k-Center Clustering with Outliers in the Sliding-Window Model

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 = ⟨ p₁,p₂,p₃,…⟩ 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+e)-approximate solution on P(t). The algorithm uses O((kz/e^d)log σ) storage, where d is the doubling dimension of the underlying space and σ is the spread of the points in the stream. Algorithms providing a (1+e)-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+e)-approximation must use Ω((kz/e)log σ) storage.