On the complexity of approximation streaming algorithms for the k-center problem

We study approximation streaming algorithms for the kcenter problem in the fixed dimensional Euclidean space. Given an integer k >= 1 and a set S of n points in the d-dimensional Euclidean space; the k-center problem is to cover those points in S with k congruent balls with the smallest possible radius. For any epsilon > 0, we devise an O( k/epsilon(d))-space (1 + epsilon)-approximation streaming algorithm for the kcenter problem; and prove that the updating time of the algorithm is (k1-1/d/epsilon(d)) O( k/epsilon(d) log k) + 2 (O) (k(1-1/d/)epsilon(d)). On the other hand, we prove that any (1 + epsilon)approximation streaming algorithm for the k-center problem must use Omega(k/epsilon(d-1)/2)-bits memory. Our approximation streaming algorithm is obtained by first designing an off-line (1 + e)-approximation algorithm with O (n log k) + 2(O)(k(1-1/d)/epsilon(d)) time complexity, and then applying this off-line algorithm repeatedly to a sketch of the input data stream. If E is fixed, our off-line algorithm improves the best-known off-line approximation algorithm for the k-center problem by Agarwal and Procopiuc [1] that has O(n log k) + ( k/epsilon)O(k(1-1/d)) time complexity. Our approximate streaming algorithm for the k-center problem is different from another streaming algorithm by Har-Peled [161; which maintains a core set of size O( E), but does not provide approximate solution for small epsilon > 0.