Approximate nearest neighbors search without false negatives for $l_2$ for $c>sqrt{loglog{n}}$

In this paper, we report progress on answering the open problem presented by Pagh~[14], who considered the nearest neighbor search without false negatives for the Hamming distance. We show new data structures for solving the $c$-approximate nearest neighbors problem without false negatives for Euclidean high dimensional space $mathcal{R}^d$. These data structures work for any $c = omega(sqrt{log{log{n}}})$, where $n$ is the number of points in the input set, with poly-logarithmic query time and polynomial preprocessing time. This improves over the known algorithms, which require $c$ to be $Omega(sqrt{d})$. This improvement is obtained by applying a sequence of reductions, which are interesting on their own. First, we reduce the problem to $d$ instances of dimension logarithmic in $n$. Next, these instances are reduced to a number of $c$-approximate nearest neighbor search instances in $big(mathbb{R}^kbig)^L$ space equipped with metric $m(x,y) = max_{1 le i le L}(lVert x_i - y_irVert_2)$.