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Levenshtein DistanceIn information theory, the Levenshtein distance or edit distance between two strings is given by the minimum number of operations needed to transform one string into the other, where an operation is an insertion, deletion, or substitution. It is named after the Russian scientist Vladimir Levenshtein, who considered this distance in 1965. It is useful in applications that need to determine how similar two strings are, such as spell checkers. For example, the Levenshtein distance between "kitten" and "sitting" is 3, since these three edits change one into the other, and there is no way to do it with less than three edits: - kitten
- sitten (substitution of 'k' for 's')
- sittin (substitution of 'i' for 'e')
- sitting (insert 'g' at the end)
It can be considered a generalization of the Hamming distance, which is used for strings of the same length and only considers substitution edits. There are also further generalizations of the Levenshtein distance that consider, for example, exchanging two characters as an operation. The algorithm A commonly-used algorithm for computing the Levenshtein distance involves the use of an (n + 1) × (m + 1) matrix, where n and m are the lengths of the two strings. Here is pseudocode for a function LevenshteinDistance that takes two strings, str1 of length lenStr1, and str2 of length lenStr2, and computes the Levenshtein distance between them: int LevenshteinDistance ( char str11..lenStr1, char str21..lenStr2 ) // d is a table with lenStr1+1 rows and lenStr2+1 columns declare int d0..lenStr2 // i1 and i2 are used to iterate over str1 and str2 declare int i1, i2, cost for i1 from 0 to lenStr1 d0 := i1 for i2 from 0 to lenStr2 di2 := i2 for i1 from 1 to lenStr1 for i2 from 1 to lenStr2 if str1i1 = str2i2 then cost := 0 else cost := 1 di2 = minimum( d- 1, i2 + 1, // insertion d , i2 - 1 + 1, // deletion d- 1, i2 - 1 + cost // substitution ) return dlenStr2 Possible improvements Possible improvements to this algorithm include: - We can adapt the algorithm to use less space, O(m) instead of O(mn), since it only requires that the previous row and current row be stored at any one time.
- We can store the number of insertions, deletions, and substitutions separately, or even the positions at which they occur, which is always
j. - We can give different penalty costs to insertion, deletion and substitution.
- The initialization of
di,0 can be moved inside the main outer loop. - This algorithm parallelizes poorly, due to a large number of data dependencies. However, all the
costs can be computed in parallel, and the algorithm can be adapted to perform the minimum function in phases to eliminate dependencies. Proof of correctness The invariant is that we can transform the initial segment s1..i into t1..j using a minimum of di,j operations. This invariant holds since: - It is initially true on row and column 0 because
s1..i can be transformed into the empty string t1..0 by simply dropping all i characters. Similarly, we can transform s1..0 to t1..j by simply adding all j characters. - The minimum is taken over three distances, each of which is feasible:
- If we can transform
s1..i to t1..j-1 in k operations, then we can simply add tj afterwards to get t1..j in k+1 operations. - If we can transform
s1..i-1 to t1..j in k operations, then we can do the same operations on s1..i and then remove the original si at the end in k+1 operations. - If we can transform
s1..i-1 to t1..j-1 in k operations, we can do the same to s1..i and then do a substitution of tj for the original si at the end if necessary, requiring k+cost operations. - The operations required to transform
s1..n into t1..m is of course the number required to transform all of s into all of t, and so dn,m holds our result. This proof fails to validate that the number placed in di,j is in fact minimal; this is more difficult to show, and involves an argument by contradiction in which we assume di,j is smaller than the minimum of the three, and use this to show one of the three is not minimal. Upper and lower bounds The Levenshtein distance has several simple upper and lower bounds that are useful in applications which compute many of them and compare them. These include: - It is always at least the difference of the sizes of the two strings.
- It is at most the length of the longer string.
- It is zero if and only if the strings are identical.
- If the strings are the same size, the Hamming distance is an upper bound on the Levenshtein distance; otherwise the Hamming distance plus the difference in sizes is an upper bound.
- If the strings are called
s and t, the number of characters found in s but not in t is a lower bound. Sample implementations Because Haskell automatically memoizes results of previous calls, it is particularly suited to a simple recursive implementation: editDistance :: String->String->Int editDistance [] [] = 0 editDistance s [] = length s editDistance [] t = length t editDistance (s:ss) (t:ts) = minimum s == t then 0 else 1) + editDistance ss ts, 1 + editDistance ss (t:ts), 1 + editDistance (s:ss) ts Uses srfi-25 and srfi-42 (define add1 (lambda (x) (+ x 1))) (define sub1 (lambda (x) (- x 1))) (define levenshtein-distance (lambda (s1 s2) (let* ((width (add1 (string-length s1))) (height (add1 (string-length s2))) (d (make-array (shape 0 height 0 width) 0))) (do-ec (:range x width) (array-set! d 0 x x)) (do-ec (:range y height) (array-set! d y 0 y)) (do-ec (:range x (string-length s1)) (:range y (string-length s2)) (array-set! d (add1 y) (add1 x) (min (add1 (array-ref d y (add1 x))) (add1 (array-ref d (add1 y) x)) (+ (array-ref d y x) (if (eqv? (string-ref s1 x) (string-ref s2 y)) 0 1))))) (displarray d) (array-ref d (sub1 height) (sub1 width))))) See also External links
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