Approximate nearest neighbors search without false negatives for l_2 for c>√(n)
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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 = ω(√(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 Ω(√(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 (R^k)^L space equipped with metric m(x,y) = _1 < i < L( x_i - y_i_2).
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