Approximate Nearest Neighbor Search in $\ell_p$

Abstract: We present a new locality sensitive hashing (LSH) algorithm for $c$-approximate nearest neighbor search in $\ell_p$ with $1<p<2$. For a database of $n$ points in $\ell_p$, we achieve $O(dn<sup>{\rho})$ query time and $O(dn+n<sup>{1+\rho})$ space, where $\rho \le O((\ln c)<sup>2/c<sup>p)$. This improves upon the previous best upper bound $\rho\le 1/c$ by Datar et al. (SOCG 2004), and is close to the lower bound $\rho \ge 1/c<sup>p$ by O'Donnell, Wu and Zhou (ITCS 2011). The proof is a simple generalization of the LSH scheme for $\ell_2$ by Andoni and Indyk (FOCS 2006).