A lower bound for sampling disjoint sets
Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x ⊆ [n] and Bob ends up with a set y ⊆ [n], such that (x, y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant β < 1, this requires Ω(n) communication even to get within statistical distance 1 − βn of the target distribution. Previously, Ambainis, Schulman, Ta-Shma, Vazirani, and Wigderson (FOCS 1998) proved that Ω(√n) communication is required to get within some constant statistical distance ε > 0 of the uniform distribution over all pairs of disjoint sets of size √n.
Leibniz International Proceedings in Informatics, LIPIcs
Göös, M., & Watson, T. (2019). A lower bound for sampling disjoint sets. Leibniz International Proceedings in Informatics, LIPIcs, 145 https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.51