Near-Optimal Distributed Maximum Flow

We present a near-optimal distributed algorithm for (1 + o(1))-approximation of single-commodity maximum flow in undirected weighted networks that runs in (D + root n) . n(o(1)) communication rounds in the CONGEST model. Here, n and D denote the number of nodes and the network diameter, respectively. This is the first improvement over the trivial bound of O(n(2)), and it nearly matches the (Omega) over tilde (D + root n)-round complexity lower bound. The development of the algorithm entails two subresults of independent interest: (i) A (D + root n) . n(o(1))-round distributed construction of a spanning tree of average stretch n(o(1)). (ii) A (D + root n) . n(o(1))-round distributed construction of an n(o(1))-congestion approximator consisting of the cuts induced by O(logn) virtual trees. The distributed representation of the cut approximator allows for evaluation in (D + root n) n(o(1)) rounds. All our algorithms make use of randomization and succeed with high probability.