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

摘要

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 + [2π/α]; (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{π/2, π sin α/2 + 1} · C_(del) 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 C_(del) is the spanning ratio of the Delaunay triangulation, which is at most (4(3)~(1/2)))/9π. And the adjustable parameter π satisfies 0 < α < π/3.