  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.
