Generalized Sweeping Line Spanners

TL;DR

This work introduces sweeping line graphs, a broad generalization of Θ-graphs, and proves they are geometric $t$-spanners of the complete graph and of visibility graphs under line-segment and polygonal obstacle constraints. A unified inductive framework shows the same spanning bound $t = \frac{1}{\cos(\frac{\theta}{2} + \gamma) - \sin\theta}$ for $k \ge 7$ and $\gamma \in [0, \frac{\pi - 3\theta}{2})$ across unconstrained, constrained, and polygonal-obstacle settings, with a corresponding $1$-local routing algorithm achieving a path length of at most $t|uw|$. The constrained and polygonal extensions rely on adapted lemmas (e.g., convex chains and visibility edges) and demonstrate that obstacle considerations can be integrated into the same inductive proof structure. This provides a general, potentially reusable approach for proving spanner properties of geometric graphs in obstacle-rich environments, with implications for motion planning and efficient network design.

Abstract

We present sweeping line graphs, a generalization of $Θ$-graphs. We show that these graphs are spanners of the complete graph, as well as of the visibility graph when line segment constraints or polygonal obstacles are considered. Our proofs use general inductive arguments to make the step to the constrained setting. These same arguments could apply to other spanner constructions in the unconstrained setting, removing the need to find separate proofs that they are spanning in the constrained and polygonal obstacle settings.

Figures (12)

• Figure 1: The plane around $u$ is split into 10 cones.
• Figure 2: Vertex $v$ is the vertex with the projection closest to $u$.
• Figure 3: The sweeping line of a cone in a $\Theta$-graph. The sweeping line is a thick black segment inside the cone and grey dotted outside, as vertices outside the cone as ignored.
• Figure 4: The sweeping line of a cone in the sweeping line graph. For comparison to $\Theta$-graphs, the line through $u$ perpendicular to the sweeping line is shown in red.
• Figure 5: Defining the point $x_{\gamma}$ for any vertex $x$.

Theorems & Definitions (32)

• Definition 1: Sweeping line graph
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6