|Perl: the Markov chain saw|
Re: swap columns in a 2-dim arrayby tlm (Prior)
|on Jun 17, 2005 at 08:59 UTC||Need Help??|
You can easily modify your shuffling procedure to give you a shuffle in which no position remains fixed (this is called a derangement):
With this change, then all you need to shuffle the columns is this:
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:
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.
the lowliest monk