Substring

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In formal language theory and computer science, a substring is a contiguous sequence of characters within a string.^[1] 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.

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

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#edit

A string ${\displaystyle u}$ is a substring (or factor)^[2] of a string ${\displaystyle t}$ if there exists two strings ${\displaystyle p}$ and ${\displaystyle s}$ such that ${\displaystyle t=pus}$ . In particular, the empty string is a substring of every string.

Example: The string ${\displaystyle u={\texttt {ana$ is equal to substrings (and subsequences) of ${\displaystyle t={\texttt {banana$ at two different offsets:

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

The first occurrence is obtained with ${\displaystyle p={\texttt {b$ and ${\displaystyle s={\texttt {na$ , while the second occurrence is obtained with ${\displaystyle p={\texttt {ban$ and ${\displaystyle s}$ 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 ${\displaystyle u}$ is a substring of ${\displaystyle t}$ , 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#edit

A string ${\displaystyle p}$ is a prefix^[2] of a string ${\displaystyle t}$ if there exists a string ${\displaystyle s}$ such that ${\displaystyle t=ps}$ . A proper prefix of a string is not equal to the string itself;^[3] some sources^[4] 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 ${\displaystyle p\sqsubseteq t}$ denotes that ${\displaystyle p}$ is a prefix of ${\displaystyle t}$ . This defines a binary relation on strings, called the prefix relation, which is a particular kind of prefix order.

Suffix#edit

A string ${\displaystyle s}$ is a suffix^[2] of a string ${\displaystyle t}$ if there exists a string ${\displaystyle p}$ such that ${\displaystyle t=ps}$ . 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#edit

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

Superstring#edit

A superstring of a finite set ${\displaystyle P}$ of strings is a single string that contains every string in ${\displaystyle P}$ as a substring. For example, ${\displaystyle {\texttt {bcclabccefab$ is a superstring of ${\displaystyle P=\\texttt {abcc,{\texttt {efab,{\texttt {bccla\$ , and ${\displaystyle {\texttt {efabccla$ is a shorter one. Concatenating all members of ${\displaystyle P}$ , in arbitrary order, always obtains a trivial superstring of ${\displaystyle P}$ . Finding superstrings whose length is as small as possible is a more interesting problem, see Shortest common superstring.

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

References#edit

^ Stephen, Graham A. (1994-10-17). String Searching Algorithms. World Scientific. ISBN 978-981-4501-86-6.
^ a b c Lothaire, M. (1997). Combinatorics on words. Cambridge: Cambridge University Press. ISBN 0-521-59924-5.
^ Kelley, Dean (1995). Automata and Formal Languages: An Introduction. London: Prentice-Hall International. ISBN 0-13-497777-7.
^ Gusfield, Dan (1999) [1997]. Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology. US: Cambridge University Press. ISBN 0-521-58519-8.

Categories String (computer science)Formal languages

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