Find largest cuboid containing only 1's in an NxNxN binary array

Question

Given an NxNxN binary array (containing only 0\'s or 1\'s), how can we obtain the largest cuboid with a non-trivial solution i.e. in O(N^3) ?

--

It is the same p

Answer 1

Here is only O(N^4).

Lets assume you are storing cubiod in bool cuboid[N][N][N];

bool array2d[N][N];

for(int x_min = 0; x_min < N; x_min++) {
   //initializing array2d
   for(int y = 0; y < N; y++) {
      for(int z = 0; z < N; z++) {
         array2d[y][z] = true;
      }
   }

   //computation
   for(int x_max = x_min; x_max < N; x_max++) {
      // now we want to find largest cube that
      // X coordinates are equal to x_min and x_max

      // cells at y,z can be used in cube if and only if
      // there are only 1's in cuboid[x][y][z] where x_min <= x <= x_max

      // so lets compute for each cell in array2d,
      // if are only 1's in cuboid[x][y][z] where x_min <= x <= x_max
      for(int y = 0; y < N; y++) {
         for(int z = 0; z < N; z++) {
            array2d[y][z] &= cubiod[x_max][y][z];
         }
      }

      //you already know how to find largest rectangle in 2d in O(N^2)
      local_volume = (x_max - x_min + 1) * find_largest_area(array2d);

      largest_volume = max(largest_volumne, local_volume);
   }
}

You can use the same trick, to compute best solution in X dimentions. Just reduce the problem to X-1 dimensions. Complexity: O(N^(2*X-2)).