On Geometric Spanners of Euclidean and Unit Disk Graphs

We consider the problem of constructing bounded-degree planar geometric spanners of Euclidean and unit-disk graphs. It is well known that the Delaunay subgraph is a planar geometric spanner with stretch factor $C_{delapprox 2.42$; however, its degree may not be bounded. Our first result is a very simple linear time algorithm for constructing a subgraph of the Delaunay graph with stretch factor $ ho =1+2pi(kcos{frac{pi{k)^{-1$ and degree bounded by $k$, for any integer parameter $kgeq 14$. This result immediately implies an algorithm for constructing a planar geometric spanner of a Euclidean graph with stretch factor $ ho cdot C_{del$ and degree bounded by $k$, for any integer parameter $kgeq 14$. Moreover, the resulting spanner contains a Euclidean Minimum Spanning Tree (EMST) as a subgraph. Our second contribution lies in developing the structural results necessary to transfer our analysis and algorithm from Euclidean graphs to unit disk graphs, the usual model for wireless ad-hoc networks. We obtain a very simple distributed, {em strictly-localized algorithm that, given a unit disk graph embedded in the plane, constructs a geometric spanner with the above stretch factor and degree bound, and also containing an EMST as a subgraph. The obtained results dramatically improve the previous results in all aspects, as shown in the paper.