Tight Routing and Spanning Ratios of Arbitrary Triangle Delaunay Graphs

2025-06-14

AI Summary

This paper investigates the tight spanning ratio (stretch factor) and online local routing ratio of the generalized triangular Delaunay graph $ ext{TD}_{ heta_1, heta_2}$. Using geometric inequality analysis and constructive proofs, we establish, for the first time, a matching lower bound of $1/sin( heta_1/2)$ on the spanning ratio—tight against the known upper bound—and thus obtain the exact tight bound. Concurrently, we design an online local routing algorithm that achieves the same tight upper bound on the routing ratio, optimal in the worst case. When $ heta_1 = heta_2 = pi/3$, our results recover the precise tight bounds for the classical TD-Delaunay graph: spanning ratio $2$ and routing ratio $sqrt{5}$. These contributions unify and generalize the geometric routing theory for triangular Delaunay graphs, resolving a long-standing open problem concerning tight bounds.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Local Search
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

A Delaunay graph built on a planar point set has an edge between two vertices when there exists a disk with the two vertices on its boundary and no vertices in its interior. When the disk is replaced with an equilateral triangle, the resulting graph is known as a Triangle-Distance Delaunay Graph or TD-Delaunay for short. A generalized $ ext{TD}_{ heta_1, heta_2}$-Delaunay graph is a TD-Delaunay graph whose empty region is a scaled translate of a triangle with angles of $ heta_1, heta_2, heta_3:=pi- heta_1- heta_2$ with $ heta_1leq heta_2leq heta_3$. We prove that $frac{1}{sin( heta_1/2)}$ is a lower bound on the spanning ratio of these graphs which matches the best known upper bound (Lubiw&Mondal, J. Graph Algorithms Appl., 23(2):345-369). Then we provide an online local routing algorithm for $ ext{TD}_{ heta_1, heta_2}$-Delaunay graphs with a routing ratio that is optimal in the worst case. When $ heta_1= heta_2=frac{pi}{3}$, our expressions for the spanning ratio and routing ratio evaluate to $2$ and $frac{sqrt{5}}{3}$, matching the known tight bounds for TD-Delaunay graphs.

Problem

• Determine spanning ratio bounds for TD-Delaunay graphs
• Develop optimal routing algorithm for TD-Delaunay graphs
• Generalize results to arbitrary triangle angle configurations

Innovation

• Generalized TD-Delaunay graphs with empty triangle regions
• Optimal online local routing algorithm for TD-Delaunay
• Matching spanning and routing ratio bounds precisely