# Smallest-circle problem

Most of the geometric approaches for the problem look for points that lie on the boundary of the minimum circle and are based on the following simple facts:

The algorithm is rather complicated and it is reflected by its big multiplicative constant. The reduction needs to solve twice the similar problem where the center of the sought-after enclosing circle is constrained to lie on a given line. The solution of the subproblem is either the solution of the unconstrained problem or it is used to determine the half-plane where the unconstrained solution center is located.

Welzl's paper states that it is sufficient to randomly permute the input at the start, rather than performing independently random choices of *p* on each recursion.

It also states that performance is improved by dynamically re-ordering the points so that those that are found to be outside a circle are subsequently considered earlier, but this requires a change in the structure of the algorithm to store *P* as a "global".