3D clustering Algorithm
Problem Statement: I have the following problem:
There are more than a billion points in 3D space. The goal is to find the top N points which has largest number of neighbors within given distance R. Another condition is that the distance between any two points of those top N points must be greater than R. The distribution of those points are not uniform. It is very common that certain regions of the space contain a lot of points.
Goal: To find an algorithm that can scale well to many processors and has a small memory requirement.
Thoughts: Normal spatial decomposition is not sufficient for this kind of problem due to the non-uniform distribution. irregular spatial decomposition that evenly divide the number of points may help us the problem. I will really appreciate that if someone can shed some lights on how to solve this problem.
Answer
Use an Octree. For 3D data with a limited value domain that scales very well to huge data sets.
Many of the aforementioned methods such as locality sensitive hashing are approximate versions designed for much higher dimensionality where you can't split sensibly anymore.
Splitting at each level into 8 bins (2^d for d=3) works very well. And since you can stop when there are too few points in a cell, and build a deeper tree where there are a lot of points that should fit your requirements quite well.
For more details, see Wikipedia:
https://en.wikipedia.org/wiki/Octree
Alternatively, you could try to build an R-tree. But the R-tree tries to balance, making it harder to find the most dense areas. For your particular task, this drawback of the Octree is actually helpful! The R-tree puts a lot of effort into keeping the tree depth equal everywhere, so that each point can be found at approximately the same time. However, you are only interested in the dense areas, which will be found on the longest paths in the Octree without even having to look at the actual points yet!