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 [21] 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 + 2p/a (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 = max(p/2, psina/2 +1) Cdel times of the shortest path in UDG); (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), 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 4 v3/9 p. And the adjustable parameter a satisfies 0 < a < p/3. In addition, experiments are conducted to show this topology is efficient in practice, compared with other well-known topologies used in wireless ad hoc networks.Previously, only centralized method [5] of constructing bounded degree planar spanner is known, with degree bound 27 and spanning ratio t 10.02. The distributed implementation of their centralized method takes O(n2) communications in the worst case. No localized methods were known previously for constructing bounded degree planar spanner.