On Geometric Spanners of Euclidean and Unit Disk Graphs

Abstract

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.