Localized Construction of Bounded Degree and Planar Spanner for Wireless Ad Hoc Networks

We propose a novel localized algorithm that constructs a bounded degree and planar spanner for wireless ad hoc networks modeled by unit disk graph (UDG). Every node only has to know its 2-hop neighbors to find the edges in this new structure. Our method applies the Yao structure on the local Delaunay graph [1] in an ordering that are computed locally. This new structure has the following attractive properties: (1) it is a planar graph; (2) its node degree is bounded from above by a positive constant $$19+\lceil\frac{2\pi}{\alpha}\rceil$$; (3) it is a t-spanner (given any two nodes u and v, there is a path connecting them in the structure such that its length is no more than $$t\leq {\rm max}\{\frac{\pi}{2},\pi {\rm sin}\frac{\alpha}{2}+1\} $$ · Cdel times of the shortest path in the unit disk graph); (4) it can be constructed locally and is easy to maintain when the nodes move around; (5) moreover, we show that the total communication cost is O(n log n) bits, where n is the number of wireless nodes, and the computation cost of each node is at most O(d log d), where d is its 2-hop neighbors in the original unit disk graph. Here Cdel is the spanning ratio of the Delaunay triangulation, which is at most $$\frac{4\sqrt{3}}{9}\pi$$. And the adjustable parameter α satisfies 0 < α ≤ π/3.