In formal language theory and computer science, a substring is a contiguous sequence of characters within a string.[citation needed] For instance, "the best of" is a substring of "It was the best of times". In contrast, "Itwastimes" is a subsequence of "It was the best of times", but not a substring.

"string" is a substring of "substring"

Prefixes and suffixes are special cases of substrings. A prefix of a string is a substring of that occurs at the beginning of ; likewise, a suffix of a string is a substring that occurs at the end of .

The substrings of the string "apple" would be: "a", "ap", "app", "appl", "apple", "p", "pp", "ppl", "pple", "pl", "ple", "l", "le" "e", "" (note the empty string at the end).

Substring

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A string   is a substring (or factor)[1] of a string   if there exists two strings   and   such that  . In particular, the empty string is a substring of every string.

Example: The string  ana is equal to substrings (and subsequences) of  banana at two different offsets:

banana
 |||||
 ana||
   |||
   ana

The first occurrence is obtained with  b and  na, while the second occurrence is obtained with  ban and   being the empty string.

A substring of a string is a prefix of a suffix of the string, and equivalently a suffix of a prefix; for example, nan is a prefix of nana, which is in turn a suffix of banana. If   is a substring of  , it is also a subsequence, which is a more general concept. The occurrences of a given pattern in a given string can be found with a string searching algorithm. Finding the longest string which is equal to a substring of two or more strings is known as the longest common substring problem. In the mathematical literature, substrings are also called subwords (in America) or factors (in Europe). [citation needed]

Prefix

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A string   is a prefix[1] of a string   if there exists a string   such that  . A proper prefix of a string is not equal to the string itself;[2] some sources[3] in addition restrict a proper prefix to be non-empty. A prefix can be seen as a special case of a substring.

Example: The string ban is equal to a prefix (and substring and subsequence) of the string banana:

banana
|||
ban

The square subset symbol is sometimes used to indicate a prefix, so that   denotes that   is a prefix of  . This defines a binary relation on strings, called the prefix relation, which is a particular kind of prefix order.

Suffix

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A string   is a suffix[1] of a string   if there exists a string   such that  . A proper suffix of a string is not equal to the string itself. A more restricted interpretation is that it is also not empty.[1] A suffix can be seen as a special case of a substring.

Example: The string nana is equal to a suffix (and substring and subsequence) of the string banana:

banana
  ||||
  nana

A suffix tree for a string is a trie data structure that represents all of its suffixes. Suffix trees have large numbers of applications in string algorithms. The suffix array is a simplified version of this data structure that lists the start positions of the suffixes in alphabetically sorted order; it has many of the same applications.

Border

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A border is suffix and prefix of the same string, e.g. "bab" is a border of "babab" (and also of "baboon eating a kebab").[citation needed]

Superstring

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A superstring of a finite set   of strings is a single string that contains every string in   as a substring. For example,   is a superstring of  , and   is a shorter one. Concatenating all members of  , in arbitrary order, always obtains a trivial superstring of  . Finding superstrings whose length is as small as possible is a more interesting problem.

A string that contains every possible permutation of a specified character set is called a superpermutation.

See also

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References

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  1. ^ a b c Lothaire, M. (1997). Combinatorics on words. Cambridge: Cambridge University Press. ISBN 0-521-59924-5.
  2. ^ Kelley, Dean (1995). Automata and Formal Languages: An Introduction. London: Prentice-Hall International. ISBN 0-13-497777-7.
  3. ^ Gusfield, Dan (1999) [1997]. Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology. US: Cambridge University Press. ISBN 0-521-58519-8.