# 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 $S$ is a substring of $S$ that occurs at the beginning of $S$ . A suffix of a string $S$ is a substring that occurs at the end of $S$ .

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 $u$  is a substring (or factor) of a string $t$  if there exists two strings $p$  and $s$  such that $t=pus$ . In particular, the empty string is a substring of every string.

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

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


The first occurrence is obtained with $p=$ b and $s=$ na, while the second occurrence is obtained with $p=$ ban and $s$  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 $u$  is a substring of $t$ , 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 $p$  is a prefix of a string $t$  if there exists a string $s$  such that $t=ps$ . 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:

banana
|||
ban


The square subset symbol is sometimes used to indicate a prefix, so that $p\sqsubseteq t$  denotes that $p$  is a prefix of $t$ . This defines a binary relation on strings, called the prefix relation, which is a particular kind of prefix order.

## Suffix

A string $s$  is a suffix of a string $t$  if there exists a string $p$  such that $t=ps$ . 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 $P$  of strings is a single string that contains every string in $P$  as a substring. For example, ${\text{bcclabccefab}}$  is a superstring of $P=\{{\text{abcc}},{\text{efab}},{\text{bccla}}\}$ , and ${\text{efabccla}}$  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 $P$  in any order gives a trivial superstring of $P$ . A string that contains every possible permutation of a specified character set is called a superpermutation.