Locality-Sensitive Hashing without False Negatives for l_p

In this paper, we show a construction of locality-sensitive hash functions without false negatives, i.e., which ensure collision for every pair of points within a given radius R in d dimensional space equipped with l_p norm when p ∈ [1,∞]. Furthermore, we show how to use these hash functions to solve the c-approximate nearest neighbor search problem without false negatives. Namely, if there is a point at distance R, we will certainly report it and points at distance greater than cR will not be reported for c=Ω(√(d),d^1-1/p). The constructed algorithms work: - with preprocessing time 𝒪(n log(n)) and sublinear expected query time, - with preprocessing time 𝒪(poly(n)) and expected query time 𝒪(log(n)). Our paper reports progress on answering the open problem presented by Pagh [8] who considered the nearest neighbor search without false negatives for the Hamming distance.