Question

I’m looking for the most efficient algorithm to randomly choose a set of n distinct integers, where all the integers are in some range [0..maxValue].

Constraints:

• maxValue is larger than n, and possibly much larger
• I don’t care if the output list is sorted or not
• all integers must be chosen with equal probability

My initial idea was to construct a list of the integers [0..maxValue] then extract n elements at random without replacement. But that seems quite inefficient, especially if maxValue is large.

Any better solutions?

Answer 1

For small values of maxValue such that it is reasonable to generate an array of all the integers in memory then you can use a variation of the Fisher-Yates shuffle except only performing the first n steps.

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If n is much smaller than maxValue and you don’t wish to generate the entire array then you can use this algorithm:

1. Keep a sorted list l of number picked so far, initially empty.
2. Pick a random number x between 0 and maxValue – (elements in l)
3. For each number in l if it smaller than or equal to x, add 1 to x
4. Add the adjusted value of x into the sorted list and repeat.

If n is very close to maxValue then you can randomly pick the elements that aren’t in the result and then find the complement of that set.

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Here is another algorithm that is simpler but has potentially unbounded execution time:

1. Keep a set s of element picked so far, initially empty.
2. Pick a number at random between 0 and maxValue.
3. If the number is not in s, add it to s.
4. Go back to step 2 until s has n elements.

In practice if n is small and maxValue is large this will be good enough for most purposes.