Optimal Lower Bounds for Locality-Sensitive Hashing (Except When q is Tiny)

Optimal lower bounds for locality sensitive hashing

We study lower bounds for Locality Sensitive Hashing (LSH) in the strongest setting: point sets in {0, 1} 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} 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} with dist(x, y) ≥ cr have probability at most...

Tight Lower Bounds for Data-Dependent Locality-Sensitive Hashing

We prove a tight lower bound for the exponent ρ for data-dependent LocalitySensitive Hashing schemes, recently used to design efficient solutions for the c-approximate nearest neighbor search. In particular, our lower bound matches the bound of ρ ≤ 1 2c−1+o(1) for the l1 space, obtained via the recent algorithm from [Andoni-Razenshteyn, STOC’15]. In recent years it emerged that data-dependent h...

Lattice-based Locality Sensitive Hashing is Optimal

Locality sensitive hashing (LSH) was introduced by Indyk and Motwani (STOC ‘98) to give the first sublinear time algorithm for the c-approximate nearest neighbor (ANN) problem using only polynomial space. At a high level, an LSH family hashes “nearby” points to the same bucket and “far away” points to different buckets. The quality of measure of an LSH family is its LSH exponent, which helps de...

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 R, our algorithm achieves Oc(n + d logn) query time and Oc(n + d logn) 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 boun...

Locality-Sensitive Hashing