Optimal lower bounds for locality sensitive hashing (except when q is tiny)

We study lower bounds for Locality Sensitive Hashing (LSH) in the strongest setting: point sets in {0; 1}^d under the Hamming distance. Recall that H is said to be an (r; cr; p; q)-sensitive hash family if all pairs x; y ∈ {0; 1}^d with dist(x; y) ≤ r have probability at least p of collision under a randomly chosen h ∈ H, whereas all pairs x; y ∈ {0; 1}^d with dist(x; y) ≥ cr have probability at most q of collision. Typically, one considers d → ∞, with c > 1 fixed and q bounded away from 0.