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". This is not to be confused with subsequence, which is a generalization of substring. For example, "Itwastimes" is a subsequence of "It was the best of times", but not a substring.
Prefix and suffix are special cases of substring. A prefix of a string is a substring of that occurs at the beginning of . A suffix of a string is a substring that occurs at the end of .
The list of all substrings of the string "apple" would be "apple", "appl", "pple", "app", "ppl", "ple", "ap", "pp", "pl", "le", "a", "p", "l", "e", "".
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. 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 "babooneatingakebab").
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. Generally, one is interested in finding superstrings whose length is as small as possible;[clarification needed] a concatenation of all strings of in any order gives a trivial superstring of . A string that contains every possible permutation of a specified character set is called a superpermutation.
- 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.
- Media related to Substring at Wikimedia Commons