Question

Given a function R which produces true random 32 bit numbers, I would like a function that returns random integers in the range 0 to n, where n is arbitrary (less than 2^32).

The function must produce all values 0 to n with equal probability.

I would like a function that executes in constant time with no if statements or loops, so something like the Java Random.nextInt(n) function is out.

I suspect that a simple modulus will not do the job unless n is a power of 2 — am I right?

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I have accepted Jason’s answer, despite it requiring a loop of undetermined duration, since it appears to be the best method to use in practice and essentially answers my question. However I am still interested in any algorithms (even if less efficient) which would be deterministic in nature and be guaranteed to terminate, such as Mark Byers has pointed to.

Answer 1

Without discarding some of the values from the source, you can not do this. For example, a set of size 2^32 can not be partitioned into three equally sized sets. Therefore, it is impossible to do this without discarding some of the values and iterating until a non-discarded value is produced.

So, just use this (pseudocode):

rng is random number generator produces uniform integers from [0, max)
compute m = max modulo (n + 1)
do {
    draw a random number r from rng
} while(r >= max - m)
return r modulo (n + 1)

Effectively I am throwing out the top part of the distribution that causes problems. If rng is uniform on [0, max), then this algorithm will be uniform on [0, n]