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Subdivision Methods for Sum-Of-Distances Problems: Fermat-Weber Point, n-Ellipses and the Min-Sum Cluster Voronoi Diagram (Media Exposition)

Authors Ioannis Mantas , Evanthia Papadopoulou , Martin Suderland , Chee Yap

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Author Details

Ioannis Mantas
  • Faculty of Informatics, Università della Svizzera italiana (USI), Lugano, Switzerland
Evanthia Papadopoulou
  • Faculty of Informatics, Università della Svizzera italiana (USI), Lugano, Switzerland
Martin Suderland
  • Faculty of Informatics, Università della Svizzera italiana (USI), Lugano, Switzerland
Chee Yap
  • Courant Institute, New York University (NYU), NY, USA

Funding

  • Papadopoulou, Evanthia: supported by the SNF~project~200021E_201356.
  • Suderland, Martin: supported by the SNF~project~200021E_201356.
  • Yap, Chee: supported by NSF Grant No.~CCF-2008768.

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Ioannis Mantas, Evanthia Papadopoulou, Martin Suderland, and Chee Yap. Subdivision Methods for Sum-Of-Distances Problems: Fermat-Weber Point, n-Ellipses and the Min-Sum Cluster Voronoi Diagram (Media Exposition). In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 69:1-69:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.SoCG.2022.69

Abstract

Given a set P of n points, the sum of distances function of a point x is d_{P}(x) : = ∑_{p ∈ P} ||x - p||. Using a subdivision approach with soft predicates we implement and visualize approximate solutions for three different problems involving the sum of distances function in ℝ². Namely, (1) finding the Fermat-Weber point, (2) constructing n-ellipses of a given set of points, and (3) constructing the nearest Voronoi diagram under the sum of distances function, given a set of point clusters as sites.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Fermat point
  • geometric median
  • Weber point
  • Fermat distance
  • sum of distances
  • n-ellipse
  • multifocal ellipse
  • min-sum Voronoi diagram
  • cluster Voronoi diagram

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References

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