Collective Tree Spanners for Unit Disk Graphs with Applications

In this paper, we establish a novel balanced separator theorem for Unit Disk Graphs (UDGs), which mimics the well-known Lipton and Tarjanʹs planar balanced shortest paths separator theorem. We prove that, in any n-vertex UDG G, one can find two hop-shortest paths P ( s , x ) and P ( s , y ) such that the removal of the 3-hop-neighborhood of these paths (i.e., N G 3 [ P ( s , x ) ∪ P ( s , y ) ] ) from G leaves no connected component with more than 2 / 3 n vertices. This new balanced shortest-paths—3- hop-neighborhood separator theorem allows us to build, for any n-vertex UDG G, a system T ( G ) of at most 2 log 3 2 n + 2 spanning trees of G such that, for any two vertices x and y of G, there exists a tree T in T ( G ) with d T ( x , y ) ⩽ 3 ⋅ d G ( x , y ) + 12 . That is, the distances in any UDG can be approximately represented by the distances in at most 2 log 3 2 n + 2 of its spanning trees. Using these results, we propose a new compact and low delay routing labeling scheme for UDGs.