Tight query complexity bounds for learning graph partitions

Given a partition of a graph into connected components, the membership oracle asserts whether any two vertices of the graph lie in the same component or not. We prove that for n≥ k≥ 2, learning the components of an n-vertex hidden graph with k components requires at least (k-1)n-binom k2 membership queries. Our result improves on the best known information-theoretic bound of Omega(nlog k) queries, and exactly matches the query complexity of the algorithm introduced by [Reyzin and Srivastava, 2007] for this problem. Additionally, we introduce an oracle, with access to which one can learn the number of components of G in asymptotically fewer queries than learning the full partition, thus answering another question posed by the same authors. Lastly, we introduce a more applicable version of this oracle, and prove asymptotically tight bounds of widetildeTheta(m) queries for both learning and verifying an m-edge hidden graph G using it.