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 special cases 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 .
A substring (or factor) of a string T = t 1 … t n is a string T ^ = t 1 + i … t m + i , where 0 ≤ i and m + i ≤ n . A substring of a string is a prefix of a suffix of the string, and equivalently a suffix of a prefix. If 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 ( n + 1 2 ) = n ( n + 1 ) 2 non-empty substrings.
A prefix of a string T = t 1 … t n is a string T ^ = t 1 … t m , where m ≤ n . A proper prefix of a string is not equal to the string itself ( 0 ≤ m < n ); some sources 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 T ^ ⊑ T denotes that 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.
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[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.
A border is suffix and prefix of the same string, e.g. "bab" is a border of "babab" (and also of "babooneatingakebab").
Given a set of k strings P = { s 1 , s 2 , s 3 , … 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 = { 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.
