Approximate Nearest Neighbors Search Without False Negatives For \(l_2\) For \(c>\sqrt{\log\log{n}}\)
Piotr Sankowski, Piotr Wygocki . Arxiv 2017 – 0 citations
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 (Ω(\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)).