The Exact Spanning Ratio of the Parallelogram Delaunay Graph

TL;DR

This work determines the exact spanning ratio of parallelogram-based Delaunay graphs, introducing two degrees of freedom: the aspect ratio $A$ and the non-obtuse angle $\theta_0$ of the parallelogram. By adapting and unifying prior techniques through a non-orthonormal basis and an inductive, distance-ranked analysis complemented by a Crossing Lemma, the authors derive the exact bound $h(A,\theta_0)$ given by $\frac{\sqrt{2}\sqrt{1+A^2+2A\cos(\theta_0)+(A+\cos(\theta_0))\sqrt{1+A^2+2A\cos(\theta_0)}}}{\sin(\theta_0)}$, and construct point sets that realize this bound, proving tightness. The result subsumes known special cases: rectangle ($\theta_0=\frac{\pi}{2}$) and square ($A=1$), thereby making parallelograms the fifth convex shape with a proven exact bound for its Delaunay graph. The paper also introduces a robust methodological framework that may extend to other convex shapes and TD-like Delaunay graphs, potentially enabling further tight bounds in geometric spanner theory.

Abstract

Finding the exact spanning ratio of a Delaunay graph has been one of the longstanding open problems in Computational Geometry. Currently there are only four convex shapes for which the exact spanning ratio of their Delaunay graph is known: the equilateral triangle, the square, the regular hexagon and the rectangle. In this paper, we show the exact spanning ratio of the parallelogram Delaunay graph, making the parallelogram the fifth convex shape for which an exact bound is known. The worst-case spanning ratio is exactly $$\frac{\sqrt{2}\sqrt{1+A^2+2A\cos(θ_0)+(A+\cos(θ_0))\sqrt{1+A^2+2A\cos(θ_0)}}}{\sin(θ_0)} .$$ where $A$ is the aspect ratio and $θ_0$ is the non-obtuse angle of the parallelogram. Moreover, we show how to construct a parallelogram Delaunay graph whose spanning ratio matches the above mentioned spanning ratio.

Figures (9)

• Figure 1: The four scenarios considered about the slope between $a$ and $b$.
• Figure 2: Illustration of the triangles intersecting $ab$.
• Figure 3: Illustration of the length of $d_{\hat{P}_{uw}}(u,w)$ in Scenario 2.
• Figure 4: Illustration of $P(a,b)$ in Scenarios 1 and 3.
• Figure 5: Example showing the positioning of $\hat{P}_1$ if $a$ is on the S side in Scenario 1 on the left $(a)$, and Scenario 2 on the right $(b)$.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Definition 3
• Lemma 1
• proof
• Lemma 2
• proof
• Definition 4
• Lemma 3
• proof