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 $$u=$$ is equal to substrings (and subsequences) of $$t=$$  at two different offsets:

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

The first occurrence is obtained with $$p=$$ and $$s=$$, while the second occurrence is obtained with  $$p=$$  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,  is a prefix of , which is in turn a suffix of. 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  is equal to a prefix (and substring and subsequence) of the string  :

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  is equal to a suffix (and substring and subsequence) of the string  :

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, $$\text{bcclabccefab}$$ is a superstring of $$P = \{\text{abcc}, \text{efab}, \text{bccla}\}$$, and $$\text{efabccla}$$ 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.