[Paper Review] Angle-Monotone Graphs: Construction and Local Routing

Anna Lubiw, Debajyoti Mondal|arXiv (Cornell University)|Jan 19, 2018

Computational Geometry and Mesh Generation11 references3 citations

TL;DR

This paper introduces a construction of angle-monotone graphs with width $90^\circ$ and $O(n^2 \log \log n / \log n)$ edges for any point set in the plane, and presents a local routing algorithm that finds angle-monotone paths of width $(90^\circ + \alpha)$ with a routing ratio of at most $1/\cos(45^\circ + \alpha/2)$, achieving stretch factors from $\sqrt{2}$ to $2.613$. The method uses Steiner points and rotational symmetry to ensure generalized angle-monotonicity and efficient local path discovery.

ABSTRACT

A geometric graph in the plane is angle-monotone of width $γ$ if every pair of vertices is connected by an angle-monotone path of width $γ$, a path such that the angles of any two edges in the path differ by at most $γ$. 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 $α$, $0 < α< 45^\circ$, an angle-monotone graph of width $(90^\circ+α)$ with $O(\frac{n}α)$ edges. Furthermore, we give a local routing algorithm to find angle-monotone paths of width $(90^\circ+α)$ in these graphs. The routing ratio, which is the ratio of path length to Euclidean distance, is at most $1/\cos(45^\circ + \fracα{2})$, i.e., ranging from $\sqrt 2 \approx 1.414$ to $2.613$. For the special case $α= 30^\circ$, we obtain the $Θ_6$-graph and our routing algorithm achieves the known routing ratio 2 while finding angle-monotone paths of width $120^\circ$.

Motivation & Objective

• To construct sparse geometric graphs that are angle-monotone of width $90^\circ$ for any point set in the plane, without requiring planarity or Steiner points.
• To design a local routing algorithm that finds generalized angle-monotone paths of width $(90^\circ + \alpha)$ in these graphs with bounded stretch factor.
• To resolve the open question of whether every point set admits an angle-monotone graph with subquadratic edges and bounded width.
• To analyze the trade-off between path width, routing ratio, and graph size using Steiner points and rotational symmetry.

Proposed method

• Construct a base graph $G_{\text{pair}}$ using nearest neighbors and directional wedges to define potential paths.
• Introduce Steiner points along a sequence of shrinking triangles to ensure coverage of all points within a wedge, using trigonometric bounds on side length reduction.
• Use rotational symmetry by creating $t$ rotated copies of the Steiner graph $H$, each rotated by $360^\circ(i-1)/t$, to cover all angular directions.
• Define $\lambda$ as the minimum distance between non-overlapping parallel lines through any pair of points at angle $90^\circ + \gamma j/2$, which bounds the number of Steiner points via $O(\log(AC/\lambda))$.
• Apply a local routing rule that selects the next vertex based on proximity to the destination and angular alignment within a wedge of width $\gamma = 90^\circ + \alpha$.
• Prove the existence of angle-monotone paths via intersection analysis of path boundaries $P_{q,\theta_a}$ and $P_{t,b'}$, using Steiner point adjacency to bridge gaps.

Experimental results

Research questions

• RQ1 Can every point set in the plane admit an angle-monotone graph of width $90^\circ$ with subquadratic edges?
• RQ2 Is there a local routing algorithm that can find generalized angle-monotone paths of width $(90^\circ + \alpha)$ with bounded stretch factor?
• RQ3 What is the minimum width $\gamma$ such that every point set admits a planar angle-monotone graph of width $\gamma$?
• RQ4 Can angle-monotone graphs with $o(n^2)$ edges be constructed for any $\gamma > 0$ using Steiner points?

Key findings

• Every point set of size $n$ admits an angle-monotone graph of width $90^\circ$ with $O(n^2 \log \log n / \log n)$ edges, answering an open question from Dehkordi, Frati, and Gudmundsson.
• For any $\alpha \in (0^\circ, 45^\circ)$, a graph of width $(90^\circ + \alpha)$ with $O(n / \alpha)$ edges can be constructed, achieving a routing ratio of at most $1 / \cos(45^\circ + \alpha/2)$.
• The routing ratio ranges from $\sqrt{2} \approx 1.414$ (for $\alpha \to 0^\circ$) to $2.613$ (for $\alpha = 45^\circ$), with the latter matching the known bound for $\Theta_6$-graphs.
• For $\alpha = 30^\circ$, the construction yields the $\Theta_6$-graph and the routing algorithm achieves the known routing ratio of 2, confirming consistency with prior results.
• The use of Steiner points ensures that for any $\gamma > 0$, an angle-monotone graph of width $\gamma$ can be built with $O(n / \gamma \cdot \log(\mu / \lambda))$ edges, where $\mu$ is the maximum point distance and $\lambda$ is the minimum distance between non-overlapping parallel lines.
• The construction guarantees that for any two points $q$ and $t$, if $t$ lies within the wedge $W_{q,\theta_a}$, then an angle-monotone path of width $\gamma = \theta_a + \theta_c$ exists in the final graph $\mathcal{H}$, proven via path boundary intersections and Steiner point bridges.