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Starting to use nanoflann to do some point cloud nearest neighbor searching and it got me thinking about just how "approximate" ANN methods are.

If I have a (more or less) randomly distributed point cloud what is the likelihood that I get the exact nearest neighbor given a target point within the clouds bounding box? I know that it is dataset dependent... but does anyone have a good numerical study somewhere that shows trends?

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    $\begingroup$ Which algorithm(s) are you interested in in particular? Have you done research? (It's unlikely people will dig into some tool documentation for you.) $\endgroup$ – Raphael May 28 '13 at 6:47
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nanoflann is a library for building KD-trees. For the best bin first (BBF) approximate nearest neighbor algorithm (which is based on KD-trees), the authors report the following performance results:

In 12 dimensions, for example, BBF recovers the closest neighbor 94% of the time (versus 59% for restricted search) while on average examining only 200 of the 100,000 leaf nodes. For this same case, the “exact” search has to examine over 2400 leaves. [Even] when the method does not discover the exact nearest neighbor, it does not fail by much. Even up to dimension 20, the average distance of recovered neighbors is only 2% greater than the true NN distances.

However, that paper does not contain a formal analysis of the error bounds in this particular $\varepsilon$-approximation.

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