  We study the metric facility location problem with client insertions and
deletions. This setting differs from the classic dynamic facility location
problem, where the set of clients remains the same, but the metric space can
change over time. We show a deterministic algorithm that maintains a constant
factor approximation to the optimal solution in worst-case time $\tilde
O(2^{O(\kappa^2)})$ per client insertion or deletion in metric spaces while
answering queries about the cost in $O(1)$ time, where $\kappa$ denotes the
doubling dimension of the metric. For metric spaces with bounded doubling
dimension, the update time is polylogarithmic in the parameters of the problem.
