Choosing subset of farthest points in given set of points
Find the maximum extent of all points. Split into 7x7x7 voxels. For all points in a voxel find the point closest to its centre. Return these 7x7x7 points. Some voxels may contain no points, hopefully not too many.
Imagine you are given set S of n points in 3 dimensions. Distance between any 2 points is simple Euclidean distance. You want to chose subset Q of k points from this set such that they are farthest from each other. In other words there is no other subset Q’ of k points exists such that min of all pair wise distances in Q is less than that in Q’.
If n is approximately 16 million and k is about 300, how do we efficiently do this?
My guess is that this NP-hard so may be we just want to focus on approximation. One idea I can think of is using Multidimensional scaling to sort these points in a line and then use version of binary search to get points that are furthest apart on this line.