Substring

A string u is a substring (or factor) of a string t if there exists two strings p and s such that t = pus. In particular, the empty string is a substring of every string. Example: The string u=\texttt{ana} is equal to substrings (and subsequences) of t=\texttt{banana} at two different offsets: banana ||||| ana|| ||| ana The first occurrence is obtained with p=\texttt{b} and s=\texttt{na}, while the second occurrence is obtained with p=\texttt{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). == Prefix ==

A string p is a prefix of a string t if there exists a string s such that t = ps. 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 \sqsubseteq 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 = ps. 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. "\texttt{bab}" is a border of "\texttt{babab}" (and also of "\texttt{baboon}\,\,\texttt{eating}\,\,\texttt{a}\,\,\texttt{kebab}"). == 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, \texttt{bcclabccefab} is a superstring of P = \{\texttt{abcc}, \texttt{efab}, \texttt{bccla}\}, and \texttt{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. A string that contains every possible permutation of a specified character set is called a superpermutation. == See also ==