# Hamming distance

In information theory, the **Hamming distance** between two strings of equal length is the number of positions at which the corresponding symbols are different. In another way, 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.

## Contents

## 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.

## Special properties

For a fixed length *n*, the Hamming distance is a metric on the vector space of the words of length n, 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 Hamming weight of *a*−*b* for an appropriate choice of the − operator.

For **binary strings** *a* and *b* the Hamming distance is equal to the number of ones (population count) in *a* XOR *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 hypercube graph. One can also view a binary string of length *n* as a vector in by treating each symbol in the string as a real coordinate; with this embedding, the strings form the vertices of an *n*-dimensional hypercube, and the Hamming distance of the strings is equivalent to the Manhattan distance between the vertices.

## History and applications

The Hamming distance is named after Richard Hamming, who introduced it in his fundamental paper on Hamming codes *Error detecting and error correcting codes* in 1950.^{1} 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**. Hamming weight analysis of bits is used in several disciplines including information theory, coding theory, and cryptography. 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. For *q*-ary strings over an alphabet of size *q* ≥ 2 the Hamming distance is applied in case of orthogonal modulation, while the Lee distance is used for phase modulation. If *q* = 2 or *q* = 3 both distances coincide.

The Hamming distance is also used in systematics as a measure of genetic distance.^{2}

On a grid such as a chessboard, the Hamming distance is the minimum number of moves it would take a rook to move from one cell to the other.

## Algorithm example

The Python function `hamming_distance()`

computes the Hamming distance between two strings (or other iterable 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(ch1 != ch2 for ch1, ch2 in zip(s1, s2))

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 exclusive or of the two inputs, and then finds the Hamming weight of the result (the number of nonzero bits) using an algorithm of Wegner (1960) that repeatedly finds and clears the lowest-order nonzero bit.

unsigned hamdist(unsigned x, unsigned y) { unsigned dist = 0, val = x ^ y; // XOR // Count the number of set bits while(val) { ++dist; val &= val - 1; } return dist; }

## See also

- Closest string
- Damerau–Levenshtein distance
- Euclidean distance
- Mahalanobis distance
- Jaccard index
- String metric
- Sørensen similarity index
- Word ladder

## Notes

## References

- This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C".
- Hamming, Richard W. (1950), "Error detecting and error correcting codes",
*Bell System Technical Journal***29**(2): 147–160, MR 0035935. - Pilcher, C. D.; Wong, J. K.; Pillai, S. K. (March 2008), "Inferring HIV transmission dynamics from phylogenetic sequence relationships",
*PLoS Med.***5**(3): e69, doi:10.1371/journal.pmed.0050069, PMC 2267810, PMID 18351799. - Wegner, Peter (1960), "A technique for counting ones in a binary computer",
*Communications of the ACM***3**(5): 322, doi:10.1145/367236.367286.

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