Problem 2191. Order of things - 3

This problem is closely related to Problem 2189, Order of things - 1 and Problem 2190, Order of things - 2. For the details, see the description for those problems. Basically, we have to find the order in which to execute tasks of which the results and prerequisites depend on each other.

• However, this time the tasks are grouped, and groups of tasks should be executed at once.
• It may (still) be impossible to find a solution, since dependencies may be cyclic. But this may now also be due to the grouping of tasks. In any case, return an empty vector if no order is found.
• There are still multiple orders possible, return them as multiple rows of the output vector.
• The tasks within a group should of course be specified in the right order, when interdependent.

The dependencies of the tasks on each other is expressed in a matrix, where each row corresponds to the execution of a task, and each column to the dependency.

   A  B  C  D  E
A  0  0  0  0  0
B  0  0  0  0  0
C  0  1  0  0  0
D  0  1  1  0  0
E  1  0  0  0  0

The 1 on row C, column B, indicates that task C depends on task B.

The grouping of tasks is expressed as an input vector with groups assigned an integer value > 0, e.g.

 [ 1 1 2 3 3 ] 

Return the new row/column order as a numeric row-vector, referring to the rows/columns of the input matrix. If multiple orders exist, return them all as rows of a matrix. In this example:

 [ 
   1 2 3 4 5
   1 2 3 5 4
   2 1 3 4 5
   2 1 3 5 4
 ]

If no order fulfilling the dependencies exists, return an empty vector.