Minimum Enclosing Ball Revisited: Stability, Sub-linear Time Algorithms, and Extension

In this paper, we revisit the Minimum Enclosing Ball (MEB) problem and its robust version, MEB with outliers, in Euclidean space $\mathbb{R}^d$. Though the problem has been extensively studied before, most of the existing algorithms need at least linear time (in the number of input points $n$ and the dimensionality $d$) to achieve a $(1+\epsilon)$-approximation. Motivated by some recent developments on beyond worst-case analysis, we introduce the notion of stability for MEB (with outliers), which is natural and easy to understand. Under the stability assumption, we present two sampling algorithms for computing approximate MEB with sample complexities independent of the number of input points; further, we achieve the first sub-linear time single-criterion approximation algorithm for the MEB with outliers problem. Our result can be viewed as a new step along the direction of beyond worst-case analysis. We also show that our ideas can be extended to be more general techniques, a novel uniform-adaptive sampling method and a sandwich lemma, for solving the general case of MEB with outliers ({\em i.e.,} without the stability assumption) and the problem of $k$-center clustering with outliers. We achieve sub-linear time bi-criteria approximation algorithms for these problems respectively; the algorithms have sample sizes independent of the number of points $n$ and the dimensionality $d$, which significantly improve the time complexities of existing algorithms. We expect that our technique will be applicable to design sub-linear time algorithms for other shape fitting with outliers problems.