Fully Dynamic $k$-Center in Low Dimensions via Approximate Furthest Neighbors

Let $P$ be a set of points in some metric space. The approximate furthest neighbor problem is, given a second point set $C,$ to find a point $p \in P$ that is a $(1+\epsilon)$ approximate furthest neighbor from $C.$ The dynamic version is to maintain $P,$ over insertions and deletions of points, in a way that permits efficiently solving the approximate furthest neighbor problem for the current $P.$ We provide the first algorithm for solving this problem in metric spaces with finite doubling dimension. Our algorithm is built on top of the navigating net data-structure. An immediate application is two new algorithms for solving the dynamic $k$-center problem. The first dynamically maintains $(2+\epsilon)$ approximate $k$-centers in general metric spaces with bounded doubling dimension and the second maintains $(1+\epsilon)$ approximate Euclidean $k$-centers. Both these dynamic algorithms work by starting with a known corresponding static algorithm for solving approximate $k$-center, and replacing the static exact furthest neighbor subroutine used by that algorithm with our new dynamic approximate furthest neighbor one. Unlike previous algorithms for dynamic $k$-center with those same approximation ratios, our new ones do not require knowing $k$ or $\epsilon$ in advance. In the Euclidean case, our algorithm also seems to be the first deterministic solution.