这篇博客介绍了Mersenne Twister算法，它是C++11中用于生成高质量随机数的算法。相比C标准库的rand()，Mersenne Twister具有更长的周期和更好的统计特性。博客提供了如何在C++11中使用Mersenne Twister算法生成随机数的基本示例，并展示了如何设定种子和获取指定范围内的随机整数。
一下内容整理自网络资源：
我们讲的随机数其实暗指伪随机数。不少朋友可能想到C语言的rand()，可惜这个函数产生的随机数随机性非常差，而且速度很慢，相信几乎不能胜任一般的应用。
古老的LCG(linear congruential generator)代表了最好的伪随机数产生器算法。主要原因是容易理解，容易实现，而且速度快。这种算法数学上基于X(n+1) = (a * X(n) + c) % m这样的公式，其中：模m, m > 0
系数a, 0 < a < m
增量c, 0 <= c < m
原始值(种子) 0 <= X(0) < m
其中参数c, m, a比较敏感，或者说直接影响了伪随机数产生的质量。
一般而言，高LCG的m是2的指数次幂(一般2^32或者2^64)，因为这样取模操作截断最右的32或64位就可以了。多数编译器的库中使用了该理论实现其伪随机数发生器rand()。下面是部分编译器使用的各个参数值：
| Source | m | a | c | rand() / Random(L)的种子位 |
| Numerical Recipes | 2^32 | 1664525 | 1013904223 | |
| Borland C/C++ | 2^32 | 22695477 | 1 | 位30..16 in rand(), 30..0 in lrand() |
| glibc (used by GCC) | 2^32 | 1103515245 | 12345 | 位30..0 |
| ANSI C: Watcom, Digital Mars, CodeWarrior, IBM VisualAge C/C++ | 2^32 | 1103515245 | 12345 | 位30..16 |
| Borland Delphi, Virtual Pascal | 2^32 | 134775813 | 1 | 位63..32 of (seed * L) |
| Microsoft Visual/Quick C/C++ | 2^32 | 214013 | 2531011 | 位30..16 |
| Apple CarbonLib | 2^31-1 | 16807 | 0 | 见Park–Miller随机数发生器 |
LCG不能用于随机数要求高的场合，例如不能用于Monte Carlo模拟，不能用于加密应用。
LCG有一些严重的缺陷，例如如果LCG用做N维空间的点坐标，这些点最多位于m1/n超平面上(Marsaglia定理)，这是由于产生的相继X(n)值的关联所致。
另外一个问题就是如果m设置为2的指数，产生的低位序列周期远远小于整体。
一般而言，输出序列的基数b中最低n位，bk = m (k是某个整数)，最大周期bn.
有些场合LCG有很好的应用，例如内存很紧张的嵌入式中，电子游戏控制台用的小整数，使用高位可以胜任。
LCG的一种C++实现版本如下：
- //************************************************************************
- // Copyright (C) 2008 - 2009 Chipset
- //
- // This program is free software: you can redistribute it and/or modify
- // it under the terms of the GNU Affero General Public License as
- // published by the Free Software Foundation, either version 3 of the
- // License, or (at your option) any later version.
- //
- // but WITHOUT ANY WARRANTY; without even the implied warranty of
- // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- // GNU Affero General Public License for more details.
- //
- // You should have received a copy of the GNU Affero General Public License
- // along with this program. If not, see <http://www.gnu.org/licenses/>.
- //************************************************************************
- #ifndef LCRANDOM_HPP_
- #define LCRANDOM_HPP_
- #include <ctime>
- class lcrandom
- {
- public:
- explicit lcrandom(size_t s = 0) : seed(s)
- {
- if (0 == seed) seed = std::time(0);
- randomize();
- }
- void reset(size_t s)
- {
- seed = s;
- if (0 == seed) seed = std::time(0);
- randomize();
- }
- size_t rand()
- {
- //returns a random integer in the range [0, -1UL)
- randomize();
- return seed;
- }
- double real()
- {
- //returns a random real number in the range [0.0, 1.0)
- randomize();
- return (double)(seed) / -1UL;
- }
- private:
- size_t seed;
- void randomize() { seed = 1103515245UL * seed + 12345UL; }
- };
- class lcrand_help
- {
- static lcrandom r;
- public:
- lcrand_help() {}
- void operator()(size_t s) { r.reset(s); }
- size_t operator()() const { return r.rand(); }
- double operator()(double) { return r.real(); }
- };
- lcrandom lcrand_help:: r;
- extern void lcsrand(size_t s) { lcrand_help()(s); }
- extern size_t lcirand() { return lcrand_help()(); }
- extern double lcdrand() { return lcrand_help()(1.0); }
- #endif // LCRANDOM_HPP_
如果需要高质量的伪随机数，内存充足(约2kb)，Mersenne twister算法是个不错的选择。Mersenne twister产生随机数的质量几乎超过任何LCG。不过一般Mersenne twister的实现使用LCG产生种子。
Mersenne twister是Makoto Matsumoto (松本)和Takuji Nishimura (西村)于1997年开发的伪随机数产生器，基于有限二进制字段上的矩阵线性再生。可以快速产生高质量的伪随机数，修正了古老随机数产生算法的很多缺陷。 Mersenne twister这个名字来自周期长度通常取Mersenne质数这样一个事实。常见的有两个变种Mersenne Twister MT19937和Mersenne Twister MT19937-64。
Mersenne Twister有很多长处，例如：周期2^19937 - 1对于一般的应用来说，足够大了，序列关联比较小，能通过很多随机性测试。
关于Mersenne Twister比较详细的论述请参阅 http://www.cppblog.com/Chipset/archive/2009/01/19/72330.html
用Mersenne twister算法实现的伪随机数版本非常多。例如boost库中的高质量快速随机数产生器就是用Mersenne twister算法原理编写的。
C++ : mt19937 Example
C++11 introduces several pseudo-random number generators designed to replace the good-oldrand from the C standard library. I’ll show basic usage examples of std::mt19937, which provides a random number generation based on Mersenne Twister algorithm. Using the Mersenne Twister implementation that comes with C++1 has advantage overrand(), among them:
mt19937has much longer period than that ofrand, e.g. it will take its random sequence much longer to repeat itself.- It much better statistical behavior.
- Several different random number generator engines can be initiated simultaneously with different seed, compared with the single “global” seed
srand()provides.
The downside is that mt19937 is a bit less straight-forward to use. However, I hope this post will help with this point :-).
We start with a basic example:
#include <iostream>
#include <random>
using namespace std;
int main()
{
mt19937 mt_rand(time(0));
cout << mt_rand() << endl;
return 0;
}Line 7 creates a new std::mt19937 object and seeds it with time(0). This is just like calling srand(time(0)). Subsequent calls to the newly created object,mt_rand will return a random 32-bit unsigned int.
If you want integers in a specific range, instead all kinds of arithmetics, C++11 provides convinient wrappers:
mt19937::result_type seed = time(0);
auto dice_rand = std::bind(std::uniform_int_distribution<int>(1,6),mt19937(seed));mt19937::result_type seed = time(0);
auto real_rand = std::bind(std::uniform_real_distribution<double>(0,1),mt19937(seed));dice_rand() will return an integer between 1 and 6 (inclusive). In the second example each call to
real_rand() will return a double in the range [0,1) (including 0, excluding 1). Note that usage of
std::bind requires
#include <functional>.
Finally if you want to go C++11 all the way, you can also replace time(0) with a proper call forstd::chrono when seeding the random number generator:
auto seed = chrono::high_resolution_clock::now().time_since_epoch().count();
mt19937 mt_rand(seed);The above code requires adding #include <chrono> to the list of includes.
参考文献：
[1] http://blog.youkuaiyun.com/hoxily/article/details/44241387
[2] http://www.guyrutenberg.com/2014/05/03/c-mt19937-example/
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