"A technique for counting ones in a binary computer". This article incorporates public domain material from the General Services Administration document: "Federal Standard 1037C"."Inferring HIV Transmission Dynamics from Phylogenetic Sequence Relationships". "Error detecting and error correcting codes" (PDF). The following function, written in Python 3, returns the Hamming distance between two strings: 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. The Hamming distance is also used in systematics as a measure of genetic distance. One can also view a binary string of length n as a vector in R n. 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. įor binary strings a and b the Hamming distance is equal to the number of ones ( population count) in a XOR b. 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, much as the difference between two integers can be seen as a distance from zero on the number line. Hence the Hamming distance between a and c is not larger than the sum of the Hamming distances between a and b and between b and c. For example, the Hamming distance between:įor a fixed length n, the Hamming distance is a metric on the set of the words of length n (also known as a Hamming space), as it fulfills the conditions of non-negativity, symmetry, the Hamming distance of two words is 0 if and only if the two words are identical, and it satisfies the triangle inequality as well: Indeed, if we fix three words a, b and c, then whenever there is a difference between the ith letter of a and the ith letter of c, then there must be a difference between the ith letter of a and ith letter of b, or between the ith letter of b and the ith letter of c. The symbols may be letters, bits, or decimal digits, among other possibilities. The Hamming distance between two equal-length strings of symbols is the number of positions at which the corresponding symbols are different.
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