In this paper, we report progress on answering the open problem presented by Pagh~[ 11 ], 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 R^d. These data structures work for any c = \omega(\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(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(R^k\big)^L space equipped with metric m(x,y) = 1 \le i \le L(\lVert x_i - y_i\rVert_2). 0pt%DIFAUXCMD gives an algorithm for c-approximate nearest neighbor search without false negatives with the query time O(d^2) and the pre-processing time O(n^{O(\frac{1{c-1}})}), for c<2.}
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 R^d. These data structures work for any c = \omega(\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(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(R^k\big)^L space equipped with metric m(x,y) = 1 \le i \le L(\lVert x_i - y_i\rVert_2). Combining the dimension reduction with the algorithm presented in \mbox{%DIFAUXCMD motwani0pt%DIFAUXCMD gives an algorithm for c-approximate nearest neighbor search without false negatives with the query time O(d^2) and the pre-processing time O(n^{O(\frac{1{c-1}})}), for c<2.}
R
11
14
A
Combining the dimension reduction with the algorithm presented in \mbox{%DIFAUXCMD motwani