Computing Euclidean k-Center over Sliding Windows

In the Euclidean $k$-center problem in sliding window model, input points are given in a data stream and the goal is to find the $k$ smallest congruent balls whose union covers the $N$ most recent points of the stream. In this model, input points are allowed to be examined only once and the amount of space that can be used to store relative information is limited. Cohen-Addad et al.~\cite{cohen-2016} gave a $(6+\epsilon)$-approximation for the metric $k$-center problem using O($k/\epsilon \log \alpha$) points, where $\alpha$ is the ratio of the largest and smallest distance and is assumed to be known in advance. In this paper, we present a $(3+\epsilon)$-approximation algorithm for the Euclidean $1$-center problem using O($1/\epsilon \log \alpha$) points. We present an algorithm for the Euclidean $k$-center problem that maintains a coreset of size $O(k)$. Our algorithm gives a $(c+2\sqrt{3} + \epsilon)$-approximation for the Euclidean $k$-center problem using O($k/\epsilon \log \alpha$) points by using any given $c$-approximation for the coreset where $c$ is a positive real number. For example, by using the $2$-approximation~\cite{feder-greene-1988} of the coreset, our algorithm gives a $(2+2\sqrt{3} + \epsilon)$-approximation ($\approx 5.465$) using $O(k\log k)$ time. This is an improvement over the approximation factor of $(6+\epsilon)$ by Cohen-Addad et al.~\cite{cohen-2016} with the same space complexity and smaller update time per point. Moreover we remove the assumption that $\alpha$ is known in advance. Our idea can be adapted to the metric diameter problem and the metric $k$-center problem to remove the assumption. For low dimensional Euclidean space, we give an approximation algorithm that guarantees an even better approximation.