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In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding are different. In other words, it measures the minimum number of substitutions required to change one string into the other, or the minimum number of errors that could have transformed one string into the other. In a more general context, the Hamming distance is one of several for measuring the between two sequences.

A major application is in , more specifically to , in which the equal-length strings are over a .


Examples
The Hamming distance between:
  • " karolin" and " kathrin" is 3.
  • " karolin" and " kerstin" is 3.
  • 1011101 and 1001001 is 2.
  • 2173896 and 2233796 is 3.


Properties
For a fixed length n, the Hamming distance is a metric on the set of the words of length n (also known as a ), as it fulfills the conditions of non-negativity, identity of indiscernibles and symmetry, and it can be shown by complete induction that it satisfies the triangle inequality as well. The Hamming distance between two words a and b can also be seen as the of ab for an appropriate choice of the − operator.

For binary strings a and b the Hamming distance is equal to the number of ones () in a b. The metric space of length- n binary strings, with the Hamming distance, is known as the Hamming cube; it is equivalent as a metric space to the set of distances between vertices in a . One can also view a binary string of length n as a vector in \mathbb{R}^{n} by treating each symbol in the string as a real coordinate; with this embedding, the strings form the vertices of an n-dimensional , and the Hamming distance of the strings is equivalent to the Manhattan distance between the vertices.


Error detection and error correction
The Hamming distance is used to define some essential notions in , such as error detecting and error correcting codes. In particular, a code C is said to be k-errors detecting if any two codewords c1 and c2 from C that have a Hamming distance less than k coincide; otherwise, a code is k-errors detecting if, and only if, the minimum Hamming distance between any two of its codewords is at least k+1.
(2018). 9783110198164, Walter de Gruyter.

A code C is said to be k-errors correcting if, for every word w in the underlying Hamming space H, there exists at most one codeword c (from C) such that the Hamming distance between w and c is less than k. In other words, a code is k-errors correcting if, and only if, the minimum Hamming distance between any two of its codewords is at least 2 k+1. This is more easily understood geometrically as any closed balls of radius k centered on distinct codewords being disjoint. These balls are also called in this context.

Thus a code with minimum Hamming distance d between its codewords can detect at most d-1 errors and can correct ⌊( d-1)/2⌋ errors. The latter number is also called the packing radius or the error-correcting capability of the code.


History and applications
The Hamming distance is named after , who introduced the concept in his fundamental paper on Error detecting and error correcting codes in 1950. Hamming weight analysis of bits is used in several disciplines including information theory, , and .

It is used in telecommunication to count the number of flipped bits in a fixed-length binary word as an estimate of error, and therefore is sometimes called the signal distance.

(2018). 9783642361562, Springer.
For q-ary strings over an of size q ≥ 2 the Hamming distance is applied in case of the q-ary symmetric channel, while the is used for phase-shift keying or more generally channels susceptible to synchronization errors because the Lee distance accounts for errors of ±1.
(2018). 9780521845045, Cambridge University Press.
If q = 2 or q = 3 both distances coincide because any pair of elements from \mathbb{Z}/2\mathbb{Z} or \mathbb{Z}/3\mathbb{Z} differ by 1, but the distances are different for larger q.

The Hamming distance is also used in as a measure of genetic distance.

However, for comparing strings of different lengths, or strings where not just substitutions but also insertions or deletions have to be expected, a more sophisticated metric like the Levenshtein distance is more appropriate.


Algorithm example
The function hamming_distance(), implemented in Python 2.3+, computes the Hamming distance between two strings (or other objects) of equal length by creating a sequence of Boolean values indicating mismatches and matches between corresponding positions in the two inputs and then summing the sequence with False and True values being interpreted as zero and one.

def hamming_distance(s1, s2):

   """Return the Hamming distance between equal-length sequences"""
   if len(s1) != len(s2):
       raise ValueError("Undefined for sequences of unequal length")
   return sum(el1 != el2 for el1, el2 in zip(s1, s2))
     
where the zip() function merges two equal-length collections in pairs.

The following C function will compute the Hamming distance of two integers (considered as binary values, that is, as sequences of bits). The running time of this procedure is proportional to the Hamming distance rather than to the number of bits in the inputs. It computes the bitwise of the two inputs, and then finds the of the result (the number of nonzero bits) using an algorithm of that repeatedly finds and clears the lowest-order nonzero bit. Some compilers support the __builtin_popcount function which can calculate this using specialized processor hardware where available.

int hamming_distance(unsigned x, unsigned y) {

   int dist = 0;
   unsigned  val = x ^ y;
     

   // Count the number of bits set
   while (val != 0)
   {
       // A bit is set, so increment the count and clear the bit
       dist++;
       val &= val - 1;
   }
     

   // Return the number of differing bits
   return dist;
     
}

Or, a much faster hardware alternative (for compilers that support builtins) is to use popcount like so.

int hamming_distance(unsigned x, unsigned y) {

   return __builtin_popcount(x ^ y);
     
} //if your compiler supports 64-bit integers int hamming_distance(unsigned long long x, unsigned long long y) {
   return __builtin_popcountll(x ^ y);
     
}


See also


Further reading

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