Efficient Distributed Algorithms for Minimum Spanning Tree in Dense Graphs

M. Bateni, Morteza Monemizadeh, Kees Voorintholt
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Abstract

In recent years, the Massively Parallel Computation (MPC) model capturing the MapReduce framework has become the de facto standard model for large-scale data analysis, given the ubiquity of efficient and affordable cloud implementations. In this model, an input of size $m$ is initially distributed among $t$ machines, each with a local space of size $s$. Computation proceeds in synchronous rounds in which each machine performs arbitrary local computation on its data and then sends messages to other machines. In this paper, we study the Minimum Spanning Tree (MST) problem for dense graphs in the MPC model. We say a graph $G(V,\ E)$ is relatively dense if $m=\Theta(n^{1+c})$ where $n=\vert V\vert$ is the number of vertices, $m=\vert E\vert$ is the number of edges in this graph, and $0 < c\leq 1$. We develop the first work- and space-efficient MPC algorithm that with high probability computes an MST of $G$ using $\lceil\log\frac{c}{\epsilon}\rceil+1$ rounds of communication. As an MPC algorithm, our algorithm uses $t=O(n^{c-\epsilon})$ machines each one having local storage of size $s=O(n^{1+\epsilon})$ for any $0 < \epsilon\leq c$. Indeed, not only is this algorithm very simple and easy to implement, it also simultaneously achieves optimal total work, per-machine space, and number of rounds.
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密集图中最小生成树的高效分布式算法
近年来,大规模并行计算(MPC)模型捕获MapReduce框架已经成为大规模数据分析事实上的标准模型,考虑到高效和负担得起的云实现无处不在。在该模型中,大小为$m$的输入初始分布在$t$台机器上,每台机器的局部空间大小为$s$。计算以同步轮进行,每台机器对其数据执行任意本地计算,然后向其他机器发送消息。本文研究了MPC模型中密集图的最小生成树(MST)问题。我们说一个图$G(V,\ E)$是相对密集的,如果$m=\Theta(n^{1+c})$$n=\vert V\vert$是顶点的数量,$m=\vert E\vert$是这个图的边的数量,$0 < c\leq 1$。我们开发了第一个工作和空间效率高的MPC算法,该算法使用$\lceil\log\frac{c}{\epsilon}\rceil+1$轮通信以高概率计算出$G$的MST。作为MPC算法,我们的算法使用$t=O(n^{c-\epsilon})$机器,每台机器的本地存储大小为$s=O(n^{1+\epsilon})$,用于任何$0 < \epsilon\leq c$。实际上,该算法不仅非常简单且易于实现,而且还同时实现了最优的总工作量、每台机器的空间和轮数。
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