Fast Smallest-Enclosing-Ball Computation in High Dimensions
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.
KeywordsHigh Dimension Orthogonal Projection Computational Geometry Simplex Algorithm Quadratic Time
Unable to display preview. Download preview PDF.
- 2.Dyer, M.E.: A class of convex programs with applications to computational geometry. In: Proc. 8th Annu. ACM Sympos. Comput. Geom., pp. 9–15 (1992)Google Scholar
- 4.Gärtner, B.: Fast and robust smallest enclosing balls. In: Nešetřil, J. (ed.) ESA 1999. LNCS, vol. 1643, pp. 325–338. Springer, Heidelberg (1999)Google Scholar
- 5.Gärtner, B., Schönherr, S.: An efficient, exact, and generic quadratic programming solver for geometric optimization. In: Proc. 16th Annu. ACM Sympos. Comput. Geom, pp. 110–118 (2000)Google Scholar
- 7.Bulatov, Y., Jambawalikar, S., Kumar, P., Sethia, S.: Hand recognition using geometric classifiers. In: Abstract of presentation for the DIMACS Workshop on Computational Geometry. Rutgers University (2002)Google Scholar
- 8.Goel, A., Indyk, P., Varadarajan, K.R.: Reductions among high dimensional proximity problems. In: Symposium on Discrete Algorithms, pp. 769–778 (2001)Google Scholar
- 9.Kumar, P., Mitchell, J.S.B., Yıldırım, E.A.: Computing core-sets and approximate smallest enclosing hyperspheres in high dimensions. In: To appear in the Proceedings of ALENEX 2003 (2003)Google Scholar
- 10.ILOG, Inc.: ILOG CPLEX 6.5 user’s manual (1999)Google Scholar
- 11.Zhou, G., Toh, K.C., Sun, J.: Efficient algorithms for the smallest enclosing ball problem (2002) (manuscript)Google Scholar
- 12.Hopp, T.H., Reeve, C.P.: An algorithm for computing the minimum covering sphere in any dimension. Technical Report NISTIR 5831, National Institute of Standards and Technology (1996)Google Scholar