Bounding sphere

Such spheres are useful in clustering, where groups of similar data points are classified together.

In statistical analysis the scattering of data points within a sphere may be attributed to measurement error or natural (usually thermal) processes, in which case the cluster represents a perturbation of an ideal point. In some circumstances this ideal point may be used as a substitute for the points in the cluster, advantageous in reducing calculation time.

In operations research the clustering of values to an ideal point may also be used to reduce the number of inputs in order to obtain approximate values for NP-hard problems in a reasonable time. The point chosen is not usually the center of the sphere, as this can be biased by outliers, but instead some form of average location such as a least squares point is computed to represent the cluster.

Nimrod Megiddo studied the 1-center problem extensively and published on it at least five times in the 1980s.[2] In 1983, he proposed a "prune and search" algorithm which finds the optimum bounding sphere and runs in linear time if the dimension is fixed as a constant. When the dimension ${\displaystyle d}$ is taken into account, the execution time complexity is ${\displaystyle O(2^{O(d^{2})}n)}$,[3][4] which is impractical for high-dimensional applications.

In 1991, Emo Welzl proposed a much simpler randomized algorithm, generalizing a randomized linear programming algorithm by Raimund Seidel. The expected running time of Welzl’s algorithm is ${\displaystyle O((d+1)(d+1)!n)}$, which again reduces to ${\displaystyle O(n)}$ for any fixed dimension ${\displaystyle d}$. The paper provides experimental results demonstrating its practicality in higher dimensions.[5] A more recent deterministic algorithm of Timothy Chan also runs in ${\displaystyle O(n)}$ time, with a smaller (but still exponential) dependence on the dimension.[4]

The open-source Computational Geometry Algorithms Library (CGAL) contains an implementation of Welzl’s algorithm.[6]

In 1990, Jack Ritter proposed a simple algorithm to find a non-minimal bounding sphere.[7] It is widely used in various applications for its simplicity. The algorithm works in this way:

1. Pick a point ${\displaystyle x}$ from ${\displaystyle P}$, search a point ${\displaystyle y}$ in ${\displaystyle P}$, which has the largest distance from ${\displaystyle x}$;
2. Search a point ${\displaystyle z}$ in ${\displaystyle P}$, which has the largest distance from ${\displaystyle y}$. Set up an initial ball ${\displaystyle B}$, with its centre as the midpoint of ${\displaystyle y}$ and ${\displaystyle z}$, the radius as half of the distance between ${\displaystyle y}$ and ${\displaystyle z}$;
3. If all points in ${\displaystyle P}$ are within ball ${\displaystyle B}$, then we get a bounding sphere. Otherwise, let ${\displaystyle p}$ be a point outside the ball, construct a new ball covering both point ${\displaystyle p}$ and previous ball. Repeat this step until all points are covered.

Ritter's algorithm runs in time ${\displaystyle O(nd)}$ on inputs consisting of ${\displaystyle n}$ points in ${\displaystyle d}$-dimensional space, which makes it very efficient. However it gives only a coarse result which is usually 5% to 20% larger than the optimum.

Bădoiu et al. presented a ${\displaystyle 1+\varepsilon }$ approximation to the bounding sphere problem,[8] where a ${\displaystyle 1+\varepsilon }$ approximation means that the constructed sphere has radius at most ${\displaystyle (1+\varepsilon )r}$, where ${\displaystyle r}$ is the smallest possible radius of a bounding sphere.

A coreset is a small subset, that a ${\displaystyle 1+\varepsilon }$ expansion of the solution on the subset is a bounding sphere of the whole set. The coreset is constructed incrementally by adding the farthest point into the set in each iteration.

Kumar et al. improved this approximation algorithm[9] so that it runs in time ${\displaystyle O({\frac {nd}{\epsilon }}+{\frac {1}{\epsilon ^{4.5}}}\log {\frac {1}{\epsilon }})}$.

Fischer et al. (2003) proposed an exact solver, though the algorithm does not have a polynomial running time in the worst case.[10] The algorithm is purely combinatorial and implements a pivoting scheme similar to the simplex method for linear programming, used earlier in some heuristics. It starts with a large sphere that covers all points and gradually shrinks it until it cannot be shrunk further. The algorithm features correct termination rules in cases of degeneracies, overlooked by prior authors; and efficient handling of partial solutions, which produces a major speed-up. The authors verified that the algorithm is efficient in practice in low and moderately low (up to 10,000) dimensions and claim it does not exhibit numerical stability problems in its floating-point operations.[10] A C++ implementation of the algorithm is available as an open-source project.[11]

Larsson (2008) proposed the "extremal points optimal sphere" method with controllable speed to accuracy approximation to solve the bounding sphere problem. This method works by taking a set of ${\displaystyle s}$ direction vectors and projecting all points onto each vector in ${\displaystyle s}$; ${\displaystyle s}$ serves as a speed-accuracy trade-off variable. An exact solver is applied to the ${\displaystyle 2s}$ extremal points of these projections. The algorithm then iterates over the remaining points, if any, growing the sphere if necessary. For large ${\displaystyle n}$ this method is orders of magnitude faster than exact methods, while giving comparable results. It has a worst case time of ${\displaystyle O(sn)}$. [1]