【算法导论学习-009】仅使用Random(0,1)实现Random(a,b)

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《算法导论》P117课后题5.1-2

Describe an implementation of the procedure Random(a, b) that only makes calls to Random(0,1). What is the expected running time of your procedure, as a function of a and b? The probability of the result of Random(a,b) should be pure uniformly distributed, as Random(0,1)

该问题的讨论区：http://stackoverflow.com/questions/8692818/how-to-implement-randoma-b-with-only-random0-1

该问题关键点是Random(a, b)生成的随机数在[a, b]中均匀分布，讨论区有两类典型错误。

第一类错误：

for i = 1 to b-a    r = a + Random(0,1)return r

第二类错误：

r = afor i = 1 to b - a //including b-a    r += Random(0,1)return r

正确方法：

1) Find the smallest number, p, such that 2^p > b-a.

2) Perform the following algorithm:

r=0for i = 1 to p    r = 2*r + Random(0,1)

3) If r is greater than b-a, go to step 2.

4) Your result is r+a

So let's try Random(1,3).
So b-a is 2.
2^1 = 2, so p will have to be 2 so that 2^p is greater than 2.
So we'll loop two times. Let's try all possible outputs:

00 -> r=0, 0 is not > 2, so we output 0+1 or 1.01 -> r=1, 1 is not > 2, so we output 1+1 or 2.10 -> r=2, 2 is not > 2, so we output 2+1 or 3.11 -> r=3, 3 is > 2, so we repeat.

在此我用java测试如下：

/** * 创建时间：2014年8月10日 下午2:48:08 项目名称：Test *  * @author Cao Yanfeng * @since JDK 1.6.0_21 类说明： */public class RandomNumbersTest {/** * @param args */public static void main(String[] args) {// TODO Auto-generated method stubSystem.out.println(getRandomNumber(1, 3));}/* * new Random().nextInt(2)用来模拟Random(0,1)，注意new * Random().nextInt(2)表示生成≥0且＜2的数，即生成0或1 */public static int getRandomNumber(int a, int b) {int diff = b - a;int p = getP(diff);int result = 0;for (int i = 1; i <= p; i++) {int random = new Random().nextInt(2);result = 2 * result + random;/* 如果r大于b-a，重复第2）步 */if (result > diff) {result = 0;i = 1;}}return result + a;}/* 获得最小的p,使得2^p>b-a */public static int getP(int diff) {for (int i = (int) Math.ceil(Math.log(diff) / Math.log(2));; i++) {if ((Math.pow(2, i)) > diff) {return i;}}}}

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