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### Re: swap columns in a 2-dim array

by tlm (Prior)
 on Jun 17, 2005 at 08:59 UTC ( #467623=note: print w/replies, xml ) Need Help??

in reply to swap columns in a 2-dim array

You can easily modify your shuffling procedure to give you a shuffle in which no position remains fixed (this is called a derangement):

```sub fys {
my \$arr = shift;
my \$i = @{ \$arr };
while ( \$i ) {
my \$j = int rand \$i;
@\$arr[\$i,\$j] = @\$arr[\$j,\$i];
--\$i;
}
}
With this change, then all you need to shuffle the columns is this:
```@nums = map [ @{\$_}[@cols] ], @nums;

The problem with this simple solution is that it cannot generate all possible derangements. For example, the modified FY misses the derangment 1,0,3,2 of 0,1,2,3.

I looked online for algorithms to fairly sample the space of all derangments of an input list, and the best I found was based on using the standard FY until a derangement is found (i.e. a rejection method). If you need to randomly sample from the space of all possible derangements of the columns, then keep your original FY procedure, but modify the creation of @cols to this:

```my @cols = 0..11;
do {
fys( \@cols );
} until is_deranged( \@cols );
where
```sub is_deranged {
my \$arr = shift;
\$arr->[ \$_ ] == \$_ and return for 0..\$#\$arr;
return 1;
}
The probability of getting a derangement from a random sample of permutations is ≈ 1/e (i.e. about three trials required per derangement, on average). Moreover, one can optimize the FY procedure around this problem (by having it automatically restart when it encounters a "trivial" swap, i.e. \$i == \$j), which obviates the need to have a specific rejection step. Therefore this approach has essentially the same time and space growth properties as FY.

Update: Added the stuff about fair sampling, and the rejection method for obtaining a random derangement.

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