Substring

This article is about definition of a substring. For the computer function which performs this operation, see String functions (programming).

A subsequence, substring, prefix, suffix, or diafix of a string is a subset of the symbols in a string, where the order of the elements is preserved. In this context, the terms string and sequence have the same meaning.

Subsequence

Main article subsequence

A subsequence of a string $T=t_{1}t_{2}\dots t_{n}$ is a string ${\hat {T}}=t_{i_{1}}\dots t_{i_{m}}$ such that $i_{1}<\dots <i_{m}$, where $m\leq n$. Subsequence is a generalisation of substring, suffix and prefix. Finding the longest string which is equal to a subsequence of two or more strings is known as the longest common subsequence problem.

Example: The string anna is equal to a subsequence of the string banana:

banana
 || ||
 an na

Including the empty subsequence, the number of subsequences of a string of length $n$ where symbols only occur once, is simply the number of subsets of the symbols in the string, i.e. $2^{n}$.

Substring

A substring (or factor) of a string $T=t_{1}\dots t_{n}$ is a string ${\hat {T}}=t_{1+i}\dots t_{m+i}$, where $0\leq i$ and $m+i\leq n$. A substring of a string is a prefix of a suffix of the string, and equivalently a suffix of a prefix. If ${\hat {T}}$ is a substring of $T$, it is also a subsequence, which is a more general concept. Given a pattern $P$, you can find its occurrences in a string $T$ 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 $n$ 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 $n+1$ such places. So there are ${\tbinom {n+1}{2}}={\tfrac {n(n+1)}{2}}$ non-empty substrings.

Prefix

A prefix of a string $T=t_{1}\dots t_{n}$ is a string ${\widehat {T}}=t_{1}\dots t_{m}$, where $m\leq n$. A proper prefix of a string is not equal to the string itself ($0\leq m<n$); some sources in addition restrict a proper prefix to be non-empty ($0<m<n$). 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 ${\widehat {T}}\sqsubseteq T$ denotes that ${\widehat {T}}$ is a prefix of $T$. This defines a binary relation on strings, called the prefix relation.

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. See the article on string functions for more details.

Suffix

A suffix of a string $T=t_{1}\dots t_{n}$ is a string ${\hat {T}}=t_{n-m+1}\dots t_{n}$, where $m\leq n$. A proper suffix of a string is not equal to the string itself ($0\leq m<n$); again, a more restricted interpretation is that it is also not empty^[1] ($0<m<n$). 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

Border

A border is suffix and prefix of the same string, e.g. "bab" is a border of "babab".

Diafix

A diafix of a string $T=t_{1}\dots t_{n}$ is a non-empty string ${\hat {T}}=t_{m+1}\dots t_{n-m}$, where $0\leq m<n/2$. A proper diafix of a string is not equal to the string itself ($0<m<n/2$). A diafix can be seen as a special case of a substring.

Example: The string "aba" is a diafix of the string "babab".

Superstring

Given a set of $k$ strings $P=\{s_{1},s_{2},s_{3},\dots s_{k}\}$, a superstring of the set $P$ is single string that contains every string in $P$ as a substring. For example, a concatenation of the strings of $P$ in any order gives a trivial superstring of $P$. For a more interesting example, let $P=\{{\text{abcc}},{\text{efab}},{\text{bccla}}\}$. Then ${\text{bcclabccefab}}$ is a superstring of $P$, and ${\text{efabccla}}$ is another, shorter superstring of $P$. Generally, we are interested in finding superstrings whose length is small.