Efficient algorithm to return all rectangles on a surface

Question

One critical piece of code on my application involves enumerating all rectangles on a 2-D surface. The rectangles do not intersect each other and I must be able to enumerate all rectangles within a given rectangular boundary.

I already have a function to return the rectangle for a given coordinate, if it exists

GetRectangle( int row, int col )

Here is how I would call the resulting code.

foreach( var rect in GetRectangles( row, col, rowCount, colCount ) ) {
   //..  my processing code here
}

Obviously, I could call the function GetRectangle() for each of the points in the surface. I can also skip the width of the rectangle that was returned from the previous call as I know that they do not intersect. But, this is still not efficient enough.

Are you aware of such an algorithm?

UPDATE: The surface is not necessarily covered by rectangles but it can be for some special cases. So, the function GetRectangle( int row, int col ) might return null.

Think of the surface as a bitmap filled with random rectangles (which don't intersect). The task is to return all rectangles on that surface that falls within (i.e. intersects) a given frame. Hope this will clarify the question.

Thanks

Can the rectangles be rotated? edit: Ok, this is already implicitly answered, sorry. — Leif
– Leif, Commented Mar 13, 2012 at 10:52
Hi Kemal, could you be a bit more descriptive? Perhaps provide a diagram of what you mean by enumerating all rectangles on a 2D surface? — Dr. Andrew Burnett-Thompson
– Dr. Andrew Burnett-Thompson, Commented Mar 13, 2012 at 10:52
Why can't you keep track of the rectangles as they are added to the surface? What am I missing? — Chris Dunaway
– Chris Dunaway, Commented Mar 13, 2012 at 17:19

Gareth Rees · Accepted Answer · 2012-03-13 13:01:12Z

3

For best results, organize your collection of rectangles into a spatial lookup structure. In this case it sounds as though a R-tree would be a good bet. But if you can't do that, and GetRectangle is all you've got, then here is what I'd do:

The plan is to maintain a list of integers x_j for 0 ≤ j < rowCount, where x_j is the leftmost point on row j for which I have not yet found a rectangle covering it.

Start by initializing x_j := 0 for all j.
If x_j = colCount for all j, we are done. Stop.
Let J = min(j | x_j = min(x_i)). Then x_J is the topmost leftmost point that I have not yet found a rectangle that covers it.
Call GetRectangle(x_J, J). If this does not return a rectangle, then the point is uncovered. Set x_J := x_J + 1 and go to step 2.
Otherwise, we have a rectangle with top left corner at (x_J, J). Call its width w and height h. For J ≤ j < J + h, set x_j := x_j + w. Go to step 2.

Here's an example run of this algorithm. The list of x_j is shown by the numbers at the left, with the topmost leftmost highlighted in bold. The orange rectangle is the one that's discovered at that step of the algorithm.

The algorithm finds seven rectangles in seven steps

For efficient discovery of the minimum at step 3, you might want to keep the list of x_j in a heap.

edited Mar 13, 2012 at 13:01

answered Mar 13, 2012 at 11:20

Gareth Rees

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1 Comment

Kemal Erdogan Over a year ago

Hi Gareth, I am afraid, your example run was not pasted into your answer correctly. Could you please re-do the paste. Thank you

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