The dilation of the Delaunay triangulation is greater than {\pi}/2

TL;DR

This paper demonstrates that the dilation of Delaunay triangulations can exceed the conjectured maximum of π/2, providing explicit constructions and probabilistic insights into worst-case scenarios.

Contribution

The authors construct point sets with dilation surpassing previous bounds and analyze how large random sets approach worst-case dilation in the limit.

Findings

Constructed point sets with dilation > 1.58

Dilation in large random sets approaches worst-case bounds

Dilation exceeds the conjectured π/2 in specific configurations

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 {\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.