Substring

Short description: Contiguous part of a sequence of symbols

"string" is a substring of "substring"

In formal language theory and computer science, a substring is a contiguous sequence of characters within a string.[citation needed] 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).

## Contents

* 1 Substring
* 2 Prefix
* 3 Suffix
* 4 Border
* 5 Superstring
* 6 See also
* 7 References

## Substring

A string u is a substring (or factor)[1] 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). [citation needed]

## Prefix

A string p is a prefix[1] 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;[2] some sources[3] 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⊑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[1] 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.[1] 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").[citation needed]

## 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

## References

1. ↑ 1.0 1.1 1.2 Lothaire, M. (1997). Combinatorics on words. Cambridge: Cambridge University Press. ISBN 0-521-59924-5.
2. ↑ Kelley, Dean (1995). Automata and Formal Languages: An Introduction. London: Prentice-Hall International. ISBN 0-13-497777-7.
3. ↑ Gusfield, Dan (1999). Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology. US: Cambridge University Press. ISBN 0-521-58519-8.

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