The Edit distance is a string metric (function applied on strings with number output) quantifying how dissimilar two strings are to one another.

When people mention the edit distance without specifying which function to apply, they mean the Levenshtein distance!

In theory the Hamming distance can be considered an edit distance, but people would not call it that to avoid confusion. They would call it approximate matches under a Hamming distance constraint.

usually referred to as Edit distance

String metric used to measure (function applied on strings with number output) the difference between two sequences/strings.

It is the minimum number of single character edits required to change one word into another.
An edit can be:

  • Insertion
  • Deletion
  • Substitution
as a difference to the Hamming distance, the Levenshtein distance can be applied to strings of different lengths.

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