The minimum enclosing ball problem means to construct a ball of the minimum radius enclosing a given set of balls in S. We propose the concept of the diameter of a set of balls and give an approximation algorithm solve the diameter. We develop the 1+e approximation algorithm using core-sets. The time complexity of this algorithm is O(nd/e+d2/e3/2(1/e+d)log(1/e)). We prove the existence of the core-sets of size O(1/e) are unrelated to n and d.

The paper proposes the k-center clustering problem of high-dimensional space balls.The problem means as for the set B built by the multiple balls in a high-dimensional space,k balls are built to cover all the known balls in B and make the biggest radius of the k balls the smallest.We selectively select some balls from B to build sets,which is called the core set of B,and for a given e,we use the core set to propose the polynomial time 1+e approximation algorithm with ball number n and dimension d based on the k-center clustering problem of high-dimensional space balls.And the number of balls in S is O(1/e2),which is not related to the number of balls in B and the dimension of the space.