Fast and Easy Computation of Approximate Smallest Enclosing Balls

The incremental Badoiu-Clarkson algorithm finds the smallest ball enclosing n points in d dimensions with at least O(1 / √ t) 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 { √ 2d, √ 2n}. If it holds, then the algorithm finds the smallest enclosing ball with ɛ precision in at most O(nd √ dm/ɛ) time, with dm = min{d, n}. Key words: computational geometry, smallest enclosing ball, pattern recognition 1