Levenshtein distance

In 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 can be considered a generalization of the Hamming distance, which is used for strings of the same length and only considers substitution edits.

Table of contents

1 The Algorithm

1.1 Possible Improvements
1.2 Proof of Correctness

2 Upper and lower bounds
3 External links

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 it:

int LevenshteinDistance(char s[1..n], char t[1..m])
   declare int d[0..n,0..m]
   declare int i, j, cost
   for i := 0 to n
       d[i,0] := i
   for j := 0 to m
       d[0,j] := j

   for i := 1 to n
       for j := 1 to m
           if s[i] = t[j] then cost := 0
                          else cost := 1
           d[i,j] := minimum(d[i-1,j  ] + 1,    // insertion
                             d[i,  j-1] + 1,    // deletion
                             d[i-1,j-1] + cost) // substitution

   return d[n,m]

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 subtitutions separately, or even the positions at which they occur, which is always j.
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 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 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 string t[1..0] by simply dropping all i characters. Similarly, we can transform s[1..0] to t[1..j] by simply adding all j characters.
The minimum is taken over three distances, each of which is feasible:
- If we can transform s[1..i] to t[1..j-1] in k operations, then we can simply add t[j] afterwards to get t[1..j] in k+1 operations.
- If we can transform s[1..i-1] to t[1..j] in k operations, then we can do the same operations on s[1..i] and then remove the original s[i] at the end in k+1 operations.
- If we can transform s[1..i-1] to t[1..j-1] in k operations, we can do the same to s[1..i] and then do a substitution of t[j] for the original s[i] at the end if necessary, requiring k+cost operations.
The operations required to transform s[1..n] into t[1..m] is of course the number required to transform all of s into all of t, and so d[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.

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 longest 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.