Substring Explained

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, bcclabccefab is a superstring of P = \{ abcc , efab , bccla \} , and 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

• Brace notation
• Substring index
• Suffix automaton