Weakly-connected dominating sets and sparse spanners in wireless ad hoc networks

A set S is dominating if each node in the graph G = (V, E) is either in S or adjacent to at least one of the nodes in S. The subgraph weakly induced by S is the graph G' = (V, E') such that each edge in E' has at least one end point in S. The set S is a weakly-connected dominating set (WCDS) of G if S is dominating and G' is connected G' is a sparse spanner if it has linear edges. In this paper, we present two distributed algorithms for finding a WCDS in O(n) time. The first algorithm has an approximation ratio of 5, and requires O(n log n) messages. The second algorithm has a larger approximation ratio, but it requires only O(n) messages. The graph G' generated by the second algorithm forms a sparse spanner with a topological dilation of 3, and a geometric dilation of 6.