Count number of points inside a circle fast

Question

Given a set of n points on plane, I want to preprocess these points somehow faster than O(n^2) (O(nlog(n)) preferably), and then be able to answer on queries of the followi

Answer 1

depending on how much precomputing time you have, you could build a tree like this:

first node branches are x-values, below them are y-value nodes, and below them are radius value nodes. at each leaf is a hashset of points.

when you want to compute the points at x,y,r: go through your tree and go down the branch that matches your x,y values the closest. when you get down to the root level, you need to do a little match (constant time stuff), but you can find a radius such that all the points in that circle (defined by the path in the tree) are inside the circle specified by x,y,r, and another circle (same x_tree,y_tree in the tree as before, but different r_tree) such that all of the points in the original circle (specified by x,y,r) are in that circle.

from there, go through all the points in the larger of the two tree circles: if a point is in the smaller circle add it to the results, if not, run the distance check on it.

only problem is that it takes a very long time to precompute the tree. although, you can specify the amount of time you want to spend by changing how many x,y, and r branches you want to have in your tree.