How approximate are "approximate" nearest neighbor (ANN) search algorithms? 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?
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.