LSH on the Hypercube Revisited

LSH (locality sensitive hashing) had emerged as a powerful technique in nearest-neighbor search in high dimensions [IM98, HIM12]. Given a point set $P$ in a metric space, and given parameters $r$ and $\varepsilon > 0$, the task is to preprocess the point set, such that given a query point $q$, one can quickly decide if $q$ is in distance at most $\leq r$ or $\geq (1+\varepsilon)r$ from the point set $P$. Once such a near-neighbor data-structure is available, one can reduce the general nearest-neighbor search to logarithmic number of queries in such structures [IM98, Har01, HIM12]. In this note, we revisit the most basic settings, where $P$ is a set of points in the binary hypercube $\{0,1\}^d$, under the $L_1$/Hamming metric, and present a short description of the LSH scheme in this case. We emphasize that there is no new contribution in this note, except (maybe) the presentation itself, which is inspired by the authors recent work [HM17].