Levenshtein Distance

The Levenshtein distance is an algorithmic method allowing to quantify a distance between two words (more generally between 2 strings of characters). Two close words (that is to say that few things separate them spelling: they have several letters in common, in the same position) will then have a small distance, while two very different words will have a large distance. This distance is also called edit distance and is equal to the minimum number of characters to be deleted, inserted, or replaced to move from one string to another.

Levenshtein's algorithm evaluates the number of differences between the two character strings, the differences can be of 3 types: a substitution (replacement of one character by another), an insertion (addition of a new character) or a deletion (deletion of a character).

$$ \operatorname{Distance}(a,b) = \begin{cases} ||a|| & \text{ if } ||b|| = 0, \\ ||b|| & \text{ if } ||a|| = 0, \\ \operatorname{Distance}(a', b') & \text{ if } a[0] = b[0] \\ 1 + \min \begin{cases} \operatorname{Distance}(a', b) \\ \operatorname{Distance}(a, b') \\ \operatorname{Distance}(a', b') \\ \end{cases} & \text{ otherwise } \end{cases} $$

With $ ||a|| $ the size of the string, and $ a' $ the string $ a $ deprived of its first character (noted $ a[0] $)

The Damerau-Levenshtein distance is similar to the Levenshtein distance but adds a transposition of 2 adjacent characters type difference (which is a common typographical error).

The Levenshtein distance measures the similarity between two strings of characters.

Use the dCode tool by entering a word and a dictionary with which to compare the word.

All words with a similar spelling will be returned.

Calculating the distance between 2 strings of characters can make it possible to know their quantity of differences and thus also their amount of similarity. So the Levenshtein calculation algorithm can be used to determine typing errors or misspellings, words for which the proximity of the compared chains is strong.

The best known algorithm for calculating the Levenshtein distance was created by Wagner and Fischer in 1974. // Pseudo-code
function levenshteinDistance(str1, str2) {
size1 = length(str1)
size2 = length(str2)
matrix = [ size1 + 1 ] x [ size2 + 1 ]
for i from 0 to size1 { matrix[i][0] = i }
for j from 0 to size2 { matrix[0][j] = j }
for i from 1 to size1 {
for j from 1 to size2 {
if (str1[i - 1] == str2[j - 1]) cost = 0
else cost = 1
matrix[i][j] = minimum( matrix[i - 1][j] + 1, matrix[i][j - 1] + 1, matrix[i - 1][j - 1] + cost )
}
}
return matrix[size1][size2]
}