Fully Dynamic k-Center in Low Dimensions via Approximate Furthest Neighbors
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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 ∈ P that is a (1+ϵ) 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+ϵ) approximate k-centers in general metric spaces with bounded doubling dimension and the second maintains (1+ϵ) 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 ϵ in advance. In the Euclidean case, our algorithm also seems to be the first deterministic solution.
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