In information theory, linguistics, and computer science, the Levenshtein distance is a string metric for measuring the difference between two sequences. The Levenshtein distance between two words is the minimum number of single-character edits (insertions, deletions or substitutions) required to change one word into the other. It is named after Soviet mathematician Vladimir Levenshtein, who defined the metric in 1965. Levenshtein distance may also be referred to as edit distance, although that term may also denote a larger family of distance metrics. It is closely related to pairwise string alignments.
Definition The Levenshtein distance between two strings a , b {\displaystyle a,b} (of length | a | {\displaystyle |a|} and | b | {\displaystyle |b|} respectively) is given by lev ( a , b ) {\displaystyle \operatorname {lev} (a,b)} where
lev ( a , b ) = { | a | if | b | = 0 , | b | if | a | = 0 , lev ( tail ( a ) , tail ( b ) ) if head ( a ) = head ( b ) , 1 + min { lev ( tail ( a ) , b ) lev ( a , tail ( b ) ) lev ( tail ( a ) , tail ( b ) ) otherwise {\displaystyle \operatorname {lev} (a,b)={\begin{cases}|a|&{\text{ if }}|b|=0,\\|b|&{\text{ if }}|a|=0,\\\operatorname {lev} {\big (}\operatorname {tail} (a),\operatorname {tail} (b){\big )}&{\text{ if }}\operatorname {head} (a)=\operatorname {head} (b),\\1+\min {\begin{cases}\operatorname {lev} {\big (}\operatorname {tail} (a),b{\big )}\\\operatorname {lev} {\big (}a,\operatorname {tail} (b){\big )}\\\operatorname {lev} {\big (}\operatorname {tail} (a),\operatorname {tail} (b){\big )}\\\end{cases}}&{\text{ otherwise}}\end{cases}}}
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