Substring

A subsequence, substring, prefix or suffix of a string is a subset of the symbols in a string, where the order of the elements is preserved. In this context, the terms string and sequence have the same meaning.

1 Subsequence
2 Substring
3 Prefix
4 Suffix
5 Border
6 Superstring
7 References

Subsequence

Main article subsequence

A subsequence of a string $T = t_1 t_2 \dots t_n$ is a string $\hat T = t_{i_1} \dots t_{i_m}$ such that $i_1 < \dots < i_m$ , where $m \leq n$ . Subsequence is a generalisation of substring, suffix and prefix. Finding the longest string which is equal to a subsequence of two or more strings is known as the longest common subsequence problem.

Example: The string anna is equal to a subsequence of the string banana:

banana
 || ||
 an na

Including the empty subsequence, the number of subsequences of a string of length $n$ where symbols only occur once, is simply the number of subsets of the symbols in the string, i.e. $2^n$ .

Substring

A substring (or factor) of a string $T = t_1 \dots t_n$ is a string $\hat T = t_{1%2Bi} \dots t_{m%2Bi}$ , where $0 \leq i$ and $m %2B i \leq n$ . 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 $n%2B1$ such places. So there are $\tbinom{n%2B1}{2} = \tfrac{n(n%2B1)}{2}$ 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.

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 $\hat T = t_{n-m%2B1} \dots t_{n}$ , where $m \leq n$ . A proper suffix of a string is not equal to the string itself ( $0 < m \leq n$ ); again, a more restricted interpretation is that it is also not empty^[1] ( $0 < m < n$ ). 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

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.

Substring

Contents

Subsequence

Substring

Prefix

Suffix

Border

Superstring

References