A Composable Coreset for k-Center in Doubling Metrics

A set of points $P$ in a metric space and a constant integer $k$ are given. The $k$-center problem finds $k$ points as centers among $P$, such that the maximum distance of any point of $P$ to their closest centers $(r)$ is minimized. Doubling metrics are metric spaces in which for any $r$, a ball of radius $r$ can be covered using a constant number of balls of radius $r/2$. Fixed dimensional Euclidean spaces are doubling metrics. The lower bound on the approximation factor of $k$-center is $1.822$ in Euclidean spaces, however, $(1+\epsilon)$-approximation algorithms with exponential dependency on $\frac{1}{\epsilon}$ and $k$ exist. For a given set of sets $P_1,\ldots,P_L$, a composable coreset independently computes subsets $C_1\subset P_1, \ldots, C_L\subset P_L$, such that $\cup_{i=1}^L C_i$ contains an approximation of a measure of the set $\cup_{i=1}^L P_i$. We introduce a $(1+\epsilon)$-approximation composable coreset for $k$-center, which in doubling metrics has size sublinear in $|P|$. This results in a $(2+\epsilon)$-approximation algorithm for $k$-center in MapReduce with a constant number of rounds in doubling metrics for any $\epsilon>0$ and sublinear communications, which is based on parametric pruning. We prove the exponential nature of the trade-off between the number of centers $(k)$ and the radius $(r)$, and give a composable coreset for a related problem called dual clustering. Also, we give a new version of the parametric pruning algorithm with $O(\frac{nk}{\epsilon})$ running time, $O(n)$ space and $2+\epsilon$ approximation factor for metric $k$-center.