# New Distributed Algorithms in Almost Mixing Time via Transformations from Parallel Algorithms

We show that many classical optimization problems --- such as (1±ϵ)-approximate maximum flow, shortest path, and transshipment --- can be computed in (G)· n^o(1) rounds of distributed message passing, where (G) is the mixing time of the network graph G. This extends the result of Ghaffari et al.[PODC'17], whose main result is a distributed MST algorithm in (G)· 2^O(√( n n)) rounds in the CONGEST model, to a much wider class of optimization problems. For many practical networks of interest, e.g., peer-to-peer or overlay network structures, the mixing time (G) is small, e.g., polylogarithmic. On these networks, our algorithms bypass the Ω̃(√(n)+D) lower bound of Das Sarma et al. [STOC'11], which applies for worst-case graphs and applies to all of the above optimization problems. For all of the problems except MST, this is the first distributed algorithm which takes o(√(n)) rounds on a (nontrivial) restricted class of network graphs. Towards deriving these improved distributed algorithms, our main contribution is a general transformation that simulates any work-efficient PRAM algorithm running in T parallel rounds via a distributed algorithm running in T·(G)· 2^O(√( n)) rounds. Work- and time-efficient parallel algorithms for all of the aforementioned problems follow by combining the work of Sherman [FOCS'13, SODA'17] and Peng and Spielman [STOC'14]. Thus, simulating these parallel algorithms using our transformation framework produces the desired distributed algorithms. The core technical component of our transformation is the algorithmic problem of solving multi-commodity routing---that is, roughly, routing n packets each from a given source to a given destination---in random graphs. For this problem, we obtain a...

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