Finding small dominating sets in stationless mobile packet radio networks Abhay K. Parekh.

We model a snapshot of a mobile packet radio network as an undirected graph. The nodes of the graph are processors, that communicate along their incident edges by broadcast. The radios do not know the size of the network, and start out with no topological information. Our goal is to select a small subset of the nodes of the graph so that every other node is adjacent to at least one node in the subset. Such a subset is called a dominating set. Finding the smallest dominating set of a graph is known to be NP-hard. An event driven distributed algorithm is presented that picks a dominating set of size at most N /2M +1, for any network with N nodes and M links. Since conmmunication is by broadcast, the messages cannot be exclusively routed on subgraphs such as spanning trees. For the class of 6-regular graphs of diameter d, it takes O(N6d) messages to learn the entire topology of the graph. We show that for graphs of regular degree, 6, our algorithm has communication complexity of O(IDgl63 + N6), where Dg is the dominating set picked by the algorithm. The time complexity is O(IDgl). Thus the algorithm is efficient for graphs with diameter greater than 3.