The Smallest Enclosing Ball of Balls: Combinatorial Structure and Algorithms

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. 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.

The Persistent Homology of Distance Functions under Random Projection

“…We will make use of the fact that center(S) ∈ conv(S), where conv(S) denotes the convex closure of S (see Fischer and Gärtner [22] for a proof of this fact in a more general form). Note that d 1 P = d P , so we will work only with this more general class of functions d k P .…”

Unique Sink Orientations of Grids

“…asymptotically dominates the bound in (15), and an estimate of f (n, z) ≤ (z − 1)H n z ≈ z ln n z immediately follows. The next result shows that the bound is actually linear in z.…”

“…, a z ) holds. Comparing this with the expected number of edge evaluations (15) of Product, we see that for large z, both algorithms have approximately the same Figure 8: Some of the N − 3 paths where sink and source span a 2-dimensional face Proof. Assume the grid has dimensions d 1 ×d 2 ×d 3 .…”

“…This in particular yields the first combinatorial algorithms for PLCP with nontrivial runtime bounds. Cube USO are also useful as combinatorial models for geometric optimization problems, in particular the problem of computing the smallest enclosing ball of a set of points [42,18], or a set of balls [15]. In the point case, this problem can be reduced to PLCP with positive definite matrix [18], but the ball case does not fit into any known class of cube USO.…”

Approximating Smallest Enclosing Balls With Applications to Machine Learning

“…Later, the method was extended to the case of balls 10,11 . Since the smallest enclosing ball is unique and defined by at most + 1 support points a (or tangent balls as depicted in Figure 1) in strictly convex position (implying being affinely independent as well), a brute-force naïve combinatorial algorithm requires ( +2 ) time b (with linear memory).…”