Substring

"string" is a substring of "substring"

In formal language theory and computer science, 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". In contrast, "Itwastimes" is a subsequence of "It was the best of times", but not a substring.

Prefixes and suffixes are special cases of substrings. A prefix of a string $S$ is a substring of $S$ that occurs at the beginning of $S$; likewise, a suffix of a string $S$ is a substring that occurs at the end of $S$.

The substrings of the string "apple" would be: "a", "ap", "app", "appl", "apple", "p", "pp", "ppl", "pple", "pl", "ple", "l", "le" "e", "" (note the empty string at the end).

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

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

The first occurrence is obtained with $p=\mathtt{\text{𝚋}}$ and $s=\mathtt{\text{𝚗𝚊}}$, while the second occurrence is obtained with $p=\mathtt{\text{𝚋𝚊𝚗}}$ 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; for example, nan is a prefix of nana, which is in turn a suffix of banana. 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 $p⊑t$ 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. "$\mathtt{\text{𝚋𝚊𝚋}}$" is a border of "$\mathtt{\text{𝚋𝚊𝚋𝚊𝚋}}$" (and also of "$\mathtt{\text{𝚋𝚊𝚋𝚘𝚘𝚗}}\mathtt{\text{𝚎𝚊𝚝𝚒𝚗𝚐}}\mathtt{\text{𝚊}}\mathtt{\text{𝚔𝚎𝚋𝚊𝚋}}$").

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

See also

• Substring index
• Suffix automaton