Construction and Local Routing for Angle-Monotone Graphs

Abstract

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 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 $\varTheta _6$-graph and our routing algorithm achieves the known routing ratio 2 while finding angle-monotone paths of width $120^\circ$.

This work is partially supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).