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In this paper, we report progress on answering the open problem presented byPagh~[14], who considered the nearest neighbor search without false negativesfor the Hamming distance. We show new data structures for solving the$c$-approximate nearest neighbors problem without false negatives for Euclideanhigh 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 inputset, with poly-logarithmic query time and polynomial preprocessing time. Thisimproves over the known algorithms, which require $c$ to be $\Omega(\sqrt{d})$.  This improvement is obtained by applying a sequence of reductions, which areinteresting on their own. First, we reduce the problem to $d$ instances ofdimension logarithmic in $n$. Next, these instances are reduced to a number of$c$-approximate nearest neighbor search instances in $\big(\mathbb{R}^k\big)^L$space equipped with metric $m(x,y) = \max_{1 \le i \le L}(\lVert x_i -y_i\rVert_2)$.

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