In information theory and computer science, the Levenshtein distance is a string metric
for measuring the difference between two sequences.
Informally, the Levenshtein distance between two words is the minimum number of single-character
edits required to change one word into the other.
It is named after Vladimir Levenshtein, who considered this distance in 1965.
Levenshtein distance may also be referred to as edit distance, although that may also
denote a larger family of distance metrics.
It is closely related to pairwise string alignments.
Definition Mathematically, the Levenshtein distance between
two strings is given by where
where is the indicator function equal to 0 when and equal to 1 otherwise.
Note that the first element in the minimum corresponds to deletion, the second to insertion
and the third to match or mismatch, depending on whether the respective symbols are the
same.
Example For example, the Levenshtein distance between
"kitten" and "sitting" is 3, since the following three edits change one into the other, and
there is no way to do it with fewer than three edits:
kitten → sitten sitten → sittin
sittin → sitting.
Upper and lower bounds The Levenshtein distance has several simple
upper and lower bounds.
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 equal.
If the strings are the same size, the Hamming distance is an upper bound on the Levenshtein
distance.
The Levenshtein distance between two strings is no greater than the sum of their Levenshtein
distances from a third string.
Applications In approximate string matching, the objective
is to find matches for short strings in many longer texts, in situations where a small
number of differences is to be expected.
The short strings could come from a dictionary, for instance.
Here, one of the strings is typically short, while the other is arbitrarily long.
This has a wide range of applications, for instance, spell checkers, correction systems
for optical character recognition, and software to assist natural language translation based
on translation memory.
The Levenshtein distance can also be computed between two longer strings, but the cost to
compute it, which is roughly proportional to the product of the two string lengths,
makes this impractical.
Thus, when used to aid in fuzzy string searching in applications such as record linkage, the
compared strings are usually short to help improve speed of comparisons.
Relationship with other edit distance metrics
There are other popular measures of edit distance, which are calculated using a different set
of allowable edit operations.
For instance, the Damerau–Levenshtein distance allows
insertion, deletion, substitution, and the transposition of two adjacent characters;
the longest common subsequence metric allows only insertion and deletion, not substitution;
the Hamming distance allows only substitution, hence, it only applies to strings of the same
length.
Edit distance is usually defined as a parameterizable metric calculated with a specific set of allowed
edit operations, and each operation is assigned a cost.
This is further generalized by DNA sequence alignment algorithms such as the Smith–Waterman
algorithm, which make an operation's cost depend on where it is applied.
Computing Levenshtein distance Recursive
This is a straightforward, but inefficient, recursive pseudocode implementation of a LevenshteinDistance
function that takes two strings, s and t, together with their lengths, and returns the
Levenshtein distance between them:
Unfortunately, this straightforward recursive implementation is very inefficient because
it recomputes the Levenshtein distance of the same substrings many times.
A more efficient method would never repeat the same distance calculation.
For example, the Levenshtein distance of all possible prefixes might be stored in an array
d[][] where d[i][j] is the distance between the first i characters of string s and the
first j characters of string t.
The table is easy to construct one row at a time starting with row 0.
When the entire table has been built, the desired distance is d[len_s][len_t].
While this technique is significantly faster, it will consume len_s * len_t more memory
than the straightforward recursive implementation.
Iterative with full matrix
Note: This section uses 1-based strings instead of 0-based strings
Computing the Levenshtein distance is based on the observation that if we reserve a matrix
to hold the Levenshtein distances between all prefixes of the first string and all prefixes
of the second, then we can compute the values in the matrix in a dynamic programming fashion,
and thus find the distance between the two full strings as the last value computed.
This algorithm, an example of bottom-up dynamic programming, is discussed, with variants,
in the 1974 article The String-to-string correction problem by Robert A. Wagner and Michael J.
Fischer.
This is a straightforward pseudocode implementation for a function LevenshteinDistance that takes
two strings, s of length m, and t of length n, and returns the Levenshtein distance between
them:
Note that this implementation does not fit the definition precisely: it always prefers
matches, even if insertions or deletions provided a better score.
This is equivalent; it can be shown that for every optimal alignment there is another optimal
alignment that prefers matches in the sense of this implementation.
Two examples of the resulting matrix:
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.
Iterative with two matrix rows It turns out that only two rows of the table
are needed for the construction if one does not want to reconstruct the edited input strings.
The Levenshtein distance may be calculated iteratively using the following algorithm:
See also
References
External links Levenshtein in PostgreSQL
Black, Paul E., ed., "Levenshtein distance", Dictionary of Algorithms and Data Structures
[online], U.S. National Institute of Standards and Technology, retrieved 3 April 2013