故障网络的路由复杂性
Routing Complexity of Faulty Networks
摘要：
分布式计算中的一个基本问题是如何在故障网络中有效地执行路由，其中每个链路都有一定的概率发生故障。本文研究了在网络中有效找到路径的能力丧失之前，故障概率可以有多大。我们的主要结果表明，严格的上限和下界的故障概率，允许路由，无论是超立方体和$d-$维网格。我们使用渗流理论的工具表明，在$d-$维网格，一旦一个巨大的组件出现-有效的路由是可能的。当考虑超立方体时，观察到不同的行为。在超立方体中，存在一系列的故障概率，其中短路径以高概率存在，但找到它们必须涉及基本上查询整个网络。因此，超立方体的路由复杂性表现出渐近相变。关于路由复杂性的临界概率位于与关于连通性的临界概率不同的位置。最后，我们表明，一个Oracle访问链接（相对于本地路由）可能会显着降低路由问题的复杂性。我们证明了这一事实，通过提供紧密的上限和下界的随机图$G_{n,p}$中的路由的复杂性。
英文摘要：
One of the fundamental problems in distributed computing is how to efficiently perform routing in a faulty network in which each link fails with some probability. This paper investigates how big the failure probability can be, before the capability to efficiently find a path in the network is lost. Our main results show tight upper and lower bounds for the failure probability which permits routing, both for the hypercube and for the $d-$dimensional mesh. We use tools from percolation theory to show that in the $d-$dimensional mesh, once a giant component appears -- efficient routing is possible. A different behavior is observed when the hypercube is considered. In the hypercube there is a range of failure probabilities in which short paths exist with high probability, yet finding them must involve querying essentially the entire network. Thus the routing complexity of the hypercube shows an asymptotic phase transition. The critical probability with respect to routing complexity lies in a different location then that of the critical probability with respect to connectivity. Finally we show that an oracle access to links (as opposed to local routing) may reduce significantly the complexity of the routing problem. We demonstrate this fact by providing tight upper and lower bounds for the complexity of routing in the random graph $G_{n,p}$.
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