The smallest enclosing ball of balls: combinatorial structure and algorithms

We develop algorithms for computing the smallest enclosing ball of a set of n balls in d-dimensional space. Unlike previous methods, we explicitly address small cases (n ≤ d + 1), derive the necessary primitive operations and show that they can efficiently be realized with rational arithmetic. An exact implementation (along with a fast1 and robust floating-point version) is available as part of the CGAL library.2 Our algorithms are based on novel insights into the combinatorial structure of the problem. As it turns out, results for smallest enclosing balls of points do not extend as one might expect. For example, 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, David White’s adaptation of the method to the ball case—as the only available implementation so far it is mentioned in many link collections—is incorrect and may crash or, in the better case, produce wrong balls. In solving ...

• Type: Conference
• Year: 2003
• Where: COMPGEOM
• Authors: Kaspar Fischer, Bernd Gärtner
• Keywords: COMPGEOM 2003, Small Cases, Smallest Enclosing Balls, Unique Sink Orientations