Fully Dynamic k-Center Clustering in Doubling Metrics.

In the $k$-center clustering problem, we are given a set of $n$ points in a metric space and a parameter $k \leq n$. The goal is to select $k$ designated points, referred to as \emph{centers}, such that the maximum distance of any point to its closest center is minimized. This notion of clustering is of fundamental importance and has been extensively studied. We study a \emph{dynamic} variant of the $k$-center clustering problem, where the goal is to maintain a clustering with small approximation ratio while supporting an intermixed update sequence of insertions and deletions of points with small update time. Moreover, the data structure should be able to support the following queries for any given point: (1) report whether this point is a center or (2) determine the cluster this point is assigned to. We present a deterministic dynamic algorithms for the $k$-center clustering problem that achieves a $(2+\epsilon)$-approximation with $O(2^{O(\kappa)} \log\Delta \log\log\Delta \cdot \epsilon^{-1} \ln \epsilon^{-1})$ update time and $O(\log \Delta)$ query time, where $\kappa$ bounds the doubling dimension of the metric and $\Delta$ is the aspect ratio. Our running and query times are independent of the number of centers $k$, and are poly-logarithmic when the metric has constant doubling dimension and the aspect ratio is bounded by a polynomial.