Learning Graph Partitions
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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 1/2(n-k)(k-1) membership queries. This proves the optimality of the O(nk) algorithm proposed by Reyzin and Srivastava (2007) for this problem, improving on the best known information-theoretic bound of Ω(nlog k) queries. Further, we construct an oracle that 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 Θ(m) queries for both learning and verifying an m-edge hidden graph G using this oracle.
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