On dice and coins: models of computation for random generation

Abstract

To distinguish between random generation in bounded, as opposed to expected, polynomial time, a model of Probabalisic Turing Machine (PTM) with the ability to make random choices with any (small) rational bias is necessary. This ability is equivalent to that of being able to simulate rolling any k-sided die (where |k| is polynomial in the length of the input). We would like to minimize the amount of hardware required for a machine with this capability. This leads to the problem of efficiently simulating a family of dice with as few different types of biased coins as possible.

In the special case of simulaing one n-sided die, we prove that only two types of biased coins are necessary, which can be reduced to one if we allow irrationally biased coins. This simulation is efficient, taking O(log n) coin flips. For the general case we get a tight time vs. number of biases tradeoff; for example, with O(log n) different biases, we can simulate, for any i<n, an i-sided die in O(log n) coin flips.

Supported by NSF grant DCF-85-13926.

Supported by a grant from Digital Equipment Corporation.