This work concerns with proving space lower bounds for graph problems in the streaming model. It is known that computing the length of shortest path between two nodes in the streaming model requires $\Omega(n)$ space, where $n$ is the number of nodes in the graph. We study the problem of finding the depth of a given node in a rooted tree in the streaming model. For this problem we prove a tight single pass space lower bound and a multipass space lower bound. As this is a special case of computing shortest paths on graphs, the above lower bounds also apply to the shortest path problem in the streaming model. The results are obtained by using known communication complexity lower bounds or by constructing hard instances for the problem. Additionally, we apply the techniques used in proving the above lower bound results to prove space lower bounds (single and multipass) for other graph problems like finding min $s-t$ cut, detecting negative weight cycle and finding whether two nodes lie in the same strongly connected component.

中文翻译：

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图流问题的空间下界

这项工作涉及证明流模型中图形问题的空间下界。已知计算流模型中两个节点之间的最短路径的长度需要$ \ Omega（n）$空间，其中$ n $是图中的节点数。我们研究了在流模型中找到根树中给定节点深度的问题。对于这个问题，我们证明了紧密的单遍空间下界和多遍空间下界。由于这是计算图上最短路径的特殊情况，因此上述下限也适用于流模型中的最短路径问题。通过使用已知的通信复杂性下限或通过构造问题的硬实例来获得结果。另外，