# Substring

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 ${\displaystyle S}$ is a substring of ${\displaystyle S}$ that occurs at the beginning of ${\displaystyle S}$. A suffix of a string ${\displaystyle S}$ is a substring that occurs at the end of ${\displaystyle S}$.

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", "".

## Substring

A string ${\displaystyle u}$ is a substring (or factor)[1] 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 ${\displaystyle u=}$ana is equal to substrings (and subsequences) of ${\displaystyle t=}$banana at two different offsets:

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

The first occurrence is obtained with ${\displaystyle p=}$b and ${\displaystyle s=}$na, while the second occurrence is obtained with ${\displaystyle p=}$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. 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).

## Prefix

A string ${\displaystyle p}$ is a prefix[1] 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;[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 ${\displaystyle p\sqsubseteq t}$ 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

A string ${\displaystyle s}$ is a suffix[1] 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. 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 "babooneatingakebab").

## Superstring

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, ${\displaystyle {\text{bcclabccefab}}}$ is a superstring of ${\displaystyle P=\{{\text{abcc}},{\text{efab}},{\text{bccla}}\}}$, and ${\displaystyle {\text{efabccla}}}$ is a shorter one. Generally, one is interested in finding superstrings whose length is as small as possible; a concatenation of all strings of ${\displaystyle P}$ in any order gives a trivial superstring of ${\displaystyle P}$.

## References

1. 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. USA: Cambridge University Press. ISBN 0-521-58519-8.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.