Question

I have a row with numbers 1:n. I’m looking to add a second row also with the numbers 1:n but these should be in a random order while satisfying the following:

1. No positions have the same number in both rows
2. No combination of numbers occurs twice

For example, in the following

Row 1:  1  2  3  4  5  6  7 ...
Row 2:  3  6  15 8  13 12 7 ...  

the number 7 occurs at the same position in both rows 1 and 2 (namely position 7; thereby not satisfying rule 1)

while in the following

Row 1:  1  2  3  4  5  6  7 ...
Row 2:  3  7  15 8  13 12 2 ...

the combination of 2+7 appears twice (in positions 2 and 7; thereby not satisfying rule 2).

It would perhaps be possible – but unnecessarily time-consuming – to do this by hand (at least up until a reasonable number), but there must be quite an elegant solution for this in MATLAB.

Answer 1

This is fairly straightforward. Create a random permutation of the nodes, but interpret the list as follows: Interpret it as a random walk around the nodes, and if node ‘b’ appears after node ‘a’, it means that node ‘b’ appears below node ‘a’ in the lists:

So if your initial random permutation is

3 2 5 1 4

Then the walk in this case is 3 -> 2 -> 5 -> 1 -> 4 and you creates the rows as follows:

Row 1:  1 2 3 4 5
Row 2:  4 5 2 3 1

This random walk will satisfy both conditions.

But do you wish to allow more than one cycle in your network? I know you don’t want two people to have each other’s hat. But what about 7 people, where 3 of them have each other’s hats and the other 4 have each other’s hats? Is this acceptable and/or desirable?