(1+?)-approximate nearest neighbor search is a special case of the nearest neighbor search problem. The solution to the (1+?)-approximate nearest neighbor search is a point or multiple points within distance (1+?) R from a query point, where R is the distance between the query point and its true nearest neighbor.
Reasons to approximate nearest neighbor search include the space and time costs of exact solutions in high-dimensional spaces (see curse of dimensionality) and that in some domains, finding an approximate nearest neighbor is an acceptable solution.