Question

I have a rectangular plane of integer dimension. Inside of this plane I have a set of non-intersecting rectangles (of integer dimension and at integer coordinates).

My question is how can I efficiently find the inverse of this set; that is the portions of the plane which are not contained in a sub-rectangle. Naturally, this collection of points forms a set of rectangles — and it is these that I am interested in.

My current, naive, solution uses a boolean matrix (the size of the plane) and works by setting a point i,j to 0 if it is contained within a sub-rectangle and 1 otherwise. Then I iterate through each element of the matrix and if it is 1 (free) attempt to ‘grow’ a rectangle outwards from the point. Uniqueness is not a concern (any suitable set of rectangles is fine).

Are there any algorithms which can solve such a problem more effectively? (I.e, without needing to resort to a boolean matrix.

Answer 1

Yes, it’s fairly straightforward. I’ve answered an almost identical question on SO before, but haven’t been able to find it yet.

Anyway, essentially you can do this:

• start with an output list containing a single output rect equal to the area of interest (some arbitrary bounding box which defines the area of interest and contains all the input rects)
• for each input rect
  • if the input rect intersects any of the rects in the output list
    • delete the old output rect and generate up to four new output
      rects which represent the difference between the intersection
      and the original output rect

Optional final step: iterate through the output list looking for pairs of rects which can be merged to a single rect (i.e. pairs of rects which share a common edge can be combined into a single rect).