What is an efficient packing algorithm for packing rectangles into a polygon?

Question

I need to put as many rectangle of same size as possible inside a polygon. The algorithm can put rectangles in different orientation(angle). But they cannot overlap. This is image is an example of the end result: image of rectangles inside larger rectangle

Any constraints on the shape of the polygon? It's also a rectangle in your illustration, which is a relatively simple case. — aaaaaaaaaaaa, Apr 09 '12 at 12:11
If that's a rectangle, then perhaps google for 2D bin packing might help. If not, your problem is not NP complete for sure.. — teodron, Apr 09 '12 at 13:34
@eBusiness, The Polygon can be in any shape but the rectangle is in fixed size. — Imran Qadir Baksh - Baloch, Apr 09 '12 at 18:04
@teodron, The Polygon can be in any shape but the rectangle is in a fixed size. — Imran Qadir Baksh - Baloch, Apr 09 '12 at 18:05

score 5 · Accepted Answer · answered May 10 '15 at 23:30

5

This is called the Pallet Loading Problem. Solving it is actually pretty hard, and we don’t know of an exact solution that always works in reasonable time. And sometimes the solution is not intuitive at all, see for instance:

solution

Here is a comprehensive list of existing algorithms (Recursive Five-block Algorithm, L-Algorithm, Recursive Partitioning Algorithm) with full source code. The same authors have a web application that lets you enter your own data and see the result.

answered May 10 '15 at 23:30

sam hocevar

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thank you for the website. those algorithms work for a rectangle within other rectangles, would you know of an algorithm which handles rectangles within polygons? – BenKoshy Feb 03 '17 at 23:46

Blau · Answer 2 · 2012-04-12T10:02:11.030

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In this sample you can find an algorithm to pack small textures into a big texture, that is quite similar to your matter... and maybe a good start to solve it.

http://create.msdn.com/en-US/education/catalog/sample/sprite_sheet

EDIT:

You can use a tree, where the leafs have empty regions, and the rest of nodes have the rectangles. You can iterate deeply to get a solution.

Algoritm

At first pass, the node contains the empty region. There are four combinations for adding the rectangle, by the orientation ( Vertical, Horizontal) and the way that region will be divided.

Add the four combinations.

Choose the first child,

Try to add next rectangle, adding every combination as before

If there are no room for the rectangle go back, and test next child of this parent.

Repeat that until there is no more rectangles to add.

enter image description here

edited Apr 12 '12 at 10:02

answered Apr 09 '12 at 14:21

Blau

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Thanks. I will check that. Is this work for a random polygon where I need to place rectangles in any direction? – Imran Qadir Baksh - Baloch Apr 09 '12 at 18:12
This seems to be for unknown length packing. I have rectangles which lengths are fixed. – Imran Qadir Baksh - Baloch Apr 11 '12 at 03:15
I think the problem would be the opposite... ;) – Blau Apr 11 '12 at 06:58
I am wondering whether there is no per-existing algorithm available for fixed size. Thanks anyway. – Imran Qadir Baksh - Baloch Apr 12 '12 at 05:36
What is useful for variable length size is useful for fixed length size... I don't understand what is your matter... – Blau Apr 12 '12 at 09:06
I think for fixed size it should be simple. Also, I am checking for a simple algorithm. BTW, Thanks for you help. – Imran Qadir Baksh - Baloch Apr 12 '12 at 09:11
the algorithm is essentially the same... the only difference could be the need to get the best solution or not... – Blau Apr 12 '12 at 09:19
OK, I will check that. – Imran Qadir Baksh - Baloch Apr 12 '12 at 09:42

What is an efficient packing algorithm for packing rectangles into a polygon?

2 Answers2

Algoritm