Stretch Factor of Long Paths in a planar Poisson-Delaunay Triangulation

TL;DR

This paper investigates the stretch factor of shortest paths in planar Poisson-Delaunay triangulations, providing bounds and experimental estimates for the expected path length and number of edges crossed as the point process intensity increases.

Contribution

It offers the first non-trivial lower bound for the expected shortest path length and bounds on the stretch factor in high-density Poisson-Delaunay triangulations.

Findings

Expected number of Delaunay edges crossed is about 2.16√n.

Expected shortest path length converges to 1.18.

Experimental estimate of the stretch factor is approximately 1.04.

Abstract

Let $X:=X_n\cup\{(0,0),(1,0)\}$, where $X_n$ is a planar Poisson point process of intensity $n$. We provide a first non-trivial lower bound for the distance between the expected length of the shortest path between $(0,0)$ and $(1,0)$ in the Delaunay triangulation associated with $X$ when the intensity of $X_n$ goes to infinity. Experimental values indicate that the correct value is about 1.04. We also prove that the expected number of Delaunay edges crossed by the line segment $[(0,0),(1,0)]$ is equivalent to $2.16\sqrt{n}$ and that the expected length of a particular path converges to 1.18 giving an upper bound on the stretch factor.