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Substring

'string' is a substring of '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 substring (or factor) of a string ${\displaystyle T=t_{1}\dots t_{n}}$ is a string ${\displaystyle {\hat {T}}=t_{1+i}\dots t_{m+i}}$, where ${\displaystyle 0\leq i}$ and ${\displaystyle m+i\leq n}$. A substring of a string is a prefix of a suffix of the string, and equivalently a suffix of a prefix. If ${\displaystyle {\hat {T}}}$ is a substring of ${\displaystyle T}$, it is also a subsequence, which is a more general concept. Given a pattern ${\displaystyle P}$, you can find its occurrences in a string ${\displaystyle T}$ 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.

Example: The string ana is equal to substrings (and subsequences) of banana at two different offsets:

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


In the mathematical literature, substrings are also called subwords (in America) or factors (in Europe).

Not including the empty substring, the number of substrings of a string of length ${\displaystyle n}$ where symbols only occur once, is the number of ways to choose two distinct places between symbols to start/end the substring. Including the very beginning and very end of the string, there are ${\displaystyle n+1}$ such places. So there are ${\displaystyle {\tbinom {n+1}{2}}={\tfrac {n(n+1)}{2}}}$ non-empty substrings.

Prefix

A prefix of a string ${\displaystyle T=t_{1}\dots t_{n}}$ is a string ${\displaystyle {\widehat {T}}=t_{1}\dots t_{m}}$, where ${\displaystyle m\leq n}$. A proper prefix of a string is not equal to the string itself (${\displaystyle 0\leq m);[1] some sources[2] in addition restrict a proper prefix to be non-empty (${\displaystyle 0). 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 ${\displaystyle {\widehat {T}}\sqsubseteq T}$ denotes that ${\displaystyle {\widehat {T}}}$ 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.

In formal language theory, the term prefix of a string is also commonly understood to be the set of all prefixes of a string, with respect to that language.

Suffix

A suffix of a string is any substring of the string which includes its last letter, including itself. 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 "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;[clarification needed] a concatenation of all strings of ${\displaystyle P}$ in any order gives a trivial superstring of ${\displaystyle P}$.

References

1. ^ Kelley, Dean (1995). Automata and Formal Languages: An Introduction. London: Prentice-Hall International. ISBN 0-13-497777-7.
2. ^ Gusfield, Dan (1999) [1997]. Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology. USA: Cambridge University Press. ISBN 0-521-58519-8.
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