Hamming weight

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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.

Examples[edit]

string Hamming weight
11101 4
11101000 4
00000000 0
hello world 11

History and usage[edit]

The Hamming weight is named after Richard Hamming although he did not originate the notion.[2] Irving S. Reed introduced a concept, equivalent to Hamming weight in the binary case, in 1954.[3]

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.[4]
  • 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.[5]

Efficient implementation[edit]

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=0110110010111010, these operations can be done:

Expression Binary Decimal Comment
A 01 10 11 00 10 11 10 10 The original number
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 every other bit from A
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 the remaining bits from A
D = B + C 01 01 10 00 01 10 01 01 1,1,2,0,1,2,1,1 list giving # of 1s in each 2-bit piece of A
E = D & 0011 0011 0011 0011 0001 0000 0010 0001 1,0,2,1 every other count from D
F = (D >> 2) & 0011 0011 0011 0011 0001 0010 0001 0001 1,2,1,1 the remaining counts from D
G = E + F 0010 0010 0011 0010 2,2,3,2 list giving # of 1s in each 4-bit piece of A
H = G & 00001111 00001111 00000010 00000010 2,2 every other count from G
I = (G >> 4) & 00001111 00001111 00000010 00000011 2,3 the remaining counts from G
J = H + I 00000100 00000101 4,5 list giving # of 1s in each 8-bit piece of A
K = J & 0000000011111111 0000000000000101 5 every other count from J
L = (J >> 8) & 0000000011111111 0000000000000100 4 the remaining counts from J
M = K + L 0000000000001001 9 the final answer

Here, the operations are as in C, 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
 
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.
//It uses 24 arithmetic operations (shift, add, and).
int popcount_1(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.
//It uses 17 arithmetic operations.
int popcount_2(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.
//It uses 12 arithmetic operations, one of which is a multiply.
int popcount_3(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,[6] 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
//It uses 3 arithmetic operations and one comparison/branch per "1" bit in x.
int popcount_4(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 */ };
static int popcount(uint32_t i)
{
    return (wordbits[i&0xFFFF] + wordbits[i>>16]);
}

Language support[edit]

Some C compilers provide intrinsics 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.[7] LLVM-GCC has included this function since version 1.5 in June, 2005.[8]

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). [9]

MySQL version of SQL language provides BIT_COUNT() as a standard function.[10]

Processor support[edit]

See also[edit]

References[edit]

  1. ^ D. E. Knuth (2009). The Art of Computer Programming Volume 4, Fascicle 1: Bitwise tricks & techniques; Binary Decision Diagrams. Addison–Wesley Professional. ISBN 0-321-58050-8.  Draft of Fascicle 1b available for download.
  2. ^ Thompson, Thomas M. (1983), From Error-Correcting Codes through Sphere Packings to Simple Groups, The Carus Mathematical Monographs #21, The Mathematical Association of America, p. 33 
  3. ^ Reed, I.S. (1954), "A Class of Multiple-Error-Correcting Codes and the Decoding Scheme", I.R.E. (I.E.E.E.), PGIT-4: 38 
  4. ^ Stoica, I., Morris, R., Liben-Nowell, D., Karger, D. R., Kaashoek, M. F., Dabek, F., and Balakrishnan, H. Chord: a scalable peer-to-peer lookup protocol for internet applications. IEEE/ACM Trans. Netw. 11, 1 (Feb. 2003), 17-32. Section 6.3: "In general, the number of fingers we need to follow will be the number of ones in the binary representation of the distance from node to query."
  5. ^ SPARC International, Inc. (1992). The SPARC architecture manual : version 8 (Version 8. ed.). Englewood Cliffs, N.J.: Prentice Hall. p. 231. ISBN 0-13-825001-4.  A.41: Population Count. Programming Note.
  6. ^ Wegner, Peter (1960), "A technique for counting ones in a binary computer", Communications of the ACM 3 (5): 322, doi:10.1145/367236.367286 
  7. ^ "GCC 3.4 Release Notes" GNU Project
  8. ^ "LLVM 1.5 Release Notes" LLVM Project.
  9. ^ "GHC 7.4.1 release notes".  GHC documentation.
  10. ^ [12.11. Bit Functions — MySQL 5.0 Reference Manual]

External links[edit]