Question

From the Python documentation for random.shuffle, which takes a list and randomizes the order if its elements:

Note that for even rather small len(x), the total number of
permutations of x is larger than the period of most random number
generators; this implies that most permutations of a long sequence can
never be generated.

Is this true of any language, since the limitation seems dependent on the random number generator? Is it possible to write a function that can generate any possible permutation of an arbitrarily long list?

Answer 1

See http://mail.python.org/pipermail/python-ideas/2009-March/003670.html . The exact length that it starts being a problem at is dependent on the PRNG, but the basic problem will always apply.

The previous question that @TryPyPy linked to is also focused on Python, but the accepted answer explains quite well why it will always happen. To paraphrase, you can imagine a naive shuffle algorithm working this way:

1. Generate a list, p, of all possible permutations of the input
2. Get a random number, x, from your PRNG
3. The shuffled list is p[x]

If p is longer than the list of all possible xs that the PRNG can produce, some of the permutations are unreachable.

Since fancy shuffle algorithms are, at their core, a way of doing this without having to generate every possible permutation before discarding all but one of them, the only way around this is to have a source of true randomness.