基于并行算法变换的几乎混合时间分布式算法
New Distributed Algorithms in Almost Mixing Time via Transformations from Parallel Algorithms
摘要：
我们表明，许多经典的优化问题-如$(1\pmε)$ -近似最大流，最短路径，转运-可以计算在$\newcommand{\tmix}{τ_{\text{mix}}}\tmix(G)\cdot n^{o(1)}$轮的分布式消息传递，其中$\tmix(G)$是混合时间的网络图$G$。这推广了Ghaffari等人的结果。[PODC'17]，其主要结果是在CONGEST模型中的$\tmix(G)\cdot 2^{O(\sqrt{\log n \log\log n})}$轮中的分布式MST算法，适用于更广泛的一类优化问题。对于许多感兴趣的实际网络，例如，在对等或覆盖网络结构中，混合时间$\tmix(G)$较小，例如，多对数的在这些网络上，我们的算法绕过了Das Sarma等人的$\tildeΩ(\sqrt n+D)$下限。[STOC 11]，其适用于最坏情况的图，并适用于所有上述优化问题。对于所有的问题，除了MST，这是第一个分布式算法，需要$o(\sqrt n)$轮（非平凡）限制类的网络图。为了获得这些改进的分布式算法，我们的主要贡献是一个通用的转换，模拟任何工作效率的PRAM算法运行在$T$并行轮通过分布式算法运行在$T\cdot \tmix(G)\cdot 2^{O(\sqrt{\log n})}$轮。工作和时间效率的并行算法，所有上述问题的结合谢尔曼[FOCS'13，SODA'17]和彭和斯皮尔曼[STOC'14]的工作。因此，模拟这些并行算法使用我们的转换框架产生所需的分布式算法。我们的转换的核心技术组件是解决\emph{multi-commodity routing}的算法问题--也就是说，粗略地说，路由$n$数据包从一个给定的源到一个给定的目的地--在随机图。对于这个问题，我们得到一个...
英文摘要：
We show that many classical optimization problems --- such as $(1\pmε)$-approximate maximum flow, shortest path, and transshipment --- can be computed in $\newcommand{\tmix}{τ_{\text{mix}}}\tmix(G)\cdot n^{o(1)}$ rounds of distributed message passing, where $\tmix(G)$ 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 $\tmix(G)\cdot 2^{O(\sqrt{\log n \log\log n})}$ 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 $\tmix(G)$ is small, e.g., polylogarithmic. On these networks, our algorithms bypass the $\tildeΩ(\sqrt n+D)$ 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 $o(\sqrt n)$ 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 $T$ parallel rounds via a distributed algorithm running in $T\cdot \tmix(G)\cdot 2^{O(\sqrt{\log n})}$ 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 $n$ packets each from a given source to a given destination---in random graphs. For this problem, we obtain a...
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