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 n1+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 n1+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 logΩ(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 n3/2) edges in O(ε-1) rounds, without any prior knowledge on the graph.