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 stretch factor in T of any pair p, p′ ∈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 φ/2 ≈ 1.5708. In this paper, we show how to construct point sets in convex position with stretch factor > 1.5810 and in general position with stretch factor > 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 stretch factor for that distribution.

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Computational Geometry
School of Computer Science

Bose, P, Devroye, L. (Luc), Löffler, M. (Maarten), Snoeyink, J. (Jack), & Verma, V. (Vishal). (2011). Almost all Delaunay triangulations have stretch factor greater than φ/2. Computational Geometry, 44(2), 121–127. doi:10.1016/j.comgeo.2010.09.009