The 2-center problem in three dimensions

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. Categories and Subject Descriptors F.2.2 [Analysis of algorithms and problem complexity]: Nonnumerical algorithms and problems—Geometrical problems and computations; I.5.3 [Pattern recognition]: Clustering—algorithms General Terms Algorithms, Theory Keywords 2-center problem, facility location, geometric optimization, intersection...

• Type: Conference
• Year: 2010
• Where: COMPGEOM
• Authors: Pankaj K. Agarwal, Rinat Ben Avraham, Micha Sharir
• Keywords: 2-center Problem, Algorithms, COMPGEOM 2010, Congruent Balls