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Unlike edit distance notions used for other purposes, diff is line-oriented rather than character-oriented, but it is like Levenshtein distance in that it tries to determine the smallest set of deletions and insertions to create one file from the other.
The diff result is displayed in a standard format, which is readable for humans and machines. This format allows to create a file basing on a given file and set of changes. Most common uage of diff is to compare two versions of the same file. Output of this operation is called "diff", sametimes also a patch (from the Unix program patch) or delta.
The output of similar file comparison utilities are also called a "diff"; like the use of the word "grep" for describing the act of searching, the word diff is used in jargon as a verb for calculating any difference.