sentences:  If there is such a need, then you must go through the trouble of solving the problem for the other half space, and then compare its result to the former result, and then return the proper result.The performance of this algorithm is nearer to logarithmic time than linear time when the query point is near the cloud, because as the distance between the query point and the closest point-cloud point nears zero, the algorithm needs only perform a look-up using the query point as a key to get the correct result.An approximate nearest neighbor search algorithm is allowed to return points, whose distance from the query is at most formula_1 times the distance from the query to its nearest points.