We consider the message complexity of verifying whether a given subgraph of the communication network forms a tree with specific properties both in the KT-$\rho$ (nodes know their $\rho$-hop neighborhood, including node IDs) and the KT-$0$ (nodes do not have this knowledge) models. We develop a rather general framework that helps in establishing tight lower bounds for various tree verification problems. We also consider two different verification requirements: namely that every node detects in the case the input is incorrect, as well as the requirement that at least one node detects. The results are stronger than previous ones in the sense that we assume that each node knows the number $n$ of nodes in the graph (in some cases) or an $\alpha$ approximation of $n$ (in other cases). For spanning tree verification, we show that the message complexity inherently depends on the quality of the given approximation of $n$: We show a tight lower bound of $\Omega(n^2)$ for the case $\alpha \ge \sqrt{2}$ and a much better upper bound (i.e., $O(n \log n)$) when nodes are given a tighter approximation. On the other hand, our framework also yields an $\Omega(n^2)$ lower bound on the message complexity of verifying a minimum spanning tree (MST), which reveals a polynomial separation between ST verification and MST verification. This result holds for randomized algorithms with perfect knowledge of the network size, and even when just one node detects illegal inputs, thus improving over the work of Kor, Korman, and Peleg (2013). For verifying a $d$-approximate BFS tree, we show that the same lower bound holds even if nodes know $n$ exactly, however, the lower bound is sensitive to $d$, which is the stretch parameter.

翻译：我们研究了在KT-$\rho$（节点知晓其$\rho$跳邻域信息，包括节点ID）和KT-$0$（节点不具备此知识）两种模型下，验证通信网络的给定子图是否构成具有特定性质的树所需的消息复杂度。我们建立了一个较为通用的理论框架，该框架有助于为各类树验证问题建立紧的下界。同时，我们考虑了两种不同的验证要求：即要求每个节点在输入错误时都能检测到，以及要求至少一个节点能够检测到。我们的结果强于先前研究，因为我们假设每个节点已知图中节点数$n$（在某些情况下）或$n$的$\alpha$近似值（在其他情况下）。对于生成树验证，我们证明消息复杂度本质上取决于所给定的$n$近似值的质量：当$\alpha \ge \sqrt{2}$时，我们给出了$\Omega(n^2)$的紧下界；而当节点获得更精确的近似值时，我们给出了更好的上界（即$O(n \log n)$）。另一方面，我们的框架还推导出验证最小生成树（MST）消息复杂度的$\Omega(n^2)$下界，这揭示了生成树验证与最小生成树验证之间存在多项式级分离。该结果适用于已知精确网络规模的随机化算法，且即使在仅要求单个节点检测非法输入的情况下依然成立，从而改进了Kor、Korman和Peleg（2013）的工作。对于验证$d$近似BFS树，我们证明即使节点精确知晓$n$，相同的下界仍然成立，但该下界对拉伸参数$d$具有敏感性。