  We provide a general framework for getting expected linear time constant
factor approximations (and in many cases FPTAS's) to several well known
problems in Computational Geometry, such as $k$-center clustering and farthest
nearest neighbor. The new approach is robust to variations in the input
problem, and yet it is simple, elegant and practical. In particular, many of
these well studied problems which fit easily into our framework, either
previously had no linear time approximation algorithm, or required rather
involved algorithms and analysis. A short list of the problems we consider
include farthest nearest neighbor, $k$-center clustering, smallest disk
enclosing $k$ points, $k$th largest distance, $k$th smallest $m$-nearest
neighbor distance, $k$th heaviest edge in the MST and other spanning forest
type problems, problems involving upward closed set systems, and more. Finally,
we show how to extend our framework such that the linear running time bound
holds with high probability.
