Beyond locality-sensitive hashing

We present a new data structure for the c-approximate near neighbor problem (ANN) in the Euclidean space. For n points in Rd, our algorithm achieves Oc(nρ + dlogn) query time and Oc(n1+ρ + dlogn) space, where ρ ≤ 7/(8c2) + O(1/c3) + oc(1). This is the first improvement over the result by Andoni and Indyk (FOCS 2006) and the first data structure that bypasses a locality-sensitive hashing lower bound proved by O'Donnell, Wu and Zhou (ICS 2011). By a standard reduction we obtain a data structure for the Hamming space and ℓ1 norm with ρ ≤ 7/(8c)+ O(1/c3/2)+ oc(1), which is the first improvement over the result of Indyk and Motwani (STOC 1998).