings of the twenty-sixth annual symposium on Computational geometry
Year:
2010

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Abstract

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.