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 xĝ† [n] and Bob ends up with a set yĝ† [n], such that (x,y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant β \u3c 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 I \u3e 0 of the uniform distribution over all pairs of disjoint sets of size n