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The Tight Spanning Ratio of the Rectangle Delaunay Triangulation

Andrè van Renssen, Yuan Sha, Yucheng Sun, Sampson Wong

TL;DR

This paper studies the spanning ratio of rectangle-based generalized Delaunay triangulations, introducing rectangle Delaunay triangulations constructed from axis-aligned rectangle homothets. It develops an inductive geometric framework using concepts such as potential, inductive rectangles, and maximal high/low paths to bound the length of shortest paths between vertex pairs. The main result provides a tight upper bound on the spanning ratio for rectangles with aspect ratio $A$: $t \le \sqrt{2}\sqrt{1 + A^2 + A\sqrt{1 + A^2}}$, matching the known lower bound. This extends tight bounds to all rectangles, clarifies the role of orientation, and suggests future work on local routing algorithms within rectangle Delaunay triangulations.

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.

The Tight Spanning Ratio of the Rectangle Delaunay Triangulation

TL;DR

This paper studies the spanning ratio of rectangle-based generalized Delaunay triangulations, introducing rectangle Delaunay triangulations constructed from axis-aligned rectangle homothets. It develops an inductive geometric framework using concepts such as potential, inductive rectangles, and maximal high/low paths to bound the length of shortest paths between vertex pairs. The main result provides a tight upper bound on the spanning ratio for rectangles with aspect ratio : , matching the known lower bound. This extends tight bounds to all rectangles, clarifies the role of orientation, and suggests future work on local routing algorithms within rectangle Delaunay triangulations.

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 have spanning ratio at most , which matches the known lower bound.
Paper Structure (4 sections, 7 theorems, 6 equations, 2 figures)

This paper contains 4 sections, 7 theorems, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bounding the Spanning Ratio
  4. Conclusion

Key Result

Lemma 4

If $R(u,v)$ is empty and $(u,v)$ is not an edge in the rectangle Delaunay triangulation, then $R_1$ has potential. Furthermore, for any $1 \le i < k$, if $R_i$ has potential but is not inductive (i.e., $(l_{i}, h_{i})$ is steep), then $R_{i+1}$ has potential.

Figures (2)

  • Figure 1: The triangles intersecting $uv$ and their associated rectangles and $h_i$ and $l_i$.
  • Figure 5: An example of a maximal high path (thick edges). The other edges of the triangles are shown using dashed line segments.

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 4
  • Lemma 5
  • Definition 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Theorem 10
  • ...and 1 more