Position N circles of different radii inside a larger circle without overlapping

Question

Given n circles with radii r1 ... rn, position them in such a way that no circles are overlapping and the bounding circle is of \"small\" radius.

The program takes a lis

Answer 1

Not a solution, just a brainstorming idea: IIRC one common way to get approximate solutions to the TSP is to start with a random configuration, and then applying local operations (e.g. "swapping" two edges in the path) to try and get shorter and shorter paths. (Wikipedia link)

I think something similar would be possible here:

1. Start with random center positions
2. "Optimize" these positions, so there are no overlapping circles and so the circles are as close as possible, by increasing the distance between overlapping circles and decreasing the distance between other circles, until they're tightly packed. This could be done by some kind of energy minimization, or there might be a more efficient greedy solution.
3. Apply an iterative improvement operator to the center positons
4. Goto 2, break after a maximum number of iterations or if the last iteration didn't find any improvement

The interesting question is: what kind of "iterative improvement operator" could you use in step 3? We can assume that the positions at that stage are locally optimal, but they might be improved by rearranging a large fraction of the circles. My suggestion would be to arbitrarily choose a line through the circles. Then take all the circles "left" of the line and mirror them at some axis perpendicular to that line: You would probably try multiple lines and pick the one that leads to the most compact solution.

The idea is, if some of the circles are already at or close to their optimal configuration, chances are good this operation won't disturb them.

Other possible operations I could think of:

• Take one of the circles with the highest distance from the center (one touching the boundary circle), and randomly move it somewhere else:
• Choose a set of cirlces that are close to each other (e.g. if their centers lie in an randomly chosen circle) and rotate them by a random angle.
• Another option (although a bit more complex) would be to measure the area between the circles, when they're tightly packed:

Then you could pick one of the circles adjacent to the largest between-circle-area (the red area, in the image) and swap it with another circle, or move it somewhere to the boundary.

(Response to comment:) Note that each of these "improvements" is almost guaranteed to create overlaps and/or unneccessary space between circles. But in the next iteration, step 2 will move the circles so they are tightly packed and non-overlapping again. This way, I can have one step for local optimizations (without caring about global ones), and one for global optimizations (which might create locally suboptimal solutions). This is far easier than having one complex step that does both.