Question

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How to pack squares into circles?

I have a problem where I need to fit a bunch of different sized rectangles within a circle. All the rectangles must fit in the circle without overlapping each other and without overflowing the circle.

Assuming the rectangles can fit inside the circle how would one develop an algorithm to distribute them inside the circle?

All I can think of is to randomly distribute the rectangles over and over and test whether the conditions are met (brute-force).

Answer 1

As others have mentioned, an optimal solution (in say minimal area or uniform distriubtion) is likely to be NP-hard. Nevertheless, depending on your needs there are some great algorithms for packing differently sized rectangles into other rectangles. For example: Fast Optimizing Rectangle Packing Algorithm for Building CSS Sprites:

This article describes a fast algorithm to pack a series of rectangles of varying widths and heights into a single enclosing rectangle, with no overlap and in a way that minimizes the amount of wasted space in the enclosing rectangle. […] shows step by step how the algorithm arrives at the optimal enclosing rectangle.

Note that in the above procedure the bounding rectangle is allowed to vary (nor am I convinced that the solution is the optimal enclosing rectangle). You can approximate a circle by breaking it up into discrete rectangles.

While not a complete solution to what you are looking for, I think this may be a good first step.