Local maxima in a point cloud

Question

I have a point cloud C, where each point has an associated value. Lets say the points are in 2-d space, so each point can be represented with the triplet (x, y, v).

I

Answer 1

Use a 2D-tree (2D instance of a kD-tree). After N.Log(N) time preprocessing, It will allow you to perform fixed-radius near-neighbor searches around all your points in about Log(N) + K time (K neighbors found on average), for a total of N.Log(N)+ K.N. It will perfectly live with the Manhattan distance.