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. Roughly speaking, an instance of MEB is stable, if the radius of the resulting ball cannot be significantly reduced by removing a small fraction of the input points. Under the stability assumption, we present two sampling algorithms for computing approximate MEB with sample complexities independent of the number of input points $n$. In particular, the second algorithm has the sample complexity even independent of the dimensionality $d$. Further, we extend the idea to achieve a sub-linear time approximation algorithm for the MEB with outliers problem. Note that most existing sub-linear time algorithms for the problems of MEB and MEB with outliers usually result in bi-criteria approximations, where the "bi-criteria" means that the solution has to allow the approximations on the radius and the number of covered points. Differently, all the algorithms proposed in this paper yield single-criterion approximations (with respect to radius). We expect that our proposed notion of stability and techniques will be applicable to design sub-linear time algorithms for other optimization problems.