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A '''random permutation''' is a [[random]] ordering of a set of objects, that is, a [[permutation]]-valued [[random variable]].  The use of random permutations is often fundamental to fields that use [[randomized algorithm]]s such as [[coding theory]], [[cryptography]], and [[simulation]].  A good example of a random permutation is the [[shuffling]] of a [[card deck|deck of cards]]: this is ideally a random permutation of the 52 cards.
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==Generating random permutations==
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===Entry-by-entry brute force method===
One method of generating a random permutation of a set of length ''n'' [[uniform distribution (discrete)|uniformly at random]] (i.e., each of the ''n''! [[permutation]]s is equally likely to appear) is to generate a sequence by taking a random number between 1 and ''n'' sequentially, ensuring that there is no repetition, and interpreting this sequence (''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) as the permutation


: <math>\begin{pmatrix}
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1 & 2 & 3 & \cdots & n \\
 
x_1 & x_2 & x_3 & \cdots & x_n \\
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\end{pmatrix},</math>
 
 
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shown here in [[permutation#Notation|two-line notation]].
 
 
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This brute-force method will require occasional retries whenever the random number picked is a repeat of a number already selected.  This can be avoided if, on the ''i''th step (when ''x''<sub>1</sub>, ..., ''x''<sub>''i'' &minus; 1</sub> have already been chosen), one chooses a number ''j'' at random between 1 and ''n'' &minus; ''i'' + 1 and sets ''x''<sub>''i''</sub> equal to the ''j''th largest of the unchosen numbers.
 
 
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===Knuth shuffles===
 
A simple [[algorithm]] to generate a permutation of ''n'' items uniformly at random without retries, known as the [[Knuth shuffle]], is to start with any permutation (for example, the [[Identity function|identity permutation]]), and then go through the positions 1 through ''n'' &minus; 1, and for each position ''i'' '''swap''' the element currently there with a randomly chosen element from positions ''i'' through ''n'', inclusive. It's easy to verify that any permutation of ''n'' elements will be produced by this algorithm with probability exactly 1/''n''!, thus yielding a uniform distribution over all such permutations.
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The initializaton to the identity permutation and the shuffling may be combined, as in the following example. It requires a function <code>uniform(m)</code> which returns a random integer between 0 and ''m'' inclusive.
</ul>
 
<source lang="c">
unsigned uniform(unsigned m); /* Returns a random integer 0 <= uniform(m) <= m */
 
unsigned permute(unsigned permutation[], unsigned n)
{
    unsigned i;
    for (i = 0; i < n; i++) {
        unsigned j = uniform(i);
        permutation[i] = permutation[j];
        permutation[j] = i;
    }
}
</source>
 
Note that the first assignment to <code>permutation[i]</code> might be copying an uninitialized value, if <code>j</code> happens to be equal to <code>i</code>. However, in this case, it is immediately overwritten with a defined value on the next line.
 
It is also important to note that the <code>uniform()</code> function can ''not'' be implemented simply as <code>random() % (m+1)</code> unless a bias in the results is acceptable.
 
==Statistics on random permutations==
===Fixed points===
{{main|Rencontres numbers}}
The [[probability distribution]] of the number of [[Fixed point (mathematics)|fixed points]] of a uniformly distributed random permutation approaches a [[Poisson distribution]] with [[expected value]] 1 as ''n'' grows. In particular, it is an elegant application of the [[inclusion-exclusion principle]] to show that the probability that there are no fixed points approaches 1/''e''. The first ''n'' [[moment (mathematics)|moments]] of this distribution are exactly those of the Poisson distribution.
 
== Randomness testing ==
As with all random processes, the quality of the resulting distribution of an implementation of a randomized algorithm such as the Knuth shuffle (i.e., how close it is to the desired uniform distribution) depends on the quality of the underlying source of randomness, such as a [[pseudorandom number generator]]. There are many possible [[randomness test]]s for random permutations, such as some of the [[Diehard tests]]. A typical example of such a test is to take some [[permutation statistic]] for which the distribution is known and test whether the distribution of this statistic on a set of randomly generated permutations closely approximates the true distribution.
 
== See also ==
* [[Ewens's sampling formula]] — a connection with [[population genetics]]
* [[Golomb–Dickman constant]]
* [[Faro shuffle|Perfect shuffle]]
* [[Random permutation statistics]]
* [[Shuffle#Shuffling algorithms|Shuffling algorithms]] — random sort method, iterative exchange method
 
== External links ==
* [http://mathworld.wolfram.com/RandomPermutation.html Random permutation] at [[MathWorld]]
* [http://www.techuser.net/randpermgen.html Random permutation generation] -- detailed and practical explanation of Knuth shuffle algorithm and its variants for generating ''k''-permutations (permutations of ''k'' elements chosen from a list) and ''k''-subsets (generating a subset of the elements in the list without replacement) with pseudocode
 
[[Category:Permutations]]
[[Category:Randomness]]

Latest revision as of 08:17, 30 December 2014

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