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 optimal, Δ being the maximum degree of the input network. This is best-possible if NP ⊈ D TIME [nO(log log n)] and if the processors are limited to polynomial-time computation. We then show how to construct dominating sets which satisfy 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(log n) in the 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.

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