Fast Deterministic Constructions of Linear-Size Spanners and Skeletons

In the distributed setting, the only existing constructions of sparse skeletons, (i.e., subgraphs with O(n) edges) either use randomization or large messages, or require Ω(D) time, where D is the hop-diameter of the input graph G. We devise the first deterministic distributed algorithm in the CONGEST model (i.e., uses small messages) for constructing linear-size skeletons in time 2O(√ log n·log log n). We can also compute a linear-size spanner with stretch polylog(n) in low deterministic polynomial time, i.e., O(nρ) for an arbitrarily small constant ρ > 0, in the CONGEST model. Yet another algorithm that we devise runs in O(log n) κ−1time, for a parameter κ = 1, 2, . . . , and constructs an O(log n) κ−1spanner with O(n 1+ 1κ ) edges. All our distributed algorithms are lightweight from the computational perspective, i.e., none of them employs any heavy computations.