Approximate Nearest Neighbor Search in ${\mathrm{\ell }}_p$

We present a new locality sensitive hashing (LSH) algorithm for $c$-approximate nearest neighbor search in ${\mathrm{\ell }}_p$ with $1<p<2$. For a database of $n$ points in ${\mathrm{\ell }}_p$, we achieve $O(d{n}^{\rho })$ query time and $O(dn+n^{1+\rho })$ space, where $\rho \le O((\ln c{)}^2\mathrm{/}{c}^p)$. This improves upon the previous best upper bound $\rho \le 1\mathrm{/}c$ by Datar et al. (SOCG 2004), and is close to the lower bound $\rho \ge 1\mathrm{/}{c}^p$ by O’Donnell, Wu and Zhou (ITCS 2011). The proof is a simple generalization of the LSH scheme for ${\mathrm{\ell }}_2$ by Andoni and Indyk (FOCS 2006).