To randomly shuffle an array, with no bias towards any particular permutation, there is the Knuth Fischer-Yeats algorithm. In Python:

#!/usr/bin/env python
import sys
from random import randrange

def KFYShuffle(items):
    i = len(items) - 1
    while i > 0:
        j = randrange(i+1)  # 0 <= j <= i
        items[j], items[i] = items[i], items[j]
        i = i - 1
    return items

print KFYShuffle(range(int(sys.argv[1])))

There is also Sattolo’s algorithm, which produces random cycles. In Python:

#!/usr/bin/env python
import sys
from random import randrange

def SattoloShuffle(items):
    i = len(items)
    while i > 1:
        i = i - 1
        j = randrange(i)  # 0 <= j <= i-1
        items[j], items[i] = items[i], items[j]
    return items

print SattoloShuffle(range(int(sys.argv[1])))

I’m currently writing a simulation with the following specifications for a shuffling algorithm:

1. The algorithm is unbiased. If a true random number generator was used, no permutation would be more likely than any other.
2. No number ends up at its original index. The input to the shuffle will always be A[i] = i for i from 0 to N-1
3. Permutations are produced that are not cycles, but still meet specification 2.

The cycles produced by Sattolo’s algorithm meet specification 2, but not specification 1 or 3. I’ve been working at creating an algorithm that meets these specifications, what I came up with was equivalent to Sattolo’s algorithm.

Does anyone have an algorithm for this problem?