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

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 approximation algorithm for MEB with outliers. We also show that our idea can be extended to the general case of MEB with outliers ({\em i.e.,} without the stability assumption), and obtain a sub-linear time bi-criteria approximation algorithm. Our results can be viewed as a new step along the direction of beyond worst-case analysis.