Find the nearest dot in a 2D space

Question

Yesterday I read a problem which can be translated into the following problem with slight modification:

The coordinate of a dot is expressed by (x, y) in a 2D space.

Answer 1

Practically we can sort the points in x-coordinate, then start from the point whose x difference is most closed to the target, and we scan to both left and right, and stop scanning to one direction as soon as the next point's x difference in that direct is already bigger than smallest distance found so far. Divide Conquer is still linear if you divide to two half but do not discard either half.