Locality-sensitive Hashing without False Negatives

We consider a new construction of locality-sensitive hash functions for Hamming space that is covering in the sense that is it guaranteed to produce a collision for every pair of vectors within a given radius r. The construction is efficient in the sense that the expected number of hash collisions between vectors at distance cr, for a given c > 1, comes close to that of the best possible data independent LSH without the covering guarantee, namely, the seminal LSH construction of Indyk and Motwani (FOCS ’98). The efficiency of the new construction essentially matches their bound if cr = log(n)/k, where n is the number of points in the data set and k ∈ N, and differs from it by at most a factor ln(4) < 1.4 in the exponent for general values of cr. As a consequence, LSH-based similarity search in Hamming space can avoid the problem of false negatives at little or no cost in efficiency.