  In fully dynamic clustering problems, a clustering of a given data set in a
metric space must be maintained while it is modified through insertions and
deletions of individual points. In this paper, we resolve the complexity of
fully dynamic $k$-center clustering against both adaptive and oblivious
adversaries. Against oblivious adversaries, we present the first algorithm for
fully dynamic $k$-center in an arbitrary metric space that maintains an optimal
$(2+\epsilon)$-approximation in $O(k \cdot \mathrm{polylog}(n,\Delta))$
amortized update time. Here, $n$ is an upper bound on the number of active
points at any time, and $\Delta$ is the aspect ratio of the metric space.
Previously, the best known amortized update time was $O(k^2\cdot
\mathrm{polylog}(n,\Delta))$, and is due to Chan, Gourqin, and Sozio (2018).
Moreover, we demonstrate that our runtime is optimal up to
$\mathrm{polylog}(n,\Delta)$ factors. In fact, we prove that even offline
algorithms for $k$-clustering tasks in arbitrary metric spaces, including
$k$-medians, $k$-means, and $k$-center, must make at least $\Omega(n k)$
distance queries to achieve any non-trivial approximation factor. This implies
a lower bound of $\Omega(k)$ which holds even for the insertions-only setting.
  We also show deterministic lower and upper bounds for adaptive adversaries,
demonstrate that an update time sublinear in $k$ is possible against oblivious
adversaries for metric spaces which admit locally sensitive hash functions
(LSH) and give the first fully dynamic $O(1)$-approximation algorithms for the
closely related $k$-sum-of-radii and $k$-sum-of-diameter problems.
