On the Complexity of Approximation Streaming Algorithms for the k-Center Problem

Abstract

We study approximation streaming algorithms for the k- center 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 Ɛ > 0, we devise an O( k/Ɛd)-space (1 + Ɛ)-approximation streaming algorithm for the k- center problem, and prove that the updating time of the algorithm is O(k/Ɛd log k) + 2O(k1-1/d/Ɛd). On the other hand, we prove that any (1 + Ɛ)- approximation streaming algorithm for the k-center problem must use Ω(k/Ɛ(d-1)/2)-bits memory. Our approximation streaming algorithm is obtained by first designing an off-line (1+Ɛ)-approximation algorithm with O(n log k) + 2O(k1-1/d/Ɛd) time complexity, and then applying this off-line algorithm repeatedly to a sketch of the input data stream. If Ɛ 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/Ɛ)O(k1-1/d) time complexity. Our approximate streaming algorithm for the k-center problem is different from another streaming algorithm by Har-Peled [16], which maintains a core set of size O(k/Ɛd), but does not provide approximate solution for small Ɛ > 0.