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PODoes Repeating a Biased Random Shuffle Reduce the Bias?
 

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  1. PODoes Repeating a Biased Random Shuffle Reduce the Bias?
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    <p>I'd like to produce fast random shuffles repeatedly with minimal bias.</p> <p>It's known that the <a href="http://en.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle" rel="noreferrer">Fisher-Yates shuffle</a> is unbiased as long as the underlying random number generator (RNG) is unbiased.</p> <pre><code>To shuffle an array a of n elements: for i from n − 1 downto 1 do j ← random integer with 0 ≤ j ≤ i exchange a[j] and a[i] </code></pre> <p>But what if the RNG is biased (but fast)?</p> <p>Suppose I want to produce many random permutations of an array of 25 elements. If I use the Fisher-Yates algorithm with a biased RNG, then my permutation will be biased, but I believe this assumes that the 25-element array starts from the same state before each application of the shuffle algorithm. One problem, for example, is if the RNG only has a period of 2^32 ~ 10^9 we can not produce every possible permutation of the 25 elements because this is 25! ~ 10^25 permutations.</p> <p>My general question is, if I leave the shuffled elements shuffled before starting each new application of the Fisher-Yates shuffle, would this reduce the bias and/or allow the algorithm to produce every permutation?</p> <p>My guess is it would generally produce better results, but it seems like if the array being repeatedly shuffled had a number of elements that was related to the underlying RNG that the permutations could actually repeat more often than expected.</p> <p>Does anyone know of any research that addresses this?</p> <p>As a sub-question, what if I only want repeated permutations of 5 of the 25 elements in the array, so I use the Fisher-Yates algorithm to select 5 elements and stop before doing a full shuffle? (I use the 5 elements on the end of the array that got swapped.) Then I start over using the previous partially shuffled 25-element array to select another permutation of 5. Again, it seems like this would be better than starting from the original 25-element array if the underlying RNG had a bias. Any thoughts on this?</p> <p>I think it would be easier to test the partial shuffle case since there are only 6,375,600 possible permutations of 5 out of 25 elements, so are there any simple tests to use to check for biases?</p>
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    1. CODid you ever test this to get the answer?
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    2. COI didn't do any formal tests on the bias. I believe with a biased RNG, there would be a bias from one shuffle to the next, but maybe not noticeable when looking at the totals of each permutation selected over millions of permutations. I ended up using a Lagged Fibonacci Generator (LFG) and leaving the deck in the shuffled state to start the next permutation. I think this is unbiased, and it's fast enough for me. So I went with the fast RNG and optimizing everything as much as possible.
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