Substring

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

"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 $S$ 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. A substring of a string is a prefix of a suffix of the string, and equivalently a suffix of a prefix. If $s$ is a substring of $t$, it is also a subsequence, which is a more general concept. Given a pattern, you can find its occurrences in a string 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 $u=$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 $s=$ne, while the second occurence is obtained with $p=$ban and $s$ being the empty string.

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

Prefix

See also: String operations § Prefixes

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^[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 $P$ 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; a concatenation of all strings of $P$ in any order gives a trivial superstring of $P$.

See also

• Brace notation
• Substring index
• Suffix automaton

External links

• Media related to Substring at Wikimedia Commons