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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)

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## Cite As

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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