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 $t = p u s$ . 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 of a string $t$ if there exists a string $s$ such that $t = p s$ . 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 $p ⊑ t$ 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 $t = p s$ . 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 " $𝚋𝚊𝚋𝚘𝚘𝚗 𝚎𝚊𝚝𝚒𝚗𝚐 𝚊 𝚔𝚎𝚋𝚊𝚋$ ").^{[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, see Shortest common superstring.

A string that contains every possible permutation of a specified character set is called a superpermutation.

References

↑ Stephen, Graham A. (1994-10-17). String Searching Algorithms. World Scientific. ISBN 978-981-4501-86-6.
1 2 3 Lothaire, M. (1997). Combinatorics on words. Cambridge: Cambridge University Press. ISBN 0-521-59924-5.
↑ 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. US: Cambridge University Press. ISBN 0-521-58519-8.

[1] Stephen, Graham A. (1994-10-17). String Searching Algorithms. World Scientific. ISBN 978-981-4501-86-6.

[Lot97-2] 1 2 3 Lothaire, M. (1997). Combinatorics on words. Cambridge: Cambridge University Press. ISBN 0-521-59924-5.

[Kel95-3] Kelley, Dean (1995). Automata and Formal Languages: An Introduction. London: Prentice-Hall International. ISBN 0-13-497777-7.

[Gus97-4] Gusfield, Dan (1999) [1997]. Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology. US: Cambridge University Press. ISBN 0-521-58519-8.

Substring

Substring

Prefix

Suffix

Border

Superstring

See also

References