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

This research is supported by Louisiana Board of Regents fund under contract number LEQSF(2004-07)-RD-A-35, and in part by NSF Grant CNS-0521585.