THE SMALLEST ENCLOSING BALL OF BALLS: COMBINATORIAL STRUCTURE AND ALGORITHMS

Abstract

We develop algorithms for computing the exact smallest enclosing ball of a set of n balls in d-dimensional space. Unlike previous methods, we explicitly address small cases (n≤d+2), derive the necessary primitive operations and show that they can efficiently be realized with rational arithmetic. An implementation (along with a fast and robust floating-point version) is available as part of the CGAL library. We show that Welzl's randomized linear-time algorithm for computing the ball spanned by a set of points fails to work for balls. Consequently, the existing adaptations of the method to the ball case are incorrect. In solving the small cases we may assume that the ball centers are affinely independent. Via a geometric transformation and suitable generalization, it fits into the combinatorial model of unique sink orientations whose rich structure has recently received considerable attention. One consequence is that Welzl's algorithm does work for small instances; moreover, there is a variety of pivoting methods for unique sink orientations which have the potential of being fast in practice even for high dimensions. As a by-product, we show that the problem of finding the smallest enclosing ball of balls with a fixed point on the boundary is equivalent to the problem of finding the minimum-norm point in the convex hull of a union of balls.