Substring Online Tutorials

A string ${\displaystyle u}$  is a substring (or factor) of a string ${\displaystyle t}$  if there exists two strings ${\displaystyle p}$  and ${\displaystyle s}$  such that ${\displaystyle t=pus}$ . In particular, the empty string is a substring of every string.

Example: The string ${\displaystyle u=}$ ana is equal to substrings (and subsequences) of ${\displaystyle t=}$ banana at two different offsets:

banana
 |||||
 ana||
   |||
   ana

The first occurrence is obtained with ${\displaystyle p=}$ b and ${\displaystyle s=}$ na, while the second occurrence is obtained with ${\displaystyle p=}$ ban and ${\displaystyle s}$  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 ${\displaystyle u}$  is a substring of ${\displaystyle t}$ , 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 ${\displaystyle p}$  is a prefix of a string ${\displaystyle t}$  if there exists a string ${\displaystyle s}$  such that ${\displaystyle t=ps}$ . A proper prefix of a string is not equal to the string itself;^[2] some sources^[3] 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 ${\displaystyle p⊑<!--\; ⊑\; -->t}$  denotes that ${\displaystyle p}$  is a prefix of ${\displaystyle t}$ . This defines a binary relation on strings, called the prefix relation, which is a particular kind of prefix order.

A string ${\displaystyle s}$  is a suffix^[1] of a string ${\displaystyle t}$  if there exists a string ${\displaystyle p}$