非均匀圆盘的最佳覆盖

Question

What kind of algorithm can I use to search for an optimal (minimum area) covering of a limited region of the XY plane with n discs ( x _j , y _j , r _j ) ?

I've found many investigations on fixed radius discs, but nothing about variable radius.

n is fixed but the discs can be placed freely (they're not in assigned positions and their centers are not required to be inside the region). The region is in general non-connected and non-simply connected (can be composed by multiple parts and can have holes). In my specific case is defined by multiple closed polygons (using odd-even filling rule).

To recap:

Input:

• a limited area of the XY plane (eg described as a collection of closed polygons with odd-even filling rule)

• an integer number n > 0

Output:

• a list of n discs described by center x[i], y[i] and radius r[i] so that every point of the area is contained in at least one disc

Minimizing:

• the area of the plane covered by the union of the discs

Example

In this example the input was the "A" shape. Ten points were placed manually and minimal circles covering the intersection of the area with the Voronoi cells were computed.

I'm currently investigating the approach based on just looking for the centers x[i], y[i] and computing the radiuses r[i] with this algorithm (search space is reduced by ℝ ⁿ and always produces an acceptable solution).

Answer 1

This is a really cool problem! I'm glad I stumbled on this one. I fully realize that this is over a year old, so you probably don't care about it anymore, but I'll answer it either way because I like riddles and this was a fun one (assuming my solution even works!).

What I would do seems similar to the Voronoi diagram suggestion:

1. Start with a hierarchical clustering solution to the problem. It won't have minimal area, but it will cover everything with N disks.

   a. Note: I wouldn't use K-means because K-Means tends to get caught in local minima quite easily.

2. You could probably then use gradient descent to move the centers of the disks (with the loss being the total area of each disk -- computed as the miximal distance to the points within this "cluster") to get you a more optimal solution.

I think some caveats here would be if you have a few lone points, they might lead to some undesirable solutions.

There's obviously no proof that this would work. What do you think? Also, what did you end up using?