Deterministic Distributed Construction of $T$-Dominating Sets in Time $T$

A k-dominating set is a set D of nodes of a graph such that, for each node v, there exists a node w ∈ D at distance at most k from v. Our aim is the deterministic distributed construction of small T -dominating sets in time T in networks modeled as undirected n-node graphs and under the LOCAL communication model. For any positive integer T , if b is the size of a pairwise disjoint collection of balls of radii at least T in a graph, then b is an obvious lower bound on the size of a T -dominating set. Our first result shows that, even on rings, it is impossible to construct a T -dominating set of size s asymptotically b (i.e., such that s/b→ 1) in time T . In the range of time T ∈ Θ(log∗ n), the size of a T -dominating set turns out to be very sensitive to multiplicative constants in running time. Indeed, it follows from [9], that for time T = γ log∗ n with large constant γ, it is possible to construct a T -dominating set whose size is a small fraction of n. By contrast, we show that, for time T = α log∗ n for small constant α, the size of a T -dominating set must be a large fraction of n. Finally, when T ∈ o(log∗ n), the above lower bound implies that, for any constant x < 1, it is impossible to construct a T -dominating set of size smaller than xn, even on rings. On the positive side, we provide an algorithm that constructs a T -dominating set of size n − Θ(T ) on all graphs.