How to generate a random number from a random bit generator and guarantee termination?

Question

Assuming I have a function that returns a random bit, is it possible to write a function that uniformly generates a random number within a certain range and always terminates?

I know how to do this so that it should (and probably will) terminate. I was just wondering if it's possible to write one that is guaranteed to terminate (and it doesn't have to be particularly efficient. What complexity would it have?

Here is a code for the not always terminating version

int random(int n)
{
  while(true)
  {
    int r = 0;
    for (int i = 0; i < ceil(log(n)); i++)
    {
      r = r<<1;
      r = r|getRandomBit();
    }

    if(r<n)
    {
      return r;
    }
  }
}

Answer 1

If the size of the range isn't a power of 2, you can't get an exactly uniform distribution except through what amounts to rejection sampling. You can get as close as you like to uniform, however, by sampling once from a large range, and dividing the smaller range into it.

For instance, while you can't uniformly sample between 1 and 10, you can quite easily sample between 1 and 1024 by picking 10 random bits, and figure out some way of equitably dividing that into 10 intervals of about the same size.

Choosing additional bits has the effect of halving the largest error (from true uniformity) you have to see in your choices... so the error decreases exponentially as you choose more bits.