Fast Algorithms for Geometric Consensuses

Authors Sariel Har-Peled, Mitchell Jones

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Sariel Har-Peled
  • Dept. of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL, USA
Mitchell Jones
  • Dept. of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL, USA


The authors thank Joachim Gudmundsson for bringing the problem of computing the yolk to our attention. The second author thanks Sampson Wong for discussions on computing the yolk in higher dimensions. We also thank Timothy Chan for useful comments (in particular, the improved algorithm for the yolk in 3D, see Remark 26).

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Sariel Har-Peled and Mitchell Jones. Fast Algorithms for Geometric Consensuses. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Let P be a set of n points in ℝ^d in general position. A median hyperplane (roughly) splits the point set P in half. The yolk of P is the ball of smallest radius intersecting all median hyperplanes of P. The egg of P is the ball of smallest radius intersecting all hyperplanes which contain exactly d points of P. We present exact algorithms for computing the yolk and the egg of a point set, both running in expected time O(n^(d-1) log n). The running time of the new algorithm is a polynomial time improvement over existing algorithms. We also present algorithms for several related problems, such as computing the Tukey and center balls of a point set, among others.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Geometric optimization
  • centerpoint
  • voting games


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