Motivated by the increasing need for fast processing of large-scale graphs, we study a number of fundamental graph problems in a message-passing model for distributed computing, called $k$-machine model, where we have $k$ machines that jointly perform computations on $n$-node graphs. The graph is assumed to be partitioned in a balanced fashion among the $k$ machines, a common implementation in many real-world systems. Communication is point-to-point via bandwidth-constrained links, and the goal is to minimize the round complexity, i.e., the number of communication rounds required to finish a computation. We present a generic methodology that allows to obtain efficient algorithms in the $k$-machine model using distributed algorithms for the classical CONGEST model of distributed computing. Using this methodology, we obtain algorithms for various fundamental graph problems such as connectivity, minimum spanning trees, shortest paths, maximal independent sets, and finding subgraphs, showing that many of these problems can be solved in $\tilde{O}(n/k)$ rounds; this shows that one can achieve speedup nearly linear in $k$. To complement our upper bounds, we present lower bounds on the round complexity that quantify the fundamental limitations of solving graph problems distributively. We first show a lower bound of $\Omega(n/k)$ rounds for computing a spanning tree of the input graph. This result implies the same bound for other fundamental problems such as computing a minimum spanning tree, breadth-first tree, or shortest paths tree. We also show a $\tilde \Omega(n/k^2)$ lower bound for connectivity, spanning tree verification and other related problems. The latter lower bounds follow from the development and application of novel results in a random-partition variant of the classical communication complexity model.
翻译:由于对快速处理大比例图的需求日益增加,我们研究了一个分布式计算的信息传递模式中的一些基本图表问题,称为$k$-机器模型,我们拥有一个叫做$k$-机器模型的元美元机器,在美元新元图表中共同进行计算。该图表假定以平衡的方式在美元机器中进行分割,这是许多现实世界系统中的一种共同实施。通信是通过带宽限制的链接的点对点,目标是将完成计算所需的信息传递模式的周期复杂性降至最低,即完成一个计算所需的通信周期数量。我们提出了一个通用方法,以便利用经典的 CONEST 计算模式的分布式算法,在美元机器模型中获得高效的算法。我们用这种方法,为各种基本图表问题,例如连接、最小的树圈、最短的路径、最高级的独立设置和找到分数模型,表明这些问题中的很多都可以在$ltildel/O}最短的轨道中解决。(n/k)美元基本树流路的周期;这显示,可以实现最快速的计算,在美元模型中显示一个最低的直径直径直值的计算, 将显示我们直径直值的硬的硬的路径的计算结果。