Planar Hop Spanners for Unit Disk Graphs
Résumé
The simplest model
of a wireless network graph is the Unit Disk Graph (UDG): an edge
exists in UDG if the Euclidean distance between its endpoints is
$\le 1$. The problem of constructing planar spanners of Unit Disk
Graphs with respect to the Euclidean distance has received
considerable attention from researchers in computational geometry
and ad-hoc wireless networks. In this paper, we present an algorithm
that, given a set $X$ of terminals in the plane, constructs a planar
hop spanner with constant stretch factor for the Unit Disk Graph
defined by $X$. Our algorithm improves on previous constructions in
the sense that (i) it ensures the planarity of the whole spanner
while previous algorithms ensure only the planarity of a backbone
subgraph; (ii) the hop stretch factor of our spanner is
significantly smaller.