On Sampling Edges Almost Uniformly

We consider the problem of sampling an edge almost uniformly from an unknown graph, G = (V, E). Access to the graph is provided via queries of the following types: (1) uniform vertex queries, (2) degree queries, and (3) neighbor queries. We describe an algorithm that returns a random edge e ∈ E using Õ(n / √(ε m)) queries in expectation, where n = |V| is the number of vertices, and m = |E| is the number of edges, such that each edge e is sampled with probability (1 ±ε)/m. We prove that our algorithm is optimal in the sense that any algorithm that samples an edge from an almost-uniform distribution must perform Ω(n / √(m)) queries.