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

We show that many classical optimization problems --- such as -approximate maximum flow, shortest path, and transshipment --- can be computed in rounds of distributed message passing, where is the mixing time of the network graph . This extends the result of Ghaffari et al.\ [PODC'17], whose main result is a distributed MST algorithm in 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 is small, e.g., polylogarithmic. On these networks, our algorithms bypass the 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 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 parallel rounds via a distributed algorithm running in 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 \emph{multi-commodity routing}---that is, roughly, routing packets each from a given source to a given destination---in random graphs. For this problem, we obtain a...
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