Minimum Enclosing Balls and Ellipsoids in General Dimensions

In this doctoral thesis, we study the problem of computing the ball of smallest radius enclosing a given set of points in any number of dimensions. Variations of this problem arise in several branches of computer science, such as computer graphics, artificial intelligence, and operations research. Applications range from collision detection for three-dimensional models in video games and computer-aided design, to high-dimensional clustering and classification in machine learning and data mining. We also consider the related and more challenging problem of finding the enclosing ellipsoid of minimum volume. Such ellipsoids can provide more descriptive data representations in the aforementioned applications, and they find further utility in, for example, optimal design of experiments and trimming of outliers in statistics. The contributions of this thesis consist of practical methods for the efficient solution of these two problems, with a primary focus on problem instances involving a large number of points. We introduce new algorithms to compute arbitrarily fine approximations of the minimum enclosing ball or ellipsoid in general dimensions. In our experimental evaluations, these algorithms exhibit running times that are highly competitive with, and often markedly superior to, those of earlier algorithms from the literature. Moreover, we present a new out-of-core algorithm to compute the exact minimum enclosing ball for massive, low-dimensional point sets residing in secondary storage. In addition to these solution methods, we develop acceleration techniques that can further improve their performance, either by using pruning heuristics to reduce the amount of work performed in each iteration, or by utilizing parallel hardware features of modern processors and graphics processing units. These techniques are also applicable to several existing algorithms.