On the Locality of Distributed Sparse Spanner Construction
Résumé
The paper presents a deterministic distributed algorithm that, given k>0, 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 and provide no performance guarantees, or perform in log^{Omega(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 eps>0, constructs a (1+eps,2)-spanner of O(eps^{-2} n^{3/2}) edges in O(eps^{-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 in {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 + eps} edges in at most n^{eps} expected rounds must stretch some distances by an additive factor of n^{Omega(eps)}. 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 polynomial number of rounds in expectation.