Substring

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This article is about definition of a substring. For the computer function which performs this operation, see String functions (programming).

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.

Contents

[edit] 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

[edit] 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).

[edit] 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 and not empty[1] (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 (mathematics) for more details.

[edit] 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 and not empty[2] (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

[edit] Superstring

Definition: Given a set of k strings P = {s1,s2,s3,---sk}, 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 = {abcc, efab, bccla}. Then bcclabccefab is a superstring of P, and efabccla is another, shorter superstring of P. Generally, we are interested in finding superstrings whose length is small.

[edit] References

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