TY - JOUR

T1 - On dice and coins

T2 - Models of computation for random generation

AU - Feldman, David

AU - Impagliazzo, Russell

AU - Naor, Mom

AU - Nisan, Noam

AU - Rudich, Steven

AU - Shamir, Adi

PY - 1993/6

Y1 - 1993/6

N2 - To examine the concept of random generation in bounded, as opposed to expected, polynomial time, a model of a probabilistic 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 a few different types of biased coins as possible. In the special case of simulating 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.

AB - To examine the concept of random generation in bounded, as opposed to expected, polynomial time, a model of a probabilistic 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 a few different types of biased coins as possible. In the special case of simulating 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.

UR - http://www.scopus.com/inward/record.url?scp=0003582942&partnerID=8YFLogxK

U2 - 10.1006/inco.1993.1028

DO - 10.1006/inco.1993.1028

M3 - Article

AN - SCOPUS:0003582942

SN - 0890-5401

VL - 104

SP - 159

EP - 174

JO - Information and Computation

JF - Information and Computation

IS - 2

ER -