Algorithms for Almost-Uniform Generation with an Unbiased Binary Source

Abstract

We consider the problem of uniform generation of random integers in the range [1, n] given only a binary source of randomness. Standard models of randomized algorithms (e.g. probabilistic Turing machines) assume the availability of a random binary source that can generate independent random bits in unit time with uniform probability. This makes the task trivial if n is a power of 2. However, exact uniform generation algorithms with bounded run time do not exist if n is not a power of 2.

We analyze several almost-uniform generation algorithms and discuss the tradeoff between the distance of the generated distribution from the uniform distribution, and the number of operations required per random number generated. In particular, we present a new algorithm which is based on a circulant, symmetric, rapidly mixing Markov chain. For a given positive integer N, the algorithm produces an integer i in the range [1, n] with probability p _i = p _i(N) using O(N log n) bit operations such that | p _i − 1/n | < c β^N, for some constant c, where

$$\beta = \frac{{2^{\frac{1}{4}} }}{\pi }\left( {\sqrt {2\sqrt 2 - \sqrt {5 - \sqrt 5 } } } \right) \approx 0.4087.$$

This rate of convergence is superior to the estimates obtainable by commonly used methods of bounding the mixing rate of Markov chains such as conductance, direct canonical paths, and couplings.

Part of the work was done at the International Computer Science Institute (ICSI), Berkeley, California