Tree Algorithms for Unbiased Coin Tossing with a Biased Coin

Abstract

We give new algorithms for simulating a flip of an unbiased coin by flipping a coin of unknown bias. We are interested in efficient algorithms, where the expected number of flips is our measure of efficiency. Other authors have represented algorithms as lattices, but by representing them instead as trees we are able to produce an algorithm more efficient than any previously appearing. We also prove a conjecture of Hoeffding and Simons that there is no optimal algorithm. Further, we consider generalizations where the input is a sequence of iid discrete random variables and the output is a uniform random variable with $N$ possible outcomes. In this setting we provide an algorithm significantly superior to those previously published.