Graphical algorithm for fitting in a square between a set of squares

Question

Let's say I have n number of equally sized and equally rotated squared boxes inside a limited area in a 2D coordinate system (floating point coordinates). The boxes should not overlap. Now I want to find a free space for one more box. I need some tips for an algorithm to solve this. Any ideas?

Answer 1

There ought to be a scan line algorithm for this. You say the boxes are equally rotated, so you should be able to rotate the co-ordinate system, if necessary, so that the edges of the boxes are parallel to the x and y coordinates. I would then sort the boxes in order of y coordinate.

Now try placing a box in the lowest possible position. Read from the sorted boxes to find all the boxes low enough to interfere with your placement and create an ordered set (eg red-black tree or similar container class) of these boxes. Now scan along this set of boxes and see if there is a gap big enough to place a box. If not, use the original sorted list of boxes to find and remove the lowest box, so you can consider putting the new box in just above that lowest box, so it cannot interfere with this. Add more boxes from the sorted list to cover all boxes high enough to interfere with this new possible height of box. Keep track of where you have removed boxes from the list and check there to see if a gap big enough to hold a box has opened up. If not, repeat the exercise until you find a gap or run out of space at the top of the possible area.

This looks like cost N log N for the initial sort, and then a cost of at most log N per box to insert and delete boxes from the ordered set. Checking for gaps is no more expensive than this, because you only check for a gap in a location where you have just removed a box. So I think the total cost is N log N.