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.

翻译：设$P$为某个度量空间中的点集。近似最远邻问题定义为：给定另一个点集$C$，寻找点$p \in P$使得其是$C$的$(1+\epsilon)$近似最远邻。动态版本要求在支持点的插入与删除操作下维护$P$，并能够高效求解当前$P$的近似最远邻问题。本文首次提出在具有有限加倍维数的度量空间中求解该问题的算法。该算法基于导航网数据结构构建。其直接应用包括两种求解动态$k$-中心问题的新算法：第一种算法能在具有有界加倍维数的一般度量空间中动态维护$(2+\epsilon)$近似$k$-中心，第二种算法则维护$(1+\epsilon)$近似欧几里得$k$-中心。这两种动态算法均以已知的相应静态近似$k$-中心算法为起点，并用我们提出的新型动态近似最远邻子程序替代原算法中使用的静态精确最远邻子程序。与先前具有相同近似比的动态$k$-中心算法不同，我们的新算法无需预先知道$k$或$\epsilon$的值。在欧几里得情形下，该算法似乎也是首个确定性解法。