On Geometric Spanners of Euclidean Graphs and their Applications in Wireless Networks

We consider the problem of constructing a bounded-degree planar geometric spanner for a unit disk graph modeling a wireless network. The related problem of constructing a planar geometric spanner of a Euclidean graph has been extensively studied in the literature. It is well known that the Delaunay subgraph is a planar geometric spanner with stretch factor ; however, its degree may not be bounded. Significant work has been done on developing algorithms for constructing bounded-degree planar geometric spanners of Euclidean graphs. Our first result presents a very simple linear time algorithm for constructing a subgraph of the Delaunay graph with stretch factor ! " # $ % and degree bounded by , for any integer parameter '& ( . This result immediately implies an algorithm for constructing a planar geometric spanner of a Euclidean graph with stretch factor *)+ and degree bounded by , for any integer parameter ,& ( . Our second contribution lies in developing the structural results necessary to transfer our analysis and algorithm from Euclidean graphs to unit disk graphs. We obtain a very simple strictly-localized algorithm that, given a unit disk graph embedded in the plane, constructs a planar geometric spanner with the above stretch factor and degree bound. The two results dramatically improve the previous results in all aspects, as shown in the paper.