In distributed computing, many papers try to evaluate the message complexity of a distributed system as a function of the number of nodes $n$n. But what about the cost of building the distributed system itself? Assuming that we want to reliably connect $n$n nodes, how does the total number of nodes of the network evolve with $n$n? Addressing such a question lies at the heart of achieving scalability in cloud computing. In this paper, we give the explicit description of a distributed system of which any two of the $n$n nodes, for any $n$n, remain connected (by a path of alive nodes and channels) with probability at least $\mu$μ, despite the very fact that (a) every other node or channel has an independent probability $\lambda$λ of failing, and (b) the number of channels connected to every node is physically bounded by a constant. We show however that if we also require any two of the $n$n nodes to maintain a balanced message throughput with a constant probability, then $O(n \log ^{1+\epsilon } n)$O(nlog1+εn) additional intermediary nodes are sufficient, where $\epsilon > 0$ε>0 is an arbitrarily small constant.

中文翻译：

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扩展可靠互连拓扑的成本

在分布式计算中，许多论文试图将分布式系统的消息复杂度评估为节点数量的函数 $n$n. 但是费用呢建造分布式系统本身？假设我们想要可靠地连接$n$n 节点，网络的节点总数如何随着 $n$n? 解决这样一个问题是在云计算中实现可扩展性的核心。在本文中，我们给出了一个分布式系统的明确描述，其中任意两个$n$n 节点，对于任何 $n$n, 保持联系 （通过活动节点和通道的路径）至少有概率 $\亩$μ，尽管 (a) 每个其他节点或通道都有一个独立的概率 $\lambda$λ失败，以及 (b) 连接到每个节点的通道数量在物理上受到一个常数的限制。然而，我们证明如果我们还需要任何两个$n$n 节点维护一个 平衡的消息吞吐量 以常数概率，那么 $O(n \log ^{1+\epsilon } n)$哦(n日志1+εn) 额外的中间节点就足够了，其中 $\epsilon > 0$ε>0 是一个任意小的常数。