Localized Spanners for Wireless Networks

We present a new efficient localized algorithm to construct, for any given quasi-unit disk graph G=(V,E) and any e > 0, a (1+e)-spanner for G of maximum degree O(1) and total weight O(w(MST)), where w(MST) denotes the weight of a minimum spanning tree for V. We further show that similar localized techniques can be used to construct, for a given unit disk graph G = (V, E), a planar Cdel(1+e)(1+pi/2)-spanner for G of maximum degree O(1) and total weight O(w(MST)). Here Cdel denotes the stretch factor of the unit Delaunay triangulation for V. Both constructions can be completed in O(1) communication rounds, and require each node to know its own coordinates.