Levenshtein distance
In information theory and computer science, the Levenshtein distance is a string metric which is one way to measure edit distance. The Levenshtein 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 of a single character. It is named after Vladimir Levenshtein, who considered this distance in 1965.[1] 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 fewer than three edits:
- kitten → sitten (substitution of 's' for 'k')
- sitten → sittin (substitution of 'i' for 'e')
- sittin → 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, like in the Damerau-Levenshtein distance algorithm.
The algorithm
A commonly-used bottom-up dynamic programming 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. This algorithm is based on the Wagner-Fischer algorithm for edit distance. Here is pseudocode for a function LevenshteinDistance that takes two strings, s of length m, and t of length n, and computes the Levenshtein distance between them:
int LevenshteinDistance(char s[1..m], char t[1..n]) // d is a table with m+1 rows and n+1 columns declare int d[0..m, 0..n] for i from 0 to m d[i, 0] := i for j from 1 to n d[0, j] := j for i from 1 to m for j from 1 to n if s[i] = t[j] then cost := 0 else cost := 1 d[i, j] := minimum( d[i-1, j] + 1, // deletion d[i, j-1] + 1, // insertion d[i-1, j-1] + cost // substitution ) return d[m, n]
Two examples of the resulting matrix (the minimum steps to be taken are highlighted):
|
|
The invariant maintained throughout the algorithm is that we can transform the initial segment s[1..i]
into t[1..j]
using a minimum of d[i,j]
operations. At the end, the bottom-right element of the array contains the answer.
This algorithm is essentially part of a solution to the Longest common subsequence problem (LCS), in the particular case of 2 input lists.
Proof of correctness
As mentioned earlier, the invariant is that we can transform the initial segment s[1..i]
into t[1..j]
using a minimum of d[i,j]
operations. This invariant holds since:
- It is initially true on row and column 0 because
s[1..i]
can be transformed into the empty stringt[1..0]
by simply dropping alli
characters. Similarly, we can transforms[1..0]
tot[1..j]
by simply adding allj
characters. - The minimum is taken over three distances, each of which is feasible:
- If we can transform
s[1..i]
tot[1..j-1]
ink
operations, then we can simply addt[j]
afterwards to gett[1..j]
ink+1
operations. - If we can transform
s[1..i-1]
tot[1..j]
ink
operations, then we can do the same operations ons[1..i]
and then remove the originals[i]
at the end ink+1
operations. - If we can transform
s[1..i-1]
tot[1..j-1]
ink
operations, we can do the same tos[1..i]
and then do a substitution oft[j]
for the originals[i]
at the end if necessary, requiringk+cost
operations.
- If we can transform
- The operations required to transform
s[1..n]
intot[1..m]
is of course the number required to transform all ofs
into all oft
, and sod[n,m]
holds our result.
This proof fails to validate that the number placed in d[i,j]
is in fact minimal; this is more difficult to show, and involves an argument by contradiction in which we assume d[i,j]
is smaller than the minimum of the three, and use this to show one of the three is not minimal.
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
d[i,0]
can be moved inside the main outer loop. - This algorithm parallelizes poorly, due to a large number of data dependencies. However, all the
cost
values can be computed in parallel, and the algorithm can be adapted to perform theminimum
function in phases to eliminate dependencies.
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.
- If the strings are called
s
andt
, the number of characters (not counting duplicates) found ins
but not int
is a lower bound.
See also
- agrep
- Bitap algorithm
- Damerau-Levenshtein distance
- diff
- Dynamic time warping
- Fuzzy string searching
- Jaro-Winkler distance
- Levenshtein automaton
- Longest common subsequence problem
- Lucene (an open source search engine that implements edit distance.)
- Metric space
- Needleman-Wunsch algorithm
- Smith-Waterman algorithm
References
- ↑ В.И. Левенштейн (1965) Двоичные коды с исправлением выпадений, вставок и замещений символов. Доклады Академий Наук СССР 163.4:845–848. Appeared in English as: V. I. Levenshtein, Binary codes capable of correcting deletions, insertions, and reversals. Soviet Physics Doklady 10 (1966):707–710.
External links
- Java, C++ and VB implementations of the algorithm
- NIST's Dictionary of Algorithms and Data Structures: Levenshtein Distance
- Levenshtein Distance visualized
- edit-distance implementation in Scheme
- Inbuilt function in PHP, and example implementation
- Continuous variants, spike train metrics, and applications to neurophysiology
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