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detecting 3D array configurations


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#1 radioteeth   Prime Members   -  Reputation: 1106

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Posted 04 June 2014 - 11:43 AM

 Hi all.

 

 I've been wracking my brain trying to find a simple solution to this problem, but everything I come up with seems overly complicated.

 

 I have a 2x2x2 array where each cell can be on or off. This yields 256 possible states for the array. The problem I'm having is finding a way to detect a specific set of configurations - including their multiple possible orientations AND mirrored versions - without having to resort to creating a look up table. There are no requirements that disallow a simple look up table to determine which configuration the cells are in, except that I'd rather have a solution that could be expanded to larger arrays.

 

 The closest to a solution I've devised thus far involves counting how many adjacent cells (eg: 0, 0, 0 - 1, 0, 0), diagonal cells (0, 0, 0 - 1, 1, 0) and opposing cells (0, 0, 0 - 1, 1, 1) there are, and comparing those numbers to the numbers involved in each configuration. This overcomes checking for various orientations and mirrored versions of specific cell configurations, but requires that cells are all manually inspected and compared against other cells, which seems unavoidable, but I keep getting this nagging feeling that I could somehow represent the array in a way that allows for an XOR or two against template configurations to count these relationships more quickly.

 

 This is a very abstract problem without a clear and concise method of solving it. I've tried looking at it from many different angles and have exhausted myself for the time being. I'm merely seeking any insight anybody may have about an effective and efficient solution beyond what I have come up with on my own so far.

 

 Thanks.



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#2 ferrous   Members   -  Reputation: 2075

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Posted 04 June 2014 - 02:35 PM

Can you just flatten the array?  Then you basically have a bitfield or an integer, and just compare the integers.  EDIT:  Whoops, that doesn't count mirrored versions.  Though you could compute that as well, I think, by flipping those sections in integer. and testing for equality.

 

EDIT2:  For rotation, I think you might be able to get away by rotating the bitfield.  IE 10010010, rotated right by 1 is 01001001.  Might be worthwhile.

 

EDIT3:  Are mirrors a derivative of rotations?  They are at least in the 2D cube case.  In which case, I believe you basically just loop and rotate one of the bitfields through 8 times to see if it matches the other?  


Edited by ferrous, 04 June 2014 - 02:59 PM.


#3 SeanMiddleditch   Members   -  Reputation: 6458

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Posted 04 June 2014 - 03:01 PM

This is roughly a set/subset problem. Take the shape you're trying to draw. This is a 3D array itself. Compare this array to the one you're testing. Take as input a matrix that maps coordinates. You then check the source array with matrices [1,1,1], [-1,1,1] (mirrored X), [1,-1,1], etc. If the template array is able to be smaller than the checked array (say the shape is a 1x2x2 shape) then first "trim" the input array to its contents. That is, if I only fill up the left half of a 2x2x2 array than I only have a 1x2x2 array. If the sizes are different after trimming, the shapes cannot possibly match and there's no further work to be done.
 
array3d trim (array3d input)
  min = minimum occupied cell coordinates (input)
  max = maximum occupied cell coordinates (input)
  size = max - min
  return new array3d(size, input[min to max])
end

bool match_permutation (array3d test_in, array3d test_for, matrix3d transform)
  if dimensions of test_in != transform * dimensions of test_for
    return false
  end

  for each x in test_in
    for each y in test_in
      for each z in test_in
        if test_for[x,y,z] != test_for[transform * (x,y,z)]
          return false
        end
      end
    end
  end

  return true
end

bool contains(array3d test_in, array3d test_for)
  trimmed = trim(test_in)
  if match_permutation(test_in, test_for, identity)
    return true
  end
  if match_permutation(test_in, test_for, [-1, 1, 1])
    return true
  end
  ... also [1, -1, 1] and [1, 1, -1] and [-1, -1, 1] and [1, -1, -1] and [-1, 1, -1] and [-1, -1, -1] ...
  return false
end





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