**汉明权重**是一串符号中非零符号的个数。因此它等同于同样长度的全零符号串的汉明距离。在最为常见的数据位符号串中，它是1的个数。

|  |  |  |
| --- | --- | --- |
| 字符 | 字符串 | 汉明权重 |
| 0,1 | 11101 | 4 |
| 0,1 | 11101000 | 4 |
| 0,1 | 00000000 | 0 |
| ' ',a-z | hello world | 11 |

历史及应用
-----

汉明权重是以理查德·衛斯里·漢明的名字命名的，它在包括信息论、编码理论、密码学等多个领域都有应用。

高效实现
----

在密码学以及其它应用中经常需要计算数据位中1的个数，针对如何高效地实现人们已经广泛地进行了研究。一些处理器使用单个的命令进行计算，另外一些根据数据位向量使用并行运算进行处理。对于没有这些特性的处理器来说，已知的最好解决办法是按照树状进行相加。例如，要计算二进制数A=0110110010111010中1的个数，这些运算可以表示为：

|  |  |  |  |
| --- | --- | --- | --- |
| 符号 | 二进制 | 十进制 | 注释 |
| A | 01 10 11 00 10 11 10 10 |  | 原始数据 |
| B = A & 01 01 01 01 01 01 01 01 | 01 00 01 00 00 01 00 00 | 1,0,1,0,0,1,0,0 | A隔一位检验 |
| C = (A >> 1) & 01 01 01 01 01 01 01 01 | 00 01 01 00 01 01 01 01 | 0,1,1,0,1,1,1,1 | A中剩余的数据位 |
| D = B + C | 01 01 10 00 01 10 01 01 | 1,1,2,0,1,2,1,1 | A中每个双位段中1的个数列表 |
| E = D & 0011 0011 0011 0011 | 00 01 00 00 00 10 00 01 | 1,0,2,1 | D中数据隔一位检验 |
| F = (D >> 2) & 0011 0011 0011 0011 | 00 01 00 10 00 01 00 01 | 1,2,1,1 | D中剩余数据的计算 |
| G = E + F | 00 10 00 10 00 11 00 10 | 2,2,3,2 | A中4位数据段中1的个数列表 |
| H = G & 00001111 00001111 | 00 00 00 10 00 00 00 10 | 2,2 | G中数据隔一位检验 |
| I = (G >> 4) & 00001111 00001111 | 00 00 00 10 00 00 00 11 | 2,3 | G中剩余数据的计算 |
| J = H + I | 00 00 01 00 00 00 01 01 | 4,5 | A中8位数据段中1的个数列表 |
| K = J & 0000000011111111 | 00 00 00 00 00 00 01 01 | 5 | J中隔一位检验 |
| L = (J >> 8) & 0000000011111111 | 00 00 00 00 00 00 01 00 | 4 | J中剩余数据的检验 |
| M = K + L | 00 00 00 00 00 00 10 01 | 9 | 最终答案 |

这里的运算是用C语言表示的，所以X >> Y表示X右移Y位，X & Y表示X与Y的位与，+表示普通的加法。基于上面所讨论的思想的这个问题的最好算法列在这里：

```
//types and constants used in the functions below

typedef unsigned \_\_int64 uint64; //assume this gives 64-bits
const uint64 m1 = 0x5555555555555555; //binary: 0101...
const uint64 m2 = 0x3333333333333333; //binary: 00110011..
const uint64 m4 = 0x0f0f0f0f0f0f0f0f; //binary: 4 zeros, 4 ones ...
const uint64 m8 = 0x00ff00ff00ff00ff; //binary: 8 zeros, 8 ones ...
const uint64 m16 = 0x0000ffff0000ffff; //binary: 16 zeros, 16 ones ...
const uint64 m32 = 0x00000000ffffffff; //binary: 32 zeros, 32 ones ...
const uint64 hff = 0xffffffffffffffff; //binary: all ones
const uint64 h01 = 0x0101010101010101; //the sum of 256 to the power of 0,1,2,3...

//This is a naive implementation, shown for comparison,
//and to help in understanding the better functions.
//It uses 24 arithmetic operations (shift, add, and).
int popcount\_1(uint64 x) {
 x = (x & m1 ) + ((x >> 1) & m1 ); //put count of each 2 bits into those 2 bits 
 x = (x & m2 ) + ((x >> 2) & m2 ); //put count of each 4 bits into those 4 bits 
 x = (x & m4 ) + ((x >> 4) & m4 ); //put count of each 8 bits into those 8 bits 
 x = (x & m8 ) + ((x >> 8) & m8 ); //put count of each 16 bits into those 16 bits 
 x = (x & m16) + ((x >> 16) & m16); //put count of each 32 bits into those 32 bits 
 x = (x & m32) + ((x >> 32) & m32); //put count of each 64 bits into those 64 bits 
 return x;
}

//This uses fewer arithmetic operations than any other known 
//implementation on machines with slow multiplication.
//It uses 17 arithmetic operations.
int popcount\_2(uint64 x) {
 x -= (x >> 1) & m1; //put count of each 2 bits into those 2 bits
 x = (x & m2) + ((x >> 2) & m2); //put count of each 4 bits into those 4 bits 
 x = (x + (x >> 4)) & m4; //put count of each 8 bits into those 8 bits 
 x += x >> 8; //put count of each 16 bits into their lowest 8 bits
 x += x >> 16; //put count of each 32 bits into their lowest 8 bits
 x += x >> 32; //put count of each 64 bits into their lowest 8 bits
 return x &0xff;
}

//This uses fewer arithmetic operations than any other known 
//implementation on machines with fast multiplication.
//It uses 12 arithmetic operations, one of which is a multiply.
int popcount\_3(uint64 x) {
 x -= (x >> 1) & m1; //put count of each 2 bits into those 2 bits
 x = (x & m2) + ((x >> 2) & m2); //put count of each 4 bits into those 4 bits 
 x = (x + (x >> 4)) & m4; //put count of each 8 bits into those 8 bits 
 return (x \* h01)>>56; //returns left 8 bits of x + (x<<8) + (x<<16) + (x<<24) + ... 
}

```

在最坏的情况下，上面的实现是所有已知算法中表现最好的。但是，如果已知大多数数据位是0的话，那么还有更快的算法。这些更快的算法是基于这样一种事实即X与X-1相*与*得到的最低位永远是0。例如：

|  |  |
| --- | --- |
| Expression | Value |
| X | 0 1 0 0 0 1 0 0 0 1 0 0 0 0 |
| X-1 | 0 1 0 0 0 1 0 0 0 0 1 1 1 1 |
| X & (X-1) | 0 1 0 0 0 1 0 0 0 0 0 0 0 0 |

减1操作将最右边的符号从0变到1，从1变到0，*与*操作将会移除最右端的1。如果最初X有N个1，那么经过N次这样的迭代运算，X将减到0。下面的算法就是根据这个原理实现的。

```
//This is better when most bits in x are 0
//It uses 3 arithmetic operations and one comparison/branch per "1" bit in x.
int popcount\_4(uint64 x) {
 uint64 count;
 for (count=0; x; count++)
 x &= x-1;
 return count;
}

//This is better if most bits in x are 0.
//It uses 2 arithmetic operations and one comparison/branch per "1" bit in x.
//It is the same as the previous function, but with the loop unrolled.
#define f(y) if ((x &= x-1) == 0) return y;
int popcount\_5(uint64 x) {
 if (x == 0) return 0;
 f( 1) f( 2) f( 3) f( 4) f( 5) f( 6) f( 7) f( 8)
 f( 9) f(10) f(11) f(12) f(13) f(14) f(15) f(16)
 f(17) f(18) f(19) f(20) f(21) f(22) f(23) f(24)
 f(25) f(26) f(27) f(28) f(29) f(30) f(31) f(32)
 f(33) f(34) f(35) f(36) f(37) f(38) f(39) f(40)
 f(41) f(42) f(43) f(44) f(45) f(46) f(47) f(48)
 f(49) f(50) f(51) f(52) f(53) f(54) f(55) f(56)
 f(57) f(58) f(59) f(60) f(61) f(62) f(63)
 return 64;
}

//Use this instead if most bits in x are 1 instead of 0
#define f(y) if ((x |= x+1) == hff) return 64-y;

```

参见
--

* 汉明距离

外部链接
----

* Aggregate Magic Algorithms （页面存档备份，存于）.优化汉明重量算法及其它算法解释（附源代码）。
* HACKMEM item 169 （页面存档备份，存于）. Population count assembly code for the PDP/6-10.
* Bit Twiddling Hacks （页面存档备份，存于）带有源代码的几种汉明重量计算方法