# Substring

A **substring** of a string is another string that occurs "in" . For example, "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 is a substring of that occurs at the *beginning* of . A suffix of a string is a substring that occurs at the *end* of .

## Substring

A substring (or factor) of a string is a string , where and . A substring of a string is a prefix of a suffix of the string, and equivalently a suffix of a prefix. If is a substring of , 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 `ana`

is equal to substrings (and subsequences) of `banana`

at two different offsets:

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

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

Not including the empty substring, the number of substrings of a string of length where symbols only occur once, is the number of ways to choose two distinct places between symbols to start/end the substring. Including the very beginning and very end of the string, there are such places. So there are non-empty substrings.

## Prefix

A prefix of a string is a string , where . A *proper prefix* of a string is not equal to the string itself ();^{[1]} some sources^{[2]} 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 denotes that is a prefix of . This defines a binary relation on strings, called the prefix relation, which is a particular kind of prefix order.

In formal language theory, the term *prefix of a string* is also commonly understood to be the set of all prefixes of a string, with respect to that language. See the article on string functions for more details.

## Suffix

A suffix of a string is any substring of the string which includes its last letter, including itself. 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

Given a set of strings , a **superstring** of the set is single string that contains every string in as a substring.
For example, a concatenation of the strings of in any order gives a trivial superstring of . For a more interesting example, let . Then is a superstring of , and is another, shorter superstring of . Generally, we are interested in finding superstrings whose length is small.