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

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

2010-06-02

arxiv.org/abs/1006.0291v1

Computer Science

Computational Geometry

12 pages, 6 figures, invited to the special edition of Computational Geometry Theory and Applications for papers from CCCG 200

Scientific paper

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 {\pi}/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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