eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-08-31
54:1
54:19
10.4230/LIPIcs.ESA.2021.54
article
Certified Approximation Algorithms for the Fermat Point and n-Ellipses
Junginger, Kolja
1
Mantas, Ioannis
1
https://orcid.org/0000-0001-8256-8107
Papadopoulou, Evanthia
1
https://orcid.org/0000-0003-0144-7384
Suderland, Martin
1
https://orcid.org/0000-0002-6604-6381
Yap, Chee
2
https://orcid.org/0000-0003-2952-3545
Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland
Courant Institute, New York University, NY, USA
Given a set A of n points in ℝ^d with weight function w: A→ℝ_{> 0}, the Fermat distance function is φ(x): = ∑_{a∈A}w(a)‖x-a‖. A classic problem in facility location dating back to 1643, is to find the Fermat point x*, the point that minimizes the function φ. We consider the problem of computing a point x̃* that is an ε-approximation of x* in the sense that ‖x̃*-x*‖<ε. The algorithmic literature has so far used a different notion based on ε-approximation of the value φ(x*). We devise a certified subdivision algorithm for computing x̃*, enhanced by Newton operator techniques. We also revisit the classic Weiszfeld-Kuhn iteration scheme for x*, turning it into an ε-approximate Fermat point algorithm. Our second problem is the certified construction of ε-isotopic approximations of n-ellipses. These are the level sets φ^{-1}(r) for r > φ(x*) and d = 2. Finally, all our planar (d = 2) algorithms are implemented in order to experimentally evaluate them, using both synthetic as well as real world datasets. These experiments show the practicality of our techniques.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol204-esa2021/LIPIcs.ESA.2021.54/LIPIcs.ESA.2021.54.pdf
Fermat point
n-ellipse
subdivision
approximation
certified
algorithms