Question

I’m looking for a way to convert a 2D array to the fewest possible rectangles like in this example:

       X
    12345678
    --------
  1|00000000
  2|00011100
  3|00111000
Y 4|00111000
  5|00111000
  6|00000000

to the corner coordinates of the rectangles:

following the (x1,y1);(x2;y2) template

rectangle #1 (4,2);(6,2)
rectangle #2 (3,3);(5,5)

There has been a similar question here before but unfortunately, the link provided in its answer is broken, and I cannot check it anymore.

I’d like to do this in C# but any kind of help is appreciated.

(It doesn’t even have to be the fewest possible rectangles, but the fewer the better 🙂 )

Thanks in advance!

Answer 1

I think that you are trying to cover a set of points in the 2D plane with the minimum required number of rectangles. An answer to Find k rectangles so that they cover the maximum number of points said that this was an NP-complete problem and linked to here (which works for me). A google search finds http://2011.cccg.ca/PDFschedule/papers/paper102.pdf.

There papers agree that rectangle covering is NP-complete but do not actually prove it, and the references for this seem to be unusually elusive – https://cstheory.stackexchange.com/questions/3957/prove-that-the-problem-of-rectilinear-picture-compression-is-np-complete

What I take from these documents is this:

1. It is unlikely that there is an affordable way of getting the absolutely best answer for large problems, so you might have to either spend a lot of time to get exact answers for problems that are in some sense small, by exhausting over all possible alternatives or perhaps using something like branch and bound, or settle for affordable methods – like greedy search, or beam search, or limited discrepancy search – which are not guaranteed to give you the absolutely best answer.

2. In this case there seem to be more restricted versions of this problem which are not NP-complete. You might possibly read a paper and find that there is some detail of your problem that means that this method applies to you. One example is "AN ALGORITHM FOR CONSTRUCTING REGIONS WITH RECTANGLES:
   INDEPENDENCE AND MINIMUM GENERATING SETS
   FOR COLLECTIONS OF INTERVALS*" by Franzblau and Kleitman – I found this in the ACM Digital Library, though – I don’t know if it is generally accessible. It works for a restricted set of polygons.