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

Abstract

We show that many classical optimization problems --- such as (1±ϵ)(1\pm\epsilon)-approximate maximum flow, shortest path, and transshipment --- can be computed in τmix(G)no(1)\newcommand{\tmix}{{\tau_{\text{mix}}}}\tmix(G)\cdot n^{o(1)} rounds of distributed message passing, where \tmix(G)\tmix(G) is the mixing time of the network graph GG. This extends the result of Ghaffari et al.\ [PODC'17], whose main result is a distributed MST algorithm in \tmix(G)2O(lognloglogn)\tmix(G)\cdot 2^{O(\sqrt{\log n \log\log 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 \tmix(G)\tmix(G) is small, e.g., polylogarithmic. On these networks, our algorithms bypass the Ω~(n+D)\tilde\Omega(\sqrt 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)o(\sqrt 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 TT parallel rounds via a distributed algorithm running in T\tmix(G)2O(logn)T\cdot \tmix(G)\cdot 2^{O(\sqrt{\log 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 \emph{multi-commodity routing}---that is, roughly, routing nn packets each from a given source to a given destination---in random graphs. For this problem, we obtain a...

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