Sampling an Edge in Sublinear Time Exactly and Optimally
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Sampling edges from a graph in sublinear time is a fundamental problem and a powerful subroutine for designing sublinear-time algorithms. Suppose we have access to the vertices of the graph and know a constant-factor approximation to the number of edges. An algorithm for pointwise ε-approximate edge sampling with complexity O(n/√(ε m)) has been given by Eden and Rosenbaum [SOSA 2018]. This has been later improved by Tětek and Thorup [STOC 2022] to O(n log(ε^-1)/√(m)). At the same time, Ω(n/√(m)) time is necessary. We close the problem, by giving an algorithm with complexity O(n/√(m)) for the task of sampling an edge exactly uniformly.
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