Tight Bounds on the Message Complexity of Distributed Tree Verification

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_ρ (nodes know their ρ-hop neighborhood, including node ids) and the KT₀ (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 α 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 Ω(n²) for the case α ≥ √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 Ω(n²) 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 bounds is sensitive to d, which is the stretch parameter. First, under the KT₀ assumption, we show a tight message complexity lower bound of Ω(n²) in the LOCAL model, when d ≤ n/(2+Ω(1)). For the KT_ρ assumption, we obtain an upper bound on the message complexity of O(nlog n) in the CONGEST model, when d ≥ (n-1)/max{2,ρ+1}, and use a novel charging argument to show that Ω((1/ρ)(n/ρ)^{1+c/ρ}) messages are required even in the LOCAL model for comparison-based algorithms. For the well-studied special case of KT₁, we obtain a tight lower bound of Ω(n²).