On the locality of distributed sparse spanner construction

The paper presents a deterministic distributed algorithm that, given k ≥ 1, constructs in k rounds a (2k-1, 0)-spanner of O(kn^1+1/k) edges for every n-node unweighted graph, (If n is not available to the nodes, then our algorithm executes in 3k - 2 rounds, and still returns a (2k - 1, 0)-spanner with O(kn^1+1/k) edges.) Previous distributed solutions achieving such optimal stretch-size trade-off either make use of randomization providing performance guarantees in expectation only, or perform in log ^Ω(1) n rounds, and all require a priori knowledge of n. Based on this algorithm, we propose a second deterministic distributed algorithm that, for every ∈ > 0, constructs a (1 + ∈, 2)-spanner of O(∈^-1n^3/2) edges in O(∈^-1) rounds, without any prior knowledge on the graph. Our algorithms are complemented with lower bounds, which hold even under the assumption that n is known to the nodes. It is shown that any (randomized) distributed algorithm requires k rounds in expectation to compute a (2k - 1, 0)-spanner of o(n^1+1/(k-1)) edges for k ∈ {2, 3, 5}. It is also shown that for every k > 1, any (randomized) distributed algorithm that constructs a spanner with fewer than n^1+1/k+∈ edges in at most n^∈ expected rounds must stretch some distances by an additive factor of n ^Ω(∈). In other words, while additive stretched spanners with O(n^1+1/k) edges may exist, e.g., for k = 2, 3, they cannot be computed distributively in a sub-polynomial number of rounds in expectation.