Faster Approximation Algorithms for k-Center via Data Reduction

Abstract

We study efficient algorithms for the Euclidean $k$-Center problem, focusing on the regime of large $k$. We take the approach of data reduction by considering $\hat{I}\pm$-coreset, which is a small subset $S$ of the dataset $P$ such that any ${\hat{I}}^2$-approximation on $S$ is an $(\hat{I}\pm +{\hat{I}}^2)$-approximation on $P$. We give efficient algorithms to construct coresets whose size is $k\cdot o(n)$, which immediately speeds up existing approximation algorithms. Notably, we obtain a near-linear time $O(1)$-approximation when $k=n^c$ for any $0<c<1$. We validate the performance of our coresets on real-world datasets with large $k$, and we observe that the coreset speeds up the well-known Gonzalez algorithm by up to $4$ times, while still achieving similar clustering cost. Technically, one of our coreset results is based on a new efficient construction of consistent hashing with competitive parameters. This general tool may be of independent interest for algorithm design in high dimensional Euclidean spaces.