  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 $\big(\mathbb{R}^k\big)^L$
space equipped with metric $m(x,y) = \max_{1 \le i \le L}(\lVert x_i -
y_i\rVert_2)$.
