The smallest enclosing ball of balls: combinatorial structure and algorithms We develop algorithms for computing the exact smallest enclosing ball of a set of n balls in d-dimensional space. Unlike previous methods, we explicitly address small cases (n≤d+2), derive the necessary primitive operations and show that they can efficiently be realized with rational arithmetic. An implementation (along with a fast and robust floating-point version) is available as part of the CGAL library. 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, the existing adaptations of the method to the ball case are incorrect. In solving the small cases we may assume that the ball centers are affinely independent. 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 variety of pivoting methods for unique sink orientations which have the potential of being fast in practice even for high dimensions. As a by-product, we show that the problem of finding the smallest enclosing ball of balls with a fixed point on the boundary is equivalent to the problem of finding the minimum-norm point in the convex hull of a union of balls (Source:

References in zbMATH (referenced in 26 articles , 1 standard article )

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  1. Dearing, P.M.; Belotti, Pietro; Smith, Andrea M.: A primal algorithm for the weighted minimum covering ball problem in $\mathbb R^n$ (2016)
  2. Liu, Ya-Feng; Diao, Rui; Ye, Feng; Liu, Hong-Wei: An efficient inexact Newton-CG algorithm for the smallest enclosing ball problem of large dimensions (2016)
  3. Nguyen Thai An; Giles, Daniel; Nguyen Mau Nam; Rector, R.Blake: The log-exponential smoothing technique and Nesterov’s accelerated gradient method for generalized Sylvester problems (2016)
  4. Botnan, Magnus Bakke; Spreemann, Gard: Approximating persistent homology in Euclidean space through collapses (2015)
  5. Courty, Nicolas; Gong, Xing; Vandel, Jimmy; Burger, Thomas: SAGA: sparse and geometry-aware non-negative matrix factorization through non-linear local embedding (2014)
  6. Foniok, Jan; Gärtner, Bernd; Klaus, Lorenz; Sprecher, Markus: Counting unique-sink orientations (2014)
  7. Akopyan, Arseniy V.: Combinatorial generalizations of Jung’s theorem (2013)
  8. Dearing, P.M.; Smith, Andrea M.: A dual algorithm for the minimum covering weighted ball problem in $\mathbbR^n$ (2013)
  9. Ryu, Joonghyun; Kim, Deok-Soo: Protein structure optimization by side-chain positioning via beta-complex (2013)
  10. Brandenberg, René; Roth, Lucia: Minimal containment under homothetics: a simple cutting plane approach (2011)
  11. Daras, Petros; Axenopoulos, Apostolos: A 3D shape retrieval framework supporting multimodal queries (2010)
  12. Liu, Yu-Shen; Yi, Jing; Zhang, Hu; Zheng, Guo-Qin; Paul, Jean-Claude: Surface area estimation of digitized 3D objects using quasi-Monte Carlo methods (2010)
  13. Dearing, P.M.; Zeck, Christiane R.: A dual algorithm for the minimum covering ball problem in $\Bbb R^n$ (2009)
  14. Nielsen, Frank; Nock, Richard: Approximating smallest enclosing balls with applications to machine learning (2009)
  15. Gärtner, Bernd; Morris, Walter D. jun.; Rüst, Leo: Unique sink orientations of grids (2008)
  16. Yildirim, E.Alper: Two algorithms for the minimum enclosing ball problem (2008)
  17. Yu, Hai; Agarwal, Pankaj K.; Poreddy, Raghunath; Varadarajan, Kasturi R.: Practical methods for shape fitting and kinetic data structures using coresets (2008)
  18. Liu, Yu-Shen; Yong, Jun-Hai; Zhang, Hui; Yan, Dong-Ming; Sun, Jia-Guang: A quasi-Monte Carlo method for computing areas of point-sampled surfaces (2006)
  19. Pan, Shaohua; Li, Xingsi: An efficient algorithm for the smallest enclosing ball problem in high dimensions (2006)
  20. Zhou, Guanglu; Toh, Kim-Chuan; Sun, Jie: Efficient algorithms for the smallest enclosing ball problem (2005)

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