The Tight Spanning Ratio of the Rectangle Delaunay Triangulation

TL;DR

This paper establishes tight bounds on the spanning ratio of rectangle Delaunay triangulations, extending known results to a broader class of shapes and providing precise bounds based on rectangle aspect ratios.

Contribution

It proves the exact spanning ratio for rectangle Delaunay triangulations with aspect ratio A, matching the known lower bound, thus extending the class of shapes with known tight bounds.

Findings

Spanning ratio for rectangle Delaunay triangulations is at most √2 √(1+A^2 + A√(A^2 + 1))

The derived bound matches the known lower bound, confirming tightness

Extends known bounds from special shapes to all rectangles with aspect ratio A

Abstract

Spanner construction is a well-studied problem and Delaunay triangulations are among the most popular spanners. Tight bounds are known if the Delaunay triangulation is constructed using an equilateral triangle, a square, or a regular hexagon. However, all other shapes have remained elusive. In this paper, we extend the restricted class of spanners for which tight bounds are known. We prove that Delaunay triangulations constructed using rectangles with aspect ratio $A$ have spanning ratio at most $\sqrt{2} \sqrt{1+A^2 + A \sqrt{A^2 + 1}}$, which matches the known lower bound.