Substring

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 is any substring of the string which includes its last letter, including itself. 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".

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.

See also

References

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