Construction and Local Routing for Angle-Monotone Graphs

A geometric graph in the plane is angle-monotone of width $\gamma$ if every pair of vertices is connected by an angle-monotone path of width $\gamma$, a path such that the angles of any two edges in the path differ by at most $\gamma$. Angle-monotone graphs have good spanning properties. We prove that every point set in the plane admits an angle-monotone graph of width $90^\circ$, hence with spanning ratio $\sqrt 2$, and a subquadratic number of edges. This answers an open question posed by Dehkordi, Frati and Gudmundsson. We show how to construct, for any point set of size $n$ and any angle $\alpha$, $0 < \alpha < 45^\circ$, an angle-monotone graph of width $(90^\circ+\alpha)$ with $O(\frac{n}{\alpha})$ edges. Furthermore, we give a local routing algorithm to find angle-monotone paths of width $(90^\circ+\alpha)$ in these graphs. The \emph{routing ratio}, which is the ratio of path length to Euclidean distance, is at most $1/\cos(45^\circ + \frac{\alpha}{2})$, i.e., ranging from $\sqrt 2 \approx 1.414$ to $2.613$. For the special case $\alpha = 30^\circ$, we obtain the full-$\Theta_6$-graph and our routing algorithm achieves the known routing ratio 2 while finding angle-monotone paths of width $120^\circ$.