geometry - Need algorithm for optimization issue of distribution points among the set of attractor points -


We have the following customization problems.

There is a k (usually, k = 3-10) on the plane, which is called an ablator: A1, .., the size on the AK and the same plane is N (N is larger, Generally 10 to -100's +) is set.

We need to tie 1 Autorator, N2 for 1 point - 2. .., KK should be tied to all the points in Kashmir: N1 + ... + nk = n.

The goal is like "the point is tied to the closest attraction and the attractor has divided the nearest points in itself":

P>

; j = 1..Nk) .

Let's add the number of points appearing by L with the number that is bound to do this: Lk = dist (Pk1, A1) + dist (Pk2, A2) + ... + dist (PkN1, A1).

Let's f be the total amount of distance: f = l1 + .. + lk .

We need to reduce the f

Does anyone offer any advice on how it can be implemented?
Or maybe there are some known algorithms to do this?

UPD: This problem can be reduced and can be solved by this and it is a matter of special problem.

This is a linear program that can be solved by normal LP solutions.

This can be modeled more specifically as a problem: Keep the attractors and the right side points to the left of a bilateral graph.

  • Connect each attractor to the source on the left side through an edge of the capability of I nii and cost 0.
  • Connect Sync to every point on the right side by the side of Capacity 1 and Cost 0.
  • Add the edges to each point with the ability 1 and at each point and with the distance of the edges.

    The minimum-cost maximum flow in this article is a solution to your problem. For example, you can solve it with Disstrostal as a method.

    Unfortunately, I do not know the theoretical limit of this approach to the specific properties of the graph (the total capacity n on bilateral and left). . I feel that the worst case for the least steady path (poliloid) is quadratic, but it should be very fast in practice. Perhaps a person may have more experience with linear programming and network optimization to tell what would be the best algorithm to solve the cost-flow problem arising.

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