Frugal conversion of uniformly distributed random numbers from one range to another

Question

Is there a way to convert uniformly distributed random numbers of one range to uniformly distributed random numbers of another range frugally?

Let me

Answer 1

Your goal is ultimately to roll a k-sided die given only a p-sided die, without wasting randomness.

In this sense, by Lemma 3 in "Simulating a dice with a dice" by B. Kloeckner, this waste is inevitable unless "every prime number dividing k also divides p". Thus, for example, if p is a power of 2 (and any block of random bits is the same as rolling a die with a power of 2 number of faces) and k has prime factors other than 2, the best you can do is get arbitrarily close to no waste of randomness.

Also, besides batching of bits to reduce "bit waste" (see also the Math Forum), there is also the technique of randomness extraction, discussed in Devroye and Gravel 2015-2020 and in my Note on Randomness Extraction.

See also the question: How to generate a random integer in the range [0,n] from a stream of random bits without wasting bits?, especially my answer there.