In formal language theory and computer science, a substring is a contiguous sequence of characters within a string. 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 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).
A string is a substring (or factor) 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
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).
A string is a prefix of a string if there exists a string such that . A proper prefix of a string is not equal to the string itself; some sources 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 ||| 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.
A string is a suffix 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. 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 |||| 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.
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").
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.
- ^ 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) . Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology. USA: Cambridge University Press. ISBN 0-521-58519-8.