Deterministic Distributed Sparse and Ultra-Sparse Spanners and Connectivity Certificates

This paper presents efficient distributed algorithms for a number of fundamental problems in the area of graph sparsification: • We provide the first deterministic distributed algorithm that computes an ultra-sparse spanner in polylog(n) rounds in weighted graphs. Concretely, our algorithm outputs a spanning subgraph with only n + o (n) edges in which the pairwise distances are stretched by a factor of at most O(logn · 2O(log* n)). • We provide a polylog(n)-round deterministic distributed algorithm that computes a spanner with stretch (2k - 1) and O(nk + n1+1/k log k) edges in unweighted graphs and with O(n1+1/k k) edges in weighted graphs. • We present the first polylog(n) round randomized distributed algorithm that computes a sparse connectivity certificate. For an n-node graph G, a certificate for connectivity k is a spanning subgraph H that is k-edge-connected if and only if G is k-edge-connected, and this subgraph H is called sparse if it has O(nk) edges. Our algorithm achieves a sparsity of (1 + o (1))nk edges, which is within a 2(1 + o(1)) factor of the best possible.