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  "slug": "substring",
  "title": "Substring",
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  "lead": "In formal language theory and computer science, a substring is a contiguous sequence of characters within a string. For instance, \"the best of\" is a substring of \"It was the best of times\". In contrast, \"Itwastimes\" is a subsequence of \"It was the best of times\", but not a substring. Prefixes and suffixes are special cases of substrings. A prefix of a string S is a substring of S that occurs at the beginning of S ; likewise, a suffix of a string S is a substring that occurs at the end of S . The substrings of the string \" \" would be: \" \", \" \", \" \", \" \", \" \", \" \", \" \", \" \", \" \", \" \", \" \", \" \", \" \" \" \", \"\" (note the empty string at the end).",
  "body": "In formal language theory and computer science, a substring is a contiguous sequence of characters within a string. For instance, \"the best of\" is a substring of \"It was the best of times\". In contrast, \"Itwastimes\" is a subsequence of \"It was the best of times\", but not a substring. Prefixes and suffixes are special cases of substrings. A prefix of a string S is a substring of S that occurs at the beginning of S ; likewise, a suffix of a string S is a substring that occurs at the end of S . The substrings of the string \" \" would be: \" \", \" \", \" \", \" \", \" \", \" \", \" \", \" \", \" \", \" \", \" \", \" \", \" \" \" \", \"\" (note the empty string at the end). 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). 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. 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. 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. ==References==",
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    "title": "Substring",
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    "publisher-place": "Dubai, United Arab Emirates",
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