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(k n^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(k n^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 ε > 0, constructs a (1+ε,2)-spanner of O(ε^-1 n^3/2) edges in O(ε^-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 + ε} edges in at most n^ε expected rounds must stretch some distances by an additive factor of n^Ω(ε). 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.