Fast distributed algorithms for (weakly) connected dominating sets and linear-size skeletons

Motivated by routing issues in ad hoc networks, we present polylogarithmic-time distributed algorithms for two problems. Given a network, we first show how to compute connected and weakly connected dominating sets whose size is at most O(logΔ) times the optimum, Δ being the maximum degree of the input network. This is best-possible if NP⊈DTIME[nO(loglogn)] and if the processors are required to run in polynomial-time. We then show how to construct dominating sets that have the above properties, as well as the “low stretch” property that any two adjacent nodes in the network have their dominators at a distance of at most O(logn) in the output network. (Given a dominating set S, a dominator of a vertex u is any v∈S such that the distance between u and v is at most one.) We also show our time bounds to be essentially optimal.