Hamming weight

The Hamming weight of a string is the number of symbols that are different from the zero-symbol of the alphabet used. It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the most typical case, a string of bits, this is the number of 1's in the string. In this binary case, it is also called the population count, popcount, or sideways sum.^[1] It is the digit sum of the binary representation of a given number and the ℓ₁ norm of a bit vector.

History and usage

The Hamming weight is named after Richard Hamming although he did not originate the notion.^[2] The Hamming weight of binary numbers was already used in 1899 by J. W. L. Glaisher to give a formula for the number of odd binomial coefficients in a single row of Pascal's triangle.^[3] Irving S. Reed introduced a concept, equivalent to Hamming weight in the binary case, in 1954.^[4]

Hamming weight is used in several disciplines including information theory, coding theory, and cryptography. Examples of applications of the Hamming weight include:

• In modular exponentiation by squaring, the number of modular multiplications required for an exponent e is log₂ e + weight(e). This is the reason that the public key value e used in RSA is typically chosen to be a number of low Hamming weight.
• The Hamming weight determines path lengths between nodes in Chord distributed hash tables.^[5]
• IrisCode lookups in biometric databases are typically implemented by calculating the Hamming distance to each stored record.
• In computer chess programs using a bitboard representation, the Hamming weight of a bitboard gives the number of pieces of a given type remaining in the game, or the number of squares of the board controlled by one player's pieces, and is therefore an important contributing term to the value of a position.
• Hamming weight can be used to efficiently compute find first set using the identity ffs(x) = pop(x ^ (~(-x))). This is useful on platforms such as SPARC that have hardware Hamming weight instructions but no hardware find first set instruction.^[6]
• The Hamming weight operation can be interpreted as a conversion from the unary numeral system to binary numbers.^[7]
• In implementation of some succinct data structures like bit vectors and wavelet trees.

Efficient implementation

The population count of a bitstring is often needed in cryptography and other applications. The Hamming distance of two words A and B can be calculated as the Hamming weight of A xor B.

The problem of how to implement it efficiently has been widely studied. Some processors have a single command to calculate it (see below), and some have parallel operations on bit vectors. For processors lacking those features, the best solutions known are based on adding counts in a tree pattern. For example, to count the number of 1 bits in the 16-bit binary number a = 0110 1100 1011 1010, these operations can be done:

Here, the operations are as in C programming language, so X >> Y means to shift X right by Y bits, X & Y means the bitwise AND of X and Y, and + is ordinary addition. The best algorithms known for this problem are based on the concept illustrated above and are given here:

//types and constants used in the functions below
//uint64_t is an unsigned 64-bit integer variable type (defined in C99 version of C language)
const uint64_t m1  = 0x5555555555555555; //binary: 0101...
const uint64_t m2  = 0x3333333333333333; //binary: 00110011..
const uint64_t m4  = 0x0f0f0f0f0f0f0f0f; //binary:  4 zeros,  4 ones ...
const uint64_t m8  = 0x00ff00ff00ff00ff; //binary:  8 zeros,  8 ones ...
const uint64_t m16 = 0x0000ffff0000ffff; //binary: 16 zeros, 16 ones ...
const uint64_t m32 = 0x00000000ffffffff; //binary: 32 zeros, 32 ones
const uint64_t hff = 0xffffffffffffffff; //binary: all ones
const uint64_t 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.
//This algorithm uses 24 arithmetic operations (shift, add, and).
int popcount64a(uint64_t 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.
//This algorithm uses 17 arithmetic operations.
int popcount64b(uint64_t 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 & 0x7f;
}

//This uses fewer arithmetic operations than any other known  
//implementation on machines with fast multiplication.
//This algorithm uses 12 arithmetic operations, one of which is a multiply.
int popcount64c(uint64_t 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) + ... 
}

The above implementations have the best worst-case behavior of any known algorithm. However, when a value is expected to have few nonzero bits, it may instead be more efficient to use algorithms that count these bits one at a time. As Wegner (1960) described,^[8] the bitwise and of x with x − 1 differs from x only in zeroing out the least significant nonzero bit: subtracting 1 changes the rightmost string of 0s to 1s, and changes the rightmost 1 to a 0. If x originally had n bits that were 1, then after only n iterations of this operation, x will be reduced to zero. The following implementation is based on this principle.

//This is better when most bits in x are 0
//This is algorithm works the same for all data sizes.
//This algorithm uses 3 arithmetic operations and 1 comparison/branch per "1" bit in x.
int popcount64d(uint64_t x)
{
    int count;
    for (count=0; x; count++)
        x &= x - 1;
    return count;
}

If we are allowed greater memory usage, we can calculate the Hamming weight faster than the above methods. With unlimited memory, we could simply create a large lookup table of the Hamming weight of every 64 bit integer. If we can store a lookup table of the hamming function of every 16 bit integer, we can do the following to compute the Hamming weight of every 32 bit integer.

static uint8_t wordbits[65536] = { /* bitcounts of integers 0 through 65535, inclusive */ };
//This algorithm uses 3 arithmetic operations and 2 memory reads.
int popcount32e(uint32_t x)
{
    return wordbits[x & 0xFFFF] + wordbits[x >> 16];
}

//Optionally, the wordbits[] table could be filled using this function
int popcount32e_init(void)
{
    uint32_t i;
    uint16_t x;
    int count;
    for (i=0; i <= 0xFFFF; i++)
    {
        x = i;
        for (count=0; x; count++) // borrowed from popcount64d() above
            x &= x - 1;
        wordbits[i] = count;
    }
}

Language support

Some C compilers provide intrinsic functions that provide bit counting facilities. For example, GCC (since version 3.4 in April 2004) includes a builtin function __builtin_popcount that will use a processor instruction if available or an efficient library implementation otherwise.^[9] LLVM-GCC has included this function since version 1.5 in June, 2005.^[10]

In C++ STL, the bit-array data structure bitset has a count() method that counts the number of bits that are set.

In Java, the growable bit-array data structure BitSet has a BitSet.cardinality() method that counts the number of bits that are set. In addition, there are Integer.bitCount(int) and Long.bitCount(long) functions to count bits in primitive 32-bit and 64-bit integers, respectively. Also, the BigInteger arbitrary-precision integer class also has a BigInteger.bitCount() method that counts bits.

In Common Lisp, the function logcount, given a non-negative integer, returns the number of 1 bits. (For negative integers it returns the number of 0 bits in 2's complement notation.) In either case the integer can be a BIGNUM.

Starting in GHC 7.4, the Haskell base package has a popCount function available on all types that are instances of the Bits class (available from the Data.Bits module).^[11]

MySQL version of SQL language provides BIT_COUNT() as a standard function.^[12]

Fortran 2008 has the standard, intrinsic, elemental function popcnt returning the number of nonzero bits within an integer (or integer array), see page 380 in Metcalf, Michael; John Reid; Malcolm Cohen (2011). Modern Fortran Explained. Oxford University Press. ISBN 0-19-960142-9. 

Processor support

• Cray supercomputers early on featured a population count machine instruction, rumoured to have been specifically requested by the U.S. government National Security Agency for cryptanalysis applications.
• Some of Control Data Corporation's CYBER 70/170 series machines included a population count instruction; in COMPASS, this instruction was coded as CXi.
• AMD's Barcelona architecture introduced the abm (advanced bit manipulation) ISA introducing the POPCNT instruction as part of the SSE4a extensions.
• Intel Core processors introduced a POPCNT instruction with the SSE4.2 instruction set extension, first available in a Nehalem-based Core i7 processor, released in November 2008.
• Compaq's Alpha 21264A, released in 1999, was the first Alpha series CPU design that had the count extension (CIX).
• Donald Knuth's model computer MMIX that is going to replace MIX in his book The Art of Computer Programming has an SADD instruction. SADD a,b,c counts all bits that are 1 in b and 0 in c and writes the result to a.
• The ARM architecture introduced the VCNT instruction as part of the Advanced SIMD (NEON) extensions.
• Analog Devices' Blackfin processors feature the ONES instruction to perform a 32-bit population count.

See also

• Minimum weight
• Two's complement
• Most frequent k characters

References

External links

• Aggregate Magic Algorithms. Optimized population count and other algorithms explained with sample code.
• HACKMEM item 169. Population count assembly code for the PDP/6-10.
• Bit Twiddling Hacks Several algorithms with code for counting bits set.
• Necessary and Sufficient - by Damien Wintour - Has code in C# for various Hamming Weight implementations.
• Best algorithm to count the number of set bits in a 32-bit integer? - Stackoverflow