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 $n\ge k\ge 2$, 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 $\Omega(n\log k)$ 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 $\widetilde\Theta(m)$ queries for both learning and verifying an $m$-edge hidden graph $G$ using it.
翻译:如果将图表分割成相连接的组件,会籍官将声明,图形的任何两个顶点是否位于同一个组件中。我们证明,对于$n\ge k\ge 2$,学习一个含有美元元件的顶点隐藏图形的组件需要至少$(k-1-n-\binom k2$)的成员查询。我们的结果改进了最已知的信息-理论约束($-Omega(n\log k)查询),与[Reyzin和Srivastava,2007年]为这一问题引入的算法的查询复杂性完全吻合。此外,我们引入了一个甲骨牌,使人们能够以非象征性的方式学习和核实美元值的隐藏图形($G$)的数量,从而解答同一位作者提出的另一个问题。最后,我们引入了一个更适用的这个信箱的版本,并且证明,用它来学习和核实一个以美元为顶点的隐藏的G$G$。