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 ${\displaystyle u={\texttt {ana}}}$ is equal to substrings (and subsequences) of ${\displaystyle t={\texttt {banana}}}$ at two different offsets:

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

The first occurrence is obtained with ${\displaystyle p={\texttt {b}}}$ and ${\displaystyle s={\texttt {na}}}$, while the second occurrence is obtained with ${\displaystyle p={\texttt {ban}}}$ 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; for example, nan is a prefix of nana, which is in turn a suffix of banana. 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). ^{[citation needed]}

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.

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.^[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.

A border is suffix and prefix of the same string, e.g. "${\displaystyle {\texttt {bab}}}$" is a border of "${\displaystyle {\texttt {babab}}}$" (and also of "${\displaystyle {\texttt {baboon}}\,\,{\texttt {eating}}\,\,{\texttt {a}}\,\,{\texttt {kebab}}}$").^{[citation needed]}

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, ${\displaystyle {\texttt {bcclabccefab}}}$ is a superstring of ${\displaystyle P=\{{\texttt {abcc}},{\texttt {efab}},{\texttt {bccla}}\}}$, and ${\displaystyle {\texttt {efabccla}}}$ is a shorter one. Concatenating all members of ${\displaystyle P}$, in arbitrary order, always obtains a trivial superstring of ${\displaystyle P}$. 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.

• Substring index
• Suffix automaton