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 + √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 Ω( ^~ D + √n)-round complexity lower bound. The development of the algorithm entails two subresults of independent interest: (i) A (D + √n) · n ^{o (1)} -round distributed construction of a spanning tree of average stretch n ^{o (1)} . (ii) A (D + √n) · n ^{o (1)} -round distributed construction of an n ^{o (1)} -congestion approximator consisting of the cuts induced by O(log n) virtual trees. The distributed representation of the cut approximator allows for evaluation in (D + √n) · n ^{o (1)} rounds. All our algorithms make use of randomization and succeed with high probability.