Near-linear Time Approximation Schemes for Clustering in Doubling Metrics

Abstract

We consider the classic Facility Location, k-Median, and k-Means problems in metric spaces of doubling dimension d. We give nearly linear-time approximation schemes for each problem.

The complexity of our algorithms is (O) over tilde (2((1/epsilon)O(d2) )n), making a significant improvement over the state-of-the-art algorithms that run in time n((d/epsilon)O(d)). Moreover, we show how to extend the techniques used to get the first efficient approximation schemes for the problems of prize-collecting k-Median and k-Means and efficient bicriteria approximation schemes for k-Median with outliers, k-Means with outliers and k-Center.

Keywords

• Approximation scheme
• k-Median
• k-Means
• Euclidean spaces
• doubling dimension