The Stretch Factor of Hexagon-Delaunay Triangulations

TL;DR

This paper determines the exact stretch factor of Hexagon-Delaunay triangulations as 2, advancing understanding of geometric network efficiency by developing new techniques for analyzing regular-polygon-based triangulations.

Contribution

It introduces two novel techniques for computing tight upper bounds on the stretch factor of Hexagon-Delaunay triangulations, providing the exact value of 2.

Findings

Exact stretch factor of Hexagon-Delaunay triangulations is 2.

Developed two techniques for tight upper bound computation.

Advances understanding of geometric network efficiency.

Abstract

The problem of computing the exact stretch factor (i.e., the tight bound on the worst case stretch factor) of a Delaunay triangulation is one of the longstanding open problems in computational geometry. Over the years, a series of upper and lower bounds on the exact stretch factor have been obtained but the gap between them is still large. An alternative approach to solving the problem is to develop techniques for computing the exact stretch factor of ``easier'' types of Delaunay triangulations, in particular those defined using regular-polygons instead of a circle. Tight bounds exist for Delaunay triangulations defined using an equilateral triangle and a square. In this paper, we determine the exact stretch factor of Delaunay triangulations defined using a regular hexagon: It is 2. We think that the main contribution of this paper are the two techniques we have developed to compute tight upper bounds for the stretch factor of Hexagon-Delaunay triangulations.