Abstract. We develop a simple combinatorial algorithm for computing the smallest enclosing ball of a set of points in high dimensional Euclidean space. The resulting code is in most cases faster (sometimes significantly) than recent dedicated methods that only deliver approximate results, and it beats off-the-shelf solutions, based e.g. on quadratic programming solvers. The algorithm resembles the simplex algorithm for linear programming; it comes with a Bland-type rule to avoid cycling in presence of degeneracies and it typically requires very few iterations. We provide a fast and robust floating-point implementation whose efficiency is based on a new dynamic data structure for maintaining intermediate solutions.The code can efficiently handle point sets in dimensions up to 2,000, and it solves instances of dimension 10,000 within hours. In low dimensions, the algorithm can keep up with the fastest computational geometry codes that are available.

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 fast 1 and robust floating-point version) is available as part of the CGAL library. 2Our 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 the small cases we may assume that the ball centers are affinely independent; in this case, the problem is surprisingly well-behaved: 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 wide variety of pivoting methods for unique sink orientations which have the potential of being fast in practice even for high dimensions.

No abstract

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 fast 1 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 the small cases we may assume that the ball centers are affinely independent; in this case, the problem is surprisingly well-behaved: 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 wide variety of pivoting methods for unique sink orientations which have the potential of being fast in practice even for high dimensions.

No abstract