Substring
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.
Prefixes and suffixes are special cases of substrings. A prefix of a string [math]\displaystyle{ S }[/math] is a substring of [math]\displaystyle{ S }[/math] that occurs at the beginning of [math]\displaystyle{ S }[/math]; likewise, a suffix of a string [math]\displaystyle{ S }[/math] is a substring that occurs at the end of [math]\displaystyle{ S }[/math].
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
A string [math]\displaystyle{ u }[/math] is a substring (or factor)[1] of a string [math]\displaystyle{ t }[/math] if there exists two strings [math]\displaystyle{ p }[/math] and [math]\displaystyle{ s }[/math] such that [math]\displaystyle{ t = pus }[/math]. In particular, the empty string is a substring of every string.
Example: The string [math]\displaystyle{ u= }[/math]ana is equal to substrings (and subsequences) of [math]\displaystyle{ t= }[/math]banana at two different offsets:
banana ||||| ana|| ||| ana
The first occurrence is obtained with [math]\displaystyle{ p= }[/math]b and [math]\displaystyle{ s= }[/math]na, while the second occurrence is obtained with [math]\displaystyle{ p= }[/math]ban and [math]\displaystyle{ s }[/math] 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 [math]\displaystyle{ u }[/math] is a substring of [math]\displaystyle{ t }[/math], 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
A string [math]\displaystyle{ p }[/math] is a prefix[1] of a string [math]\displaystyle{ t }[/math] if there exists a string [math]\displaystyle{ s }[/math] such that [math]\displaystyle{ t = ps }[/math]. 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 [math]\displaystyle{ p \sqsubseteq t }[/math] denotes that [math]\displaystyle{ p }[/math] is a prefix of [math]\displaystyle{ t }[/math]. This defines a binary relation on strings, called the prefix relation, which is a particular kind of prefix order.
Suffix
A string [math]\displaystyle{ s }[/math] is a suffix[1] of a string [math]\displaystyle{ t }[/math] if there exists a string [math]\displaystyle{ p }[/math] such that [math]\displaystyle{ t = ps }[/math]. 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
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
A superstring of a finite set [math]\displaystyle{ P }[/math] of strings is a single string that contains every string in [math]\displaystyle{ P }[/math] as a substring. For example, [math]\displaystyle{ \text{bcclabccefab} }[/math] is a superstring of [math]\displaystyle{ P = \{\text{abcc}, \text{efab}, \text{bccla}\} }[/math], and [math]\displaystyle{ \text{efabccla} }[/math] is a shorter one. Concatenating all members of [math]\displaystyle{ P }[/math], in arbitrary order, always obtains a trivial superstring of [math]\displaystyle{ P }[/math]. 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
References
- ↑ 1.0 1.1 1.2 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). Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology. US: Cambridge University Press. ISBN 0-521-58519-8.
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