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Levenshtein distance (or edit distance) between two strings is the number of deletions, insertions, or substitutions required to transform source string into target string.
For example, if source is "book" and target is "back", to transform "book" to "back" you will need to change first "o" to "a", second "o" to "c", without additional deletions and insertions, thus, levenshtein distance will be 2.
Levenshtein distance is named after the Russian scientist Vladimir Levenshtein, who devised the algorithm in 1965.
This has a wide range of applications, for instance, spell checkers, correction systems for optical character recognition, etc.
You can find more information, for example, in Wikipedia