The Stretch Factor of the Delaunay Triangulation Is Less Than 1.998

TL;DR

This paper proves that the stretch factor of the Delaunay triangulation for any finite point set in the plane is less than 1.998, improving previous bounds and breaking the barrier of 2, which was previously unachieved.

Contribution

The paper establishes a new upper bound of 1.998 for the stretch factor of Delaunay triangulations, improving prior bounds and resolving a long-standing open problem.

Findings

Stretch factor of Delaunay triangulation is less than 1.998

Improves previous upper bound of 2.42 from 1989

Breaks the barrier of 2 for the first time in plane graphs

Abstract

Let $S$ be a finite set of points in the Euclidean plane. Let $D$ be a Delaunay triangulation of $S$. The {\em stretch factor} (also known as {\em dilation} or {\em spanning ratio}) of $D$ is the maximum ratio, among all points $p$ and $q$ in $S$, of the shortest path distance from $p$ to $q$ in $D$ over the Euclidean distance $||pq||$. Proving a tight bound on the stretch factor of the Delaunay triangulation has been a long standing open problem in computational geometry. In this paper we prove that the stretch factor of the Delaunay triangulation of a set of points in the plane is less than $\rho = 1.998$, improving the previous best upper bound of 2.42 by Keil and Gutwin (1989). Our bound 1.998 is better than the current upper bound of 2.33 for the special case when the point set is in convex position by Cui, Kanj and Xia (2009). This upper bound breaks the barrier 2, which is significant because previously no family of plane graphs was known to have a stretch factor guaranteed to be less than 2 on any set of points.