1 research outputs found
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 . Though
the problem has been extensively studied before, most of the existing
algorithms need at least linear time (in the number of input points and the
dimensionality ) to achieve a -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 . In particular, the second algorithm has the sample complexity
even independent of the dimensionality . 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