Levenshtein distance
Encyclopedia
In information theory
and computer science
, the Levenshtein distance is a string metric
for measuring the amount of difference between two sequences. The term edit distance
is often used to refer specifically to Levenshtein distance.
The Levenshtein distance between two strings
is defined as the minimum number of edits needed to transform one string into the other, with the allowable edit operations being insertion, deletion, or substitution of a single character. It is named after Vladimir Levenshtein
, who considered this distance in 1965.
, the objective is to find matches for short strings, for instance, strings from a dictionary, in many longer texts, in situations where a small number of differences is to be expected. Here, one of the strings is typically short, while the other is arbitrarily long. This has a wide range of applications, for instance, spell checker
s, 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.
Variations can be obtained by changing the set of allowable edit operations: for instance,
Edit distance
in general is usually defined as a parametrizable metric in which a repertoire of edit operations is available, and each operation is assigned a cost (possibly infinite). 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.
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 by flood fill
ing the matrix, 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.
A straightforward implementation, as pseudocode
for a function LevenshteinDistance that takes two strings, s of length m, and t of length n, and returns the Levenshtein distance between them:
int LevenshteinDistance(char s[1..m], char t[1..n])
{
// for all i and j, d[i,j] will hold the Levenshtein distance between
// the first i characters of s and the first j characters of t;
// note that d has (m+1)x(n+1) values
declare int d[0..m, 0..n]
for i from 0 to m
d[i, 0] := i // the distance of any first string to an empty second string
for j from 0 to n
d[0, j] := j // the distance of any second string to an empty first string
for j from 1 to n
{
for i from 1 to m
{
if s[i] = t[j] then
d[i, j] := d[i-1, j-1] // no operation required
else
d[i, j] := minimum
(
d[i-1, j] + 1, // a deletion
d[i, j-1] + 1, // an insertion
d[i-1, j-1] + 1 // a substitution
)
}
}
return d[m,n]
}
Two examples of the resulting matrix (hovering over a number reveals the operation performed to get that number):
The invariant
maintained throughout the algorithm is that we can transform the initial segment
is that we can transform the initial segment
This proof fails to validate that the number placed in
in which we assume
Information theory
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
and computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
, the Levenshtein distance is a string metric
String metric
In mathematics, string metrics are a class of metric that measure similarity or dissimilarity between two text strings for approximate string matching or comparison and in fuzzy string searching. For example the strings "Sam" and "Samuel" can be considered to be similar...
for measuring the amount of difference between two sequences. The term edit distance
Edit distance
In information theory and computer science, the edit distance between two strings of characters generally refers to the Levenshtein distance. However, according to Nico Jacobs, “The term ‘edit distance’ is sometimes used to refer to the distance in which insertions and deletions have equal cost and...
is often used to refer specifically to Levenshtein distance.
The Levenshtein distance between two strings
String (computer science)
In formal languages, which are used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set or alphabet....
is defined as the minimum number of edits needed to transform one string into the other, with the allowable edit operations being insertion, deletion, or substitution of a single character. It is named after Vladimir Levenshtein
Vladimir Levenshtein
Vladimir Iosifovich Levenshtein is a Russian scientist who did research in information theory and error-correcting codes. Among other contributions, he is known for the Levenshtein distance algorithm, which he developed in 1965....
, who considered this distance in 1965.
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 (substitution of 's' for 'k')
- sitten → sittin (substitution of 'i' for 'e')
- sittin → sitting (insertion of 'g' at the end).
Applications
In approximate string matchingApproximate string matching
In computing, approximate string matching is the technique of finding strings that match a pattern approximately...
, the objective is to find matches for short strings, for instance, strings from a dictionary, in many longer texts, in situations where a small number of differences is to be expected. Here, one of the strings is typically short, while the other is arbitrarily long. This has a wide range of applications, for instance, spell checker
Spell checker
In computing, a spell checker is an application program that flags words in a document that may not be spelled correctly. Spell checkers may be stand-alone capable of operating on a block of text, or as part of a larger application, such as a word processor, email client, electronic dictionary,...
s, correction systems for optical character recognition
Optical character recognition
Optical character recognition, usually abbreviated to OCR, is the mechanical or electronic translation of scanned images of handwritten, typewritten or printed text into machine-encoded text. It is widely used to convert books and documents into electronic files, to computerize a record-keeping...
, and software to assist natural language translation based on translation memory
Translation memory
A translation memory, or TM, is a database that stores so-called "segments", which can be sentences or sentence-like units that have previously been translated. A translation memory system stores the words, phrases and paragraphs that have already been translated, in order to aid human translators...
.
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
Record linkage
Record linkage refers to the task of finding records in a data set that refer to the same entity across different data sources...
, the compared strings are usually short to help improve speed of comparisons.
Relationship with other edit distance metrics
Levenshtein distance is not the only popular notion of edit distance.Variations can be obtained by changing the set of allowable edit operations: for instance,
- length of the longest common subsequenceLongest common subsequence problemThe longest common subsequence problem is to find the longest subsequence common to all sequences in a set of sequences . Note that subsequence is different from a substring, see substring vs. subsequence...
is the metric obtained by allowing only addition and deletion, not substitution; - the Damerau–Levenshtein distance allows addition, deletion, substitution, and the transpositionTranspositionTransposition may refer to:Mathematics* Transposition , a permutation which exchanges two elements and keeps all others fixed* Transposition, producing the transpose of a matrix AT, which is computed by swapping columns for rows in the matrix AGames* Transposition , different moves or a different...
of two adjacent characters; - the Hamming distanceHamming distanceIn information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different...
only allows substitution (and hence, only applies to strings of the same length).
Edit distance
Edit distance
In information theory and computer science, the edit distance between two strings of characters generally refers to the Levenshtein distance. However, according to Nico Jacobs, “The term ‘edit distance’ is sometimes used to refer to the distance in which insertions and deletions have equal cost and...
in general is usually defined as a parametrizable metric in which a repertoire of edit operations is available, and each operation is assigned a cost (possibly infinite). This is further generalized by DNA sequence alignment
Sequence alignment
In bioinformatics, a sequence alignment is a way of arranging the sequences of DNA, RNA, or protein to identify regions of similarity that may be a consequence of functional, structural, or evolutionary relationships between the sequences. Aligned sequences of nucleotide or amino acid residues are...
algorithms such as the Smith–Waterman algorithm, which make an operation's cost depend on where it is applied.
Computing Levenshtein distance
Computing the Levenshtein distance is based on the observation that if we reserve a matrixMatrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
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 by flood fill
Flood fill
Flood fill, also called seed fill, is an algorithm that determines the area connected to a given node in a multi-dimensional array. It is used in the "bucket" fill tool of paint programs to determine which parts of a bitmap to fill with color, and in games such as Go and Minesweeper for determining...
ing the matrix, and thus find the distance between the two full strings as the last value computed.
This algorithm, an example of bottom-up dynamic programming
Dynamic programming
In mathematics and computer science, dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It is applicable to problems exhibiting the properties of overlapping subproblems which are only slightly smaller and optimal substructure...
, is discussed, with variants, in the 1974 article The String-to-string correction problem
String-to-string correction problem
The string-to-string correction problem refers to the minimum number of edit operations necessary to change one string into another. A single edit operation may be changing a single symbol of the string into another, deleting, or inserting a symbol...
by Robert A. Wagner and Michael J. Fischer.
A straightforward implementation, as pseudocode
Pseudocode
In computer science and numerical computation, pseudocode is a compact and informal high-level description of the operating principle of a computer program or other algorithm. It uses the structural conventions of a programming language, but is intended for human reading rather than machine reading...
for a function LevenshteinDistance that takes two strings, s of length m, and t of length n, and returns the Levenshtein distance between them:
int LevenshteinDistance(char s[1..m], char t[1..n])
{
// for all i and j, d[i,j] will hold the Levenshtein distance between
// the first i characters of s and the first j characters of t;
// note that d has (m+1)x(n+1) values
declare int d[0..m, 0..n]
for i from 0 to m
d[i, 0] := i // the distance of any first string to an empty second string
for j from 0 to n
d[0, j] := j // the distance of any second string to an empty first string
for j from 1 to n
{
for i from 1 to m
{
if s[i] = t[j] then
d[i, j] := d[i-1, j-1] // no operation required
else
d[i, j] := minimum
(
d[i-1, j] + 1, // a deletion
d[i, j-1] + 1, // an insertion
d[i-1, j-1] + 1 // a substitution
)
}
}
return d[m,n]
}
Two examples of the resulting matrix (hovering over a number reveals the operation performed to get that number):
|
|
The invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
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.Proof of correctness
As mentioned earlier, the invariantInvariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
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 allicharacters. Similarly, we can transforms[1..0]tot[1..j]by simply adding alljcharacters. - If
s[i] = t[j], and we can transforms[1..i-1]tot[1..j-1]inkoperations, then we can do the same tos[1..i]and just leave the last character alone, givingkoperations. - Otherwise, the distance is the minimum of the three possible ways to do the transformation:
- If we can transform
s[1..i]tot[1..j-1]inkoperations, then we can simply addt[j]afterwards to gett[1..j]ink+1operations (insertion). - If we can transform
s[1..i-1]tot[1..j]inkoperations, then we can removes[i]and then do the same transformation, for a total ofk+1operations (deletion). - If we can transform
s[1..i-1]tot[1..j-1]inkoperations, then we can do the same tos[1..i], and exchange the originals[i]fort[j]afterwards, for a total ofk+1operations (substitution).
- If we can transform
- The operations required to transform
s[1..n]intot[1..m]is of course the number required to transform all ofsinto 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 contradictionReductio ad absurdum
In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a 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, OBig O notationIn mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...
(min(n,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 normalize the distance to the interval
[0,1]. - If we are only interested in the distance if it is smaller than a threshold k, then it suffices to compute a diagonal stripe of width 2k+1 in the matrix. In this way, the algorithm can be run in OBig O notationIn mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...
(kl) time, where l is the length of the shortest string. - We can give different penalty costs to insertion, deletion and substitution. We can also give penalty costs that depend on which characters are inserted, deleted or substituted.
- By initializing the first row of the matrix with 0, the algorithm can be used for fuzzy string search of a string in a text. This modification gives the end-position of matching substrings of the text. To determine the start-position of the matching substrings, the number of insertions and deletions can be stored separately and used to compute the start-position from the end-position.
- This algorithm parallelizesParallel computingParallel computing is a form of computation in which many calculations are carried out simultaneously, operating on the principle that large problems can often be divided into smaller ones, which are then solved concurrently . There are several different forms of parallel computing: bit-level,...
poorly, due to a large number of data dependenciesData dependencyA data dependency in computer science is a situation in which a program statement refers to the data of a preceding statement. In compiler theory, the technique used to discover data dependencies among statements is called dependence analysis.There are three types of dependencies: data, name, and...
. However, all thecostvalues can be computed in parallel, and the algorithm can be adapted to perform theminimumfunction in phases to eliminate dependencies. - By examining diagonals instead of rows, and by using lazy evaluationLazy evaluationIn programming language theory, lazy evaluation or call-by-need is an evaluation strategy which delays the evaluation of an expression until the value of this is actually required and which also avoids repeated evaluations...
, we can find the Levenshtein distance in O(m (1 + d)) time (where d is the Levenshtein distance), which is much faster than the regular dynamic programming algorithm if the distance is small.
Upper and lower bounds
The Levenshtein distance has several simple upper and lower bounds that are useful in applications which are applied with 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 distanceHamming distanceIn information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different...
is an upper bound on the Levenshtein distance.