algorithm - All k nearest neighbors in 2D, C++

Question

I need to find for each point of the data set all its nearest neighbors. The data set contains approx. 10 million 2D points. The data are close to the grid, but do not form a precise grid...

This option excludes (in my opinion) the use of KD Trees, where the basic assumption is no points have same x coordinate and y coordinate.

I need a fast algorithm O(n) or better (but not too difficult for implementation :-)) ) to solve this problem ... Due to the fact that boost is not standardized, I do not want to use it ...

Thanks for your answers or code samples...

Answer 1

I would do the following:

1. Create a larger grid on top of the points.

2. Go through the points linearly, and for each one of them, figure out which large "cell" it belongs to (and add the points to a list associated with that cell).

   (This can be done in constant time for each point, just do an integer division of the coordinates of the points.)

3. Now go through the points linearly again. To find the 10 nearest neighbors you only need to look at the points in the adjacent, larger, cells.

   Since your points are fairly evenly scattered, you can do this in time proportional to the number of points in each (large) cell.

Here is an (ugly) pic describing the situation:

The cells must be large enough for (the center) and the adjacent cells to contain the closest 10 points, but small enough to speed up the computation. You could see it as a "hash-function" where you'll find the closest points in the same bucket.

(Note that strictly speaking it's not O(n) but by tweaking the size of the larger cells, you should get close enough. :-)