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，包括节点ID）和KT-$0$（节点缺乏此类知识）两种模型下，验证通信网络给定子图是否构成具有特定属性的树结构时的消息复杂度问题。我们发展了一个通用框架，有助于建立各类树验证问题的紧下界。同时考虑两种不同的验证要求：即每个节点在输入错误时均能检测，以及至少一个节点能检测。相较于以往工作，本文结果更强之处在于假设每个节点已知图中节点数$n$（部分情形）或其$\alpha$近似值（其他情形）。对于生成树验证，我们证明消息复杂度本质上取决于给定近似值$n$的质量：当$\alpha \ge \sqrt{2}$时，给出紧下界$\Omega(n^2)$；当节点获得更优近似时，得到更好的上界（即$O(n \log n)$）。另一方面，我们的框架对最小生成树（MST）验证给出了$\Omega(n^2)$的消息复杂度下界，揭示了生成树验证与最小生成树验证之间存在多项式分离。该结论适用于具备完美网络规模知识的随机算法，且即使仅需单个节点检测非法输入时仍成立，从而改进了Kor、Korman和Peleg（2013）的工作。对于$d$近似BFS树的验证，即使节点精确知晓$n$值，相同下界依然成立，但该下界对伸缩参数$d$敏感。