Approximate Nearest Neighbor Search in $\ell_p$

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 $O(dn^ρ)$ query time and $O(dn+n^{1+ρ})$ space, where $ρ\le O((\ln c)^2/c^p)$. This improves upon the previous best upper bound $ρ\le 1/c$ by Datar et al. (SOCG 2004), and is close to the lower bound $ρ\ge 1/c^p$ 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).

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