  The $k$-center problem for a point set~$P$ asks for a collection of $k$
congruent balls (that is, balls of equal radius) that together cover all the
points in $P$ and whose radius is minimized. The $k$-center problem with
outliers is defined similarly, except that $z$ of the points in $P$ do need not
to be covered, for a given parameter $z$. We study the $k$-center problem with
outliers in data streams in the sliding-window model. In this model we are
given a possibly infinite stream $P=\langle p_1,p_2,p_3,\ldots\rangle$ of
points and a time window of length $W$, and we want to maintain a small sketch
of the set $P(t)$ of points currently in the window such that using the sketch
we can approximately solve the problem on $P(t)$.
  We present the first algorithm for the $k$-center problem with outliers in
the sliding-window model. The algorithm works for the case where the points
come from a space of bounded doubling dimension and it maintains a set $S(t)$
such that an optimal solution on $S(t)$ gives a $(1+\varepsilon)$-approximate
solution on $P(t)$. The algorithm is deterministic and uses
$O((kz/\varepsilon^d)\log \sigma)$ storage, where $d$ is the doubling dimension
of the underlying space and $\sigma$ is the spread of the points in the stream.
Algorithms providing a $(1+\varepsilon)$-approximation were not even known in
the setting without outliers or in the insertion-only setting with outliers. We
also present a lower bound showing that any algorithm that provides a
$(1+\varepsilon)$-approximation must use $\Omega((kz/\varepsilon)\log \sigma)$
storage.
