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

Assuming you have a set of points S in a cartesian plane with coordinates (x_i,y_i), given an arbitrary circle with center (x_c,y_c) and radius r you want to find all the points contained within that circle.

I will also assume that the points and the circle may move so certain static structures that can speed this up won't necessarily be appropriate.

Three things spring to mind that can speed this up:

Firstly, you can check:

(xi-xc)^2 + (yi-yc)^2 <= r^2

instead of

sqrt((xi-xc)^2 + (yi-yc)^2) <= r

Secondly, you can cull the list of points somewhat by remembering that a point can only be within the circle if:

• x_i is in the range [x_c-r,x_c+r]; and
• y_i is in the range [y_c-r,y_c+r]; and

This is known as a bounding box. You can use it as either an approximation or to cut down your list of points to a smaller subset to check accurately with the first equation.

Lastly, sort your points in either x or y order and then you can do a bisection search to find the set of points that are possibly within your bounding box, further cutting down on unnecessary checks.