产生伪随机数常用的两种算法
我们讲的随机数其实暗指伪随机数。提及随机数，不少朋友可能想到C语言的库函数rand()，rand()随机性太差，速度太慢。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
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的一种实现版本如下：
http://www.cppblog.com/Chipset/archive/2008/12/20/69918.html
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
一种实现版本如下：
1
//
************************************************************************
2 // This is a slightly modified version of Equamen mersenne twister.
3 //
4 // Copyright (C) 2009 Chipset
5 //
6 // This program is free software: you can redistribute it and/or modify
7 // it under the terms of the GNU Affero General Public License as
8 // published by the Free Software Foundation, either version 3 of the
9 // License, or (at your option) any later version.
10 //
11 // but WITHOUT ANY WARRANTY; without even the implied warranty of
12 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
13 // GNU Affero General Public License for more details.
14 //
15 // You should have received a copy of the GNU Affero General Public License
16 // along with this program. If not, see < http://www.gnu.org/licenses/ >.
17 // ************************************************************************
18
19 // Original Coyright (c) 1997 - 2002, Makoto Matsumoto and Takuji Nishimura
20 //
21 // Functions for MT19937, with initialization improved 2002/2/10.
22 // Coded by Takuji Nishimura and Makoto Matsumoto.
23 // This is a faster version by taking Shawn Cokus's optimization,
24 // Matthe Bellew's simplification, Isaku Wada's real version.
25 // C++ version by Lyell Haynes (Equamen)
26 //
27 // Redistribution and use in source and binary forms, with or without
28 // modification, are permitted provided that the following conditions
29 // are met:
30 //
31 // 1. Redistributions of source code must retain the above copyright
32 // notice, this list of conditions and the following disclaimer.
33 //
34 // 2. Redistributions in binary form must reproduce the above copyright
35 // notice, this list of conditions and the following disclaimer in the
36 // documentation and/or other materials provided with the distribution.
37 //
38 // 3. The names of its contributors may not be used to endorse or promote
39 // products derived from this software without specific prior written
40 // permission.
41 //
42 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
43 // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
44 // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
45 // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR
46 // CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
47 // EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
48 // PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
49 // PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
50 // LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
51 // NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
52 // SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
53 //
54
55 #ifndef mtrandom_HPP_
56 #define mtrandom_HPP_
57
58 #include < stddef.h >
59
60 class mtrandom
61 {
62 public :
63 mtrandom() : left( 1 ) { init(); }
64
65 explicit mtrandom(size_t seed) : left( 1 ) { init(seed); }
66
67 mtrandom(size_t * init_key, int key_length) : left( 1 )
68 {
69 int i = 1 , j = 0 ;
70 int k = N > key_length ? N : key_length;
71 init();
72 for (; k; -- k)
73 {
74 state[i] = (state[i] ^ ((state[i - 1 ] ^ (state[i - 1 ] >> 30 )) * 1664525UL )) + init_key[j] + j; // non linear
75 state[i] &= 4294967295UL ; // for WORDSIZE > 32 machines
76 ++ i;
77 ++ j;
78 if (i >= N)
79 {
80 state[ 0 ] = state[N - 1 ];
81 i = 1 ;
82 }
83 if (j >= key_length)
84 j = 0 ;
85 }
86
87 for (k = N - 1 ; k; -- k)
88 {
89 state[i] = (state[i] ^ ((state[i - 1 ] ^ (state[i - 1 ] >> 30 )) * 1566083941UL )) - i; // non linear
90 state[i] &= 4294967295UL ; // for WORDSIZE > 32 machines
91 ++ i;
92 if (i >= N)
93 {
94 state[ 0 ] = state[N - 1 ];
95 i = 1 ;
96 }
97 }
98
99 state[ 0 ] = 2147483648UL ; // MSB is 1; assuring non-zero initial array
100 }
101
102 void reset(size_t rs)
103 {
104 init(rs);
105 next_state();
106 }
107
108 size_t rand()
109 {
110 size_t y;
111 if ( 0 == -- left)
112 next_state();
113 y = * next ++ ;
114 // Tempering
115 y ^= (y >> 11 );
116 y ^= (y << 7 ) & 0x9d2c5680UL ;
117 y ^= (y << 15 ) & 0xefc60000UL ;
118 y ^= (y >> 18 );
119 return y;
120 }
121
122 double real() { return ( double )rand() / - 1UL ; }
123
124 // generates a random number on [0,1) with 53-bit resolution
125 double res53()
126 {
127 size_t a = rand() >> 5 , b = rand() >> 6 ;
128 return (a * 67108864.0 + b) / 9007199254740992.0 ;
129 }
130
131 private :
132 void init(size_t seed = 19650218UL )
133 {
134 state[ 0 ] = seed & 4294967295UL ;
135 for ( int j = 1 ; j < N; ++ j)
136 {
137 state[j] = ( 1812433253UL * (state[j - 1 ] ^ (state[j - 1 ] >> 30 )) + j);
138 // See Knuth TAOCP Vol2. 3rd Ed. P.106 for multiplier.
139 // In the previous versions, MSBs of the seed affect
140 // only MSBs of the array state[].
141 // 2002/01/09 modified by Makoto Matsumoto
142 state[j] &= 4294967295UL ; // for >32 bit machines
143 }
144 }
145
146 void next_state()
147 {
148 size_t * p = state;
149 int i;
150
151 for (i = N - M + 1 ; -- i; ++ p)
152 * p = (p[M] ^ twist(p[ 0 ], p[ 1 ]));
153
154 for (i = M; -- i; ++ p)
155 * p = (p[M - N] ^ twist(p[ 0 ], p[ 1 ]));
156 * p = p[M - N] ^ twist(p[ 0 ], state[ 0 ]);
157 left = N;
158 next = state;
159 }
160
161 size_t mixbits(size_t u, size_t v) const
162 {
163 return (u & 2147483648UL ) | (v & 2147483647UL );
164 }
165
166 size_t twist(size_t u, size_t v) const
167 {
168 return ((mixbits(u, v) >> 1 ) ^ (v & 1UL ? 2567483615UL : 0UL ));
169 }
170
171 static const int N = 624 , M = 397 ;
172 size_t state[N];
173 size_t left;
174 size_t * next;
175 };
176
177 class mtrand_help
178 {
179 static mtrandom r;
180 public :
181 mtrand_help() {}
182 void operator ()(size_t s) { r.reset(s); }
183 size_t operator ()() const { return r.rand(); }
184 double operator ()( double ) { return r.real(); }
185 };
186 mtrandom mtrand_help:: r;
187
188 extern void mtsrand(size_t s) { mtrand_help()(s); }
189 extern size_t mtirand() { return mtrand_help()(); }
190 extern double mtdrand() { return mtrand_help()( 1.0 ); }
191
192 #endif // mtrandom_HPP_
193
2 // This is a slightly modified version of Equamen mersenne twister.
3 //
4 // Copyright (C) 2009 Chipset
5 //
6 // This program is free software: you can redistribute it and/or modify
7 // it under the terms of the GNU Affero General Public License as
8 // published by the Free Software Foundation, either version 3 of the
9 // License, or (at your option) any later version.
10 //
11 // but WITHOUT ANY WARRANTY; without even the implied warranty of
12 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
13 // GNU Affero General Public License for more details.
14 //
15 // You should have received a copy of the GNU Affero General Public License
16 // along with this program. If not, see < http://www.gnu.org/licenses/ >.
17 // ************************************************************************
18
19 // Original Coyright (c) 1997 - 2002, Makoto Matsumoto and Takuji Nishimura
20 //
21 // Functions for MT19937, with initialization improved 2002/2/10.
22 // Coded by Takuji Nishimura and Makoto Matsumoto.
23 // This is a faster version by taking Shawn Cokus's optimization,
24 // Matthe Bellew's simplification, Isaku Wada's real version.
25 // C++ version by Lyell Haynes (Equamen)
26 //
27 // Redistribution and use in source and binary forms, with or without
28 // modification, are permitted provided that the following conditions
29 // are met:
30 //
31 // 1. Redistributions of source code must retain the above copyright
32 // notice, this list of conditions and the following disclaimer.
33 //
34 // 2. Redistributions in binary form must reproduce the above copyright
35 // notice, this list of conditions and the following disclaimer in the
36 // documentation and/or other materials provided with the distribution.
37 //
38 // 3. The names of its contributors may not be used to endorse or promote
39 // products derived from this software without specific prior written
40 // permission.
41 //
42 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
43 // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
44 // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
45 // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR
46 // CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
47 // EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
48 // PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
49 // PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
50 // LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
51 // NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
52 // SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
53 //
54
55 #ifndef mtrandom_HPP_
56 #define mtrandom_HPP_
57
58 #include < stddef.h >
59
60 class mtrandom
61 {
62 public :
63 mtrandom() : left( 1 ) { init(); }
64
65 explicit mtrandom(size_t seed) : left( 1 ) { init(seed); }
66
67 mtrandom(size_t * init_key, int key_length) : left( 1 )
68 {
69 int i = 1 , j = 0 ;
70 int k = N > key_length ? N : key_length;
71 init();
72 for (; k; -- k)
73 {
74 state[i] = (state[i] ^ ((state[i - 1 ] ^ (state[i - 1 ] >> 30 )) * 1664525UL )) + init_key[j] + j; // non linear
75 state[i] &= 4294967295UL ; // for WORDSIZE > 32 machines
76 ++ i;
77 ++ j;
78 if (i >= N)
79 {
80 state[ 0 ] = state[N - 1 ];
81 i = 1 ;
82 }
83 if (j >= key_length)
84 j = 0 ;
85 }
86
87 for (k = N - 1 ; k; -- k)
88 {
89 state[i] = (state[i] ^ ((state[i - 1 ] ^ (state[i - 1 ] >> 30 )) * 1566083941UL )) - i; // non linear
90 state[i] &= 4294967295UL ; // for WORDSIZE > 32 machines
91 ++ i;
92 if (i >= N)
93 {
94 state[ 0 ] = state[N - 1 ];
95 i = 1 ;
96 }
97 }
98
99 state[ 0 ] = 2147483648UL ; // MSB is 1; assuring non-zero initial array
100 }
101
102 void reset(size_t rs)
103 {
104 init(rs);
105 next_state();
106 }
107
108 size_t rand()
109 {
110 size_t y;
111 if ( 0 == -- left)
112 next_state();
113 y = * next ++ ;
114 // Tempering
115 y ^= (y >> 11 );
116 y ^= (y << 7 ) & 0x9d2c5680UL ;
117 y ^= (y << 15 ) & 0xefc60000UL ;
118 y ^= (y >> 18 );
119 return y;
120 }
121
122 double real() { return ( double )rand() / - 1UL ; }
123
124 // generates a random number on [0,1) with 53-bit resolution
125 double res53()
126 {
127 size_t a = rand() >> 5 , b = rand() >> 6 ;
128 return (a * 67108864.0 + b) / 9007199254740992.0 ;
129 }
130
131 private :
132 void init(size_t seed = 19650218UL )
133 {
134 state[ 0 ] = seed & 4294967295UL ;
135 for ( int j = 1 ; j < N; ++ j)
136 {
137 state[j] = ( 1812433253UL * (state[j - 1 ] ^ (state[j - 1 ] >> 30 )) + j);
138 // See Knuth TAOCP Vol2. 3rd Ed. P.106 for multiplier.
139 // In the previous versions, MSBs of the seed affect
140 // only MSBs of the array state[].
141 // 2002/01/09 modified by Makoto Matsumoto
142 state[j] &= 4294967295UL ; // for >32 bit machines
143 }
144 }
145
146 void next_state()
147 {
148 size_t * p = state;
149 int i;
150
151 for (i = N - M + 1 ; -- i; ++ p)
152 * p = (p[M] ^ twist(p[ 0 ], p[ 1 ]));
153
154 for (i = M; -- i; ++ p)
155 * p = (p[M - N] ^ twist(p[ 0 ], p[ 1 ]));
156 * p = p[M - N] ^ twist(p[ 0 ], state[ 0 ]);
157 left = N;
158 next = state;
159 }
160
161 size_t mixbits(size_t u, size_t v) const
162 {
163 return (u & 2147483648UL ) | (v & 2147483647UL );
164 }
165
166 size_t twist(size_t u, size_t v) const
167 {
168 return ((mixbits(u, v) >> 1 ) ^ (v & 1UL ? 2567483615UL : 0UL ));
169 }
170
171 static const int N = 624 , M = 397 ;
172 size_t state[N];
173 size_t left;
174 size_t * next;
175 };
176
177 class mtrand_help
178 {
179 static mtrandom r;
180 public :
181 mtrand_help() {}
182 void operator ()(size_t s) { r.reset(s); }
183 size_t operator ()() const { return r.rand(); }
184 double operator ()( double ) { return r.real(); }
185 };
186 mtrandom mtrand_help:: r;
187
188 extern void mtsrand(size_t s) { mtrand_help()(s); }
189 extern size_t mtirand() { return mtrand_help()(); }
190 extern double mtdrand() { return mtrand_help()( 1.0 ); }
191
192 #endif // mtrandom_HPP_
193