I have a set of k n-dimensional points:
P1(x11, x12, ..., x1n), P2(x21, x22, ..., x2n), ..., Pk(xk1, xk2, ..., xkn).
A distance D(Pa, Pb) is defined between any two points, which satisfy usual properties:
- D(Pa, Pb) >= 0
- D(Pa, Pb) = 0 <=> Pa = Pb
- D(Pa, Pb) = D(Pb, Pa)
- D(Pa, Pc) ≤ D(Pa, Pb) + D(Pb, Pc)
What is the algorithm to find areas in which these points aggregate?
More specifically, given a radius R find the smallest set β of n-dimensional "balls" B(C) such that every point lies inside at least one of the balls:
ⱯPa ƎB(C) ∈ β => D(Pa, C) ≤ R
There is definitely more than one solution (in most cases a ball could be moved a little so that it would still contain the points) and probably there is something else I am missing. But I need an algorithm that would provide at least one solution.
k = 5, n = 2, R = 2, D(Pa, Pb) = euclidean distance on a plane
P1(0,0), P2(1,1), P3(4.5, 4), P4(-4, 1), P5(4,4),
B1(0.5, 0.5) (contains first two points), B2(-4,1) (contains only 4th point), B3(4,4) (contains 3rd and 5th points)