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arXiv · 1006.0291

The dilation of the Delaunay triangulation is greater than π/2

Prosenjit Bose·Luc Devroye·Maarten Löffler·Jack Snoeyink·Vishal Verma
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Abstract

Consider the Delaunay triangulation T of a set P of points in the plane as a Euclidean graph, in which the weight of every edge is its length. It has long been conjectured that the dilation in T of any pair p, p \in P, which is the ratio of the length of the shortest path from p to p' in T over the Euclidean distance ||pp'||, can be at most π/2 \approx 1.5708. In this paper, we show how to construct point sets in convex position with dilation > 1.5810 and in general position with dilation > 1.5846. Furthermore, we show that a sufficiently large set of points drawn independently from any distribution will in the limit approach the worst-case dilation for that distribution.

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Prosenjit Bose, Luc Devroye, Maarten Löffler, Jack Snoeyink, Vishal Verma. 2010-06-02. The dilation of the Delaunay triangulation is greater than π/2. https://arxiv.org/abs/1006.0291

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