The Arboricity Captures the Complexity of Sampling Edges
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In this paper, we revisit the problem of sampling edges in an unknown graph G = (V, E) from a distribution that is (pointwise) almost uniform over E. We consider the case where there is some a priori upper bound on the arboriciy of G. Given query access to a graph G over n vertices and of average degree d and arboricity at most α, we design an algorithm that performs O(α/d·^3 n/ε) queries in expectation and returns an edge in the graph such that every edge e ∈ E is sampled with probability (1 ±ε)/m. The algorithm performs two types of queries: degree queries and neighbor queries. We show that the upper bound is tight (up to poly-logarithmic factors and the dependence in ε), as Ω(α/d) queries are necessary for the easier task of sampling edges from any distribution over E that is close to uniform in total variational distance. We also prove that even if G is a tree (i.e., α = 1 so that α/d=Θ(1)), Ω( n/ n) queries are necessary to sample an edge from any distribution that is pointwise close to uniform, thus establishing that a poly( n) factor is necessary for constant α. Finally we show how our algorithm can be applied to obtain a new result on approximately counting subgraphs, based on the recent work of Assadi, Kapralov, and Khanna (ITCS, 2019).
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