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Delaunay Triangulations in the Hilbert Metric

Authors Auguste H. Gezalyan , Soo H. Kim, Carlos Lopez, Daniel Skora, Zofia Stefankovic, David M. Mount

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Author Details

Auguste H. Gezalyan
  • Department of Computer Science, University of Maryland, College Park, MD, USA
Soo H. Kim
  • Wellesley College, MA, USA
Carlos Lopez
  • Montgomery Blair High School, Silver Spring, MD, USA
Daniel Skora
  • Indiana University, Bloomington, IN, USA
Zofia Stefankovic
  • Stony Brook University, Stony Brook, NY, USA
David M. Mount
  • Department of Computer Science, University of Maryland, College Park, MD, USA

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Auguste H. Gezalyan, Soo H. Kim, Carlos Lopez, Daniel Skora, Zofia Stefankovic, and David M. Mount. Delaunay Triangulations in the Hilbert Metric. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SWAT.2024.25

Abstract

The Hilbert metric is a distance function defined for points lying within the interior of a convex body. It arises in the analysis and processing of convex bodies, machine learning, and quantum information theory. In this paper, we show how to adapt the Euclidean Delaunay triangulation to the Hilbert geometry defined by a convex polygon in the plane. We analyze the geometric properties of the Hilbert Delaunay triangulation, which has some notable differences with respect to the Euclidean case, including the fact that the triangulation does not necessarily cover the convex hull of the point set. We also introduce the notion of a Hilbert ball at infinity, which is a Hilbert metric ball centered on the boundary of the convex polygon. We present a simple randomized incremental algorithm that computes the Hilbert Delaunay triangulation for a set of n points in the Hilbert geometry defined by a convex m-gon. The algorithm runs in O(n (log n + log³ m)) expected time. In addition we introduce the notion of the Hilbert hull of a set of points, which we define to be the region covered by their Hilbert Delaunay triangulation. We present an algorithm for computing the Hilbert hull in time O(n h log² m), where h is the number of points on the hull’s boundary.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Delaunay Triangulations
  • Hilbert metric
  • convexity
  • randomized algorithms

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