Substring Indices

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 $u=$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; 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 $p \sqsubseteq t$ 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 "baboon eating a 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, $\text$ is a superstring of $P = \$, and $\text$ 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

* Brace notation
* Substring index
* Suffix automaton

References

{{Reflist, refs=
{{cite book
, last = Gusfield
, first = Dan
, orig-year = 1997
, year = 1999
, title = Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology
, publisher = Cambridge University Press
, location = USA
, isbn = 0-521-58519-8

{{cite book
, last = Kelley
, first = Dean
, year = 1995
, title = Automata and Formal Languages: An Introduction
, publisher = Prentice-Hall International
, location = London
, isbn = 0-13-497777-7

{{cite book
, last = Lothaire
, first = M.
, year = 1997
, title = Combinatorics on words
, publisher = Cambridge University Press
, location = Cambridge
, isbn = 0-521-59924-5

String (computer science)
Formal languages