Generalized Sweeping Line Spanners

TL;DR

This paper introduces sweeping line graphs, a generalized class of spanners that maintain their spanning properties even in constrained environments with obstacles, simplifying proofs across different settings.

Contribution

It generalizes $ heta$-graphs to sweeping line graphs and provides unified proofs for their spanner properties in both unconstrained and constrained environments.

Findings

Sweeping line graphs are spanners of complete and visibility graphs.

Unified inductive proofs apply to constrained and unconstrained settings.

The approach simplifies establishing spanner properties in various environments.

Abstract

We present sweeping line graphs, a generalization of $\Theta$-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.