How to randomly shuffle a list that has more permutations than the PRNG's period?
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To shuffle a list effectively, especially when it has more permutations than the period of the Pseudorandom Number Generator (PRNG) you're using, requires careful consideration of both the permutation count and the limitations inherent in PRNGs. Understanding why this is important and how to implement a solution effectively will ensure that your shuffle is genuinely random and uniformly distributed.
Understanding Permutations and PRNGs
Permutations
The number of permutations of a list refers to the different ways in which its elements can be ordered. For a list of n items, the number of permutations is n! (n factorial). As n increases, n! grows extremely rapidly. For example:
- A list of 5 items has
5! = 120permutations. - A list of 10 items has
10! = 3,628,800permutations. - A list of 20 items has
20! ≈ 2.43 × 10^\{18\}permutations.
Pseudorandom Number Generators (PRNGs)
PRNGs are algorithms used to generate sequences of numbers that mimic the properties of random numbers. They are deterministic and based on an initial seed. Each PRNG has a certain "period," which is the maximum length of the sequence before it starts repeating.
For many PRNGs, this period is quite large, but as lists grow in size, it becomes feasible that n! may exceed the PRNG's period. For instance, a common PRNG like MT19937, used in Python's random module, has a period of 2^\{19937\}.
Problem of Insufficient PRNG Period
When the number of permutations of a list exceeds the period of a PRNG, not all permutations can be generated through conventional methods. This "birthday paradox"-like problem means some permutations have a higher probability of being produced than others, reducing randomness.
Solutions to Shuffle a Large List
True Random Numbers
Utilizing true random number sources, such as those from random.org or hardware-based TRNGs (True Random Number Generators), mitigates the period limitations of PRNGs, offering genuinely random numbers.
Cryptographically Secure PRNGs
Cryptographically secure PRNGs (CSPRNGs), like those available in libraries such as Python's secrets or openssl, possess extremely large periods and robust randomness ideal for shuffling even large lists.
