Fast and Easy Computation of Approximate Smallest Enclosing Balls

Abstract

The incremental Badoiu-Clarkson algorithm finds the smallest ball enclosing n points in d dimensions with at least O(1/radict) precision, after t iteration steps. The extremely simple incremental step of the algorithm makes it very attractive both for theoreticians and practitioners. A simplified proof for this convergence is given. This proof allows to show that the precision increases, in fact, even as O(u/t) with the number of iteration steps. Computer experiments, but not yet a proof, suggest that the u, which depends only on the data instance, is actually bounded by min{radic2d, radic2n}. If it holds, then the algorithm finds the smallest enclosing ball with epsi precision in at most 0(ndradic/dm/epsi) time, with dm = min{d, n}