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Damerau-Levenshtein Distance

The Damerau-Levenshtein distance is a string metric that measures the edit distance between two strings by counting the minimum number of operations (insertions, deletions, substitutions, and transpositions of adjacent characters) required to transform one string into another.

O(m·n)

space

O(1)

O(n)·O(n)

Cosine distance

O(n)·O(1)

Hamming distance

O(m+n)·O(m+n)

Jaccard distance

Sorensen-Dice coefficient

Overlap coefficient

O(m·n)·O(m·n)

Levenshtein distance

Damerau-Levenshtein distance

O(m·n)·O(1)

Jaro-Winkler distance

O(n)

time

O(m·n)

Mathematical Definition

The Damerau-Levenshtein distance between two strings and is defined recursively as:

where:

denotes the length of string
denotes the substring of starting from index
denotes the first character of

Properties

The Damerau-Levenshtein distance exhibits several important mathematical properties. It produces a non-negative value that ranges from 0 to infinity, where 0 indicates identical strings. The distance is symmetric, meaning the distance from string A to B is the same as from B to A. It satisfies the triangle inequality, making it a proper metric. This means that for any three strings A, B, and C, the distance from A to C cannot be greater than the sum of the distances from A to B and B to C. A key distinguishing feature of this distance is its consideration of transpositions of adjacent characters, making it particularly suitable for detecting common typing errors.

Applications

The Damerau-Levenshtein distance has become an essential tool in various text processing and analysis applications. In spell checking systems, it helps identify and correct misspelled words by finding the closest matching words in a dictionary. For DNA sequence alignment, it provides a way to measure genetic similarities and differences between sequences. In natural language processing, it's used for text normalization and fuzzy matching. The distance is also valuable in plagiarism detection systems, where it helps identify similar text passages. Additionally, it plays a crucial role in speech recognition systems, helping to match spoken words with their written forms.

Implementation

code.ts
1function damerauLevenshteinDistance(s1: string, s2: string): number {
2    // Implementation coming soon
3}

References

Damerau, F. J. (1964). A technique for computer detection and correction of spelling errors. Communications of the ACM, 7(3), 171-176.
Levenshtein, V. I. (1966). Binary codes capable of correcting deletions, insertions, and reversals. Soviet Physics Doklady, 10(8), 707-710.