Wagner–Fischer algorithm

In computer science, the Wagner–Fischer algorithm is a dynamic programming algorithm that computes the edit distance between two strings of characters.

The Wagner–Fischer algorithm has a history of multiple invention. Navarro lists the following inventors of it, with date of publication, and acknowledges that the list is incomplete:

The Wagner–Fischer algorithm computes edit distance based on the observation that if we reserve a matrix to hold the edit 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 filling the matrix, and thus find the distance between the two full strings as the last value computed.

A straightforward implementation, as pseudocode for a function EditDistance that takes two strings, s of length m, and t of length n, and returns the Levenshtein distance between them, looks as follows. Note that the input strings are one-indexed, while the matrix d is zero-indexed, and [i..k] is a closed range.

Two examples of the resulting matrix (hovering over an underlined number reveals the operation performed to get that number):

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.

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:

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.

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