Continuous Yao Graphs

TL;DR

This paper introduces continuous Yao graphs, a variation of Yao graphs, and analyzes their spanning properties, showing they are spanners for certain angles but not others, with implications for geometric network design.

Contribution

The paper defines continuous Yao graphs and proves they are spanners for angles up to 2π/3 using a new algebraic technique, expanding understanding of geometric spanners.

Findings

cY(θ) is a spanner for θ ≤ 2π/3

cY(π) is not a spanner

cY(θ) may be disconnected for θ > π

Abstract

In this paper, we introduce a variation of the well-studied Yao graphs. Given a set of points $S\subset \mathbb{R}^2$ and an angle $0 < \theta \leq 2\pi$, we define the continuous Yao graph $cY(\theta)$ with vertex set $S$ and angle $\theta$ as follows. For each $p,q\in S$, we add an edge from $p$ to $q$ in $cY(\theta)$ if there exists a cone with apex $p$ and aperture $\theta$ such that $q$ is the closest point to $p$ inside this cone. We study the spanning ratio of $cY(\theta)$ for different values of $\theta$. Using a new algebraic technique, we show that $cY(\theta)$ is a spanner when $\theta \leq 2\pi /3$. We believe that this technique may be of independent interest. We also show that $cY(\pi)$ is not a spanner, and that $cY(\theta)$ may be disconnected for $\theta > \pi$.