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.

翻译：用于一个点的 $- p$ 的 $- center 问题, 用于收集一个 $k 的 comgruent 球( 等半径的球), 共覆盖$P$的所有点, 其半径最小化 。 外端的 $k 美元 问题定义相似, 但对于一个给定的参数 $P 的点中, 却不需要覆盖 $P 的 美元 问题 。 我们研究了 livel- window 模型中数据流中的 $- center 的 $- center 问题 。 在这个模型中, 我们给一个可能无限的流 $\ ligle p_ 1, p_ 2, p_ 3,\ ldotsanglegle $, 时间窗口的美元长度为 $, 我们想要保持一个小的 $P( t) 点的草图, 这样我们就可以在 $( t) 上解决 $. we listal- listal dent 问题。