![]() ![]() 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. , x i − 1 have already been chosen), one chooses a number j at random between 1 and n − i + 1 and sets x i equal to the jth largest of the unchosen numbers.Ī simple algorithm to generate a permutation of n items uniformly at random without retries, known as the Fisher–Yates shuffle, is to start with any permutation (for example, the identity permutation), and then go through the positions 0 through n − 2 (we use a convention where the first element has index 0, and the last element has index n − 1), and for each position i swap the element currently there with a randomly chosen element from positions i through n − 1 (the end), inclusive. This can be avoided if, on the ith step (when x 1. This brute-force method will require occasional retries whenever the random number picked is a repeat of a number already selected. One method of generating a random permutation of a set of length n uniformly at random (i.e., each of the n! permutations 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 1. Generating random permutations Entry-by-entry brute force method A good example of a random permutation is the shuffling of a deck of cards: this is ideally a random permutation of the 52 cards. The use of random permutations is often fundamental to fields that use randomized algorithms such as coding theory, cryptography, and simulation. ![]() Sequence where any order is equally likelyĪ random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable. ![]()
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