Location of highest density on a sphere

Question

I have a lot of points on the surface of the sphere. How can I calculate the area/spot of the sphere that has the largest point density? I need this to be done very fast. If

Answer 1

Treating points on a sphere as 3D points might not be so bad.

Try either:

1. Select k, do approximate k-NN search in 3D for each point in the data or selected point of interest, then weight the result by their distance to the query point. Complexity may vary for different approximate k-NN algorithms.
2. Build a space-partitioning data structure like k-d Tree, then do approximate (or exact) range counting query with a ball range centered at each point in the data or selected point of interest. Complexity is O(log(n) + epsilon^(-3)) or O(epsilon^(-3)*log(n)) for each approximate range query with state of the art algorithms, where epsilon is the range error threshold w.r.t. the size of the querying ball. For exact range query, the complexity is O(n^(2/3)) for each query.