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Journey to the Center of the Point Set

Authors Sariel Har-Peled, Mitchell Jones

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LIPIcs.SoCG.2019.41.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Random walks and Markov chains
Keywords
  • Computational geometry
  • Centerpoints
  • Random walks

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Abstract

We revisit an algorithm of Clarkson et al. [K. L. Clarkson et al., 1996], that computes (roughly) a 1/(4d^2)-centerpoint in O~(d^9) time, for a point set in R^d, where O~ hides polylogarithmic terms. We present an improved algorithm that computes (roughly) a 1/d^2-centerpoint with running time O~(d^7). While the improvements are (arguably) mild, it is the first progress on this well known problem in over twenty years. The new algorithm is simpler, and the running time bound follows by a simple random walk argument, which we believe to be of independent interest. We also present several new applications of the improved centerpoint algorithm.

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Sariel Har-Peled and Mitchell Jones. Journey to the Center of the Point Set. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.SoCG.2019.41

Author Details

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

Funding

  • Har-Peled, Sariel: Supported in part by NSF AF awards CCF-1421231 and CCF-1217462.

References

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