Approximating k-Center via Farthest-First on $\delta$-Covers

The farthest-first traversal of Gonzalez is a classical $2$-approximation algorithm for solving the $k$-center problem, but its sequential nature makes it difficult to scale to very large datasets. In this work we study the effect of running farthest-first on a $\delta$-cover of the dataset rather than on the full set of points. A $\delta$-cover provides a compact summary of the data in which every point lies within distance $\delta$ of some selected center. We prove that if farthest-first is applied to a $\delta$-cover, the resulting $k$-center radius is at most twice the optimal radius plus $\delta$. In our experiments on large high-dimensional datasets, we show that restricting the input to a $\delta$-cover dramatically reduces the running time of the farthest-first traversal while only modestly increasing the $k$-center radius.