Substring

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A substring of a string $S$ is another string $S'$ that occurs "in" $S$ . 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 refinements of substring. A prefix of a string $S$ is a substring of $S$ that occurs at the beginning of $S$ . A suffix of a string $S$ is a substring that occurs at the end of $S$ .

Substring

A substring (or factor) of a string $T=t_{1}\dots t_{n}$ is a string ${\hat T}=t_{{1+i}}\dots t_{{m+i}}$ , where $0\leq i$ and $m+i\leq n$ . A substring of a string is a prefix of a suffix of the string, and equivalently a suffix of a prefix. If ${\hat T}$ is a substring of $T$ , it is also a subsequence, which is a more general concept. Given a pattern $P$ , you can find its occurrences in a string $T$ 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 $n$ 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 $n+1$ such places. So there are ${\tbinom {n+1}{2}}={\tfrac {n(n+1)}{2}}$ non-empty substrings.

Prefix

A prefix of a string $T=t_{1}\dots t_{n}$ is a string $\widehat T=t_{1}\dots t_{{m}}$ , where $m\leq n$ . A proper prefix of a string is not equal to the string itself ( $0\leq m<n$ );^[1] some sources^[2] in addition restrict a proper prefix to be non-empty ( $0<m<n$ ). 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 $\widehat T\sqsubseteq T$ denotes that $\widehat T$ is a prefix of $T$ . 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 $T=t_{1}\dots t_{n}$ is a string ${\hat T}=t_{{n-m+1}}\dots t_{{n}}$ , where $m\leq n$ . A proper suffix of a string is not equal to the string itself ( $0\leq m<n$ ); again, a more restricted interpretation is that it is also not empty ( $0<m<n$ ). 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".

Superstring

Given a set of $k$ strings $P=\{s_{1},s_{2},s_{3},\dots s_{k}\}$ , a superstring of the set $P$ is single string that contains every string in $P$ as a substring. For example, a concatenation of the strings of $P$ in any order gives a trivial superstring of $P$ . For a more interesting example, let $P=\{{\text{abcc}},{\text{efab}},{\text{bccla}}\}$ . Then ${\text{bcclabccefab}}$ is a superstring of $P$ , and ${\text{efabccla}}$ is another, shorter superstring of $P$ . Generally, we are interested in finding superstrings whose length is small.

References

↑ Kelley, Dean (1995). Automata and Formal Languages: An Introduction. London: Prentice-Hall International. ISBN 0-13-497777-7.
↑ Gusfield, Dan (1999) [1997]. Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology. USA: Cambridge University Press. ISBN 0-521-58519-8.

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