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 $p\in [1,\mathrm{\infty }]$. 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 $c=\mathrm{\Omega }(\sqrt{d},d^{1-\frac{1}{p}})$. The constructed algorithms work: - with preprocessing time $\mathcal{O}(n\log (n))$ and sublinear expected query time,

• with preprocessing time $\mathcal{O}(\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}(n))$ and expected query time $\mathcal{O}(\log (n))$. 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.