Sampling an Edge in Sublinear Time Exactly and Optimally

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 $\epsilon$-approximate edge sampling with complexity $O(n\mathrm{/}\sqrt{\epsilon m})$ has been given by Eden and Rosenbaum [SOSA 2018]. This has been later improved by Tětek and Thorup [STOC 2022] to $O(n\log ({\epsilon }^{-1})\mathrm{/}\sqrt{m})$. At the same time, $\mathrm{\Omega }(n\mathrm{/}\sqrt{m})$ time is necessary. We close the problem, by giving an algorithm with complexity $O(n\mathrm{/}\sqrt{m})$ for the task of sampling an edge exactly uniformly.