Improved Stretch Factor of Delaunay Triangulations of Points in Convex Position

Abstract

Let S be a set of n points in the plane, and let DT(S) be the planar graph of the Delaunay triangulation of S. For a pair of points $a, b \in S$, denote by |ab| the Euclidean distance between a and b. Denote by DT(a, b) the shortest path in DT(S) between a and b, and let |DT(a, b)| be the total length of DT(a, b). Dobkin et al. were the first to show that DT(S) can be used to approximate the complete graph of S in the sense that the stretch factor $\frac{|DT(a, b)|}{|a b|}$ is upper bounded by $((1 + \sqrt{5})/2) \pi \approx 5.08$. Recently, Xia improved this factor to 1.998. Amani et al. have also shown that if the points of S are in convex position, then a planar graph with these vertices can be constructed such that its stretch factor is 1.88. A set of points is said to be in convex position, if all points form the vertices of a convex polygon.

In this paper, we prove that if the points of S are in convex position, then the stretch factor of DT(S) is less than 1.83, improving upon the previously known factors, for either the Delaunay triangulation or planar graph of a point set. Our result is obtained by investigating some geometric properties of DT(S) and showing that there exists a convex chain between a and b in DT(S) such that it is either contained in a semicircle of diameter ab, or enclosed by segment ab and a simple (convex) chain that consists of a circular arc and two or three line segments.

The work by Tan was partially supported by JSPS KAKENHI Grant Number 15K00023, and the work by Jiang was partially supported by National Natural Science Foundation of China under grant 61702242.