Substring#Suffix

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 $u=$ana is equal to substrings (and subsequences) of $t=$banana at two different offsets:

banana

|||||

ana||

|||

ana

The first occurrence is obtained with $p=$b and $s=$na, while the second occurrence is obtained with $p=$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). {{citation needed|date=November 2020}}

{{see also|String operations#Prefixes}}

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 $p\; \backslash sqsubseteq\; t$ 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.{{ref|Gus97}} 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 "baboon eating a kebab").{{citation needed|date=January 2022}}

A superstring of a finite set $P$ of strings is a single string that contains every string in $P$ as a substring. For example, $\backslash text\{bcclabccefab\}$ is a superstring of $P\; =\; \backslash \{\backslash text\{abcc\},\; \backslash text\{efab\},\; \backslash text\{bccla\}\backslash \}$, and $\backslash text\{efabccla\}$ 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.

• Brace notation
• Substring index
• Suffix automaton

{{Reflist|refs=

{{cite book

| last = Gusfield

| first = Dan

| orig-year = 1997

| year = 1999

| title = Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology

| publisher = Cambridge University Press

| location = US

| isbn = 0-521-58519-8

}}

{{cite book

| last = Kelley

| first = Dean

| year = 1995

| title = Automata and Formal Languages: An Introduction

| publisher = Prentice-Hall International

| location = London

| isbn = 0-13-497777-7

}}

{{cite book

| last = Lothaire

| first = M.

| year = 1997

| title = Combinatorics on words

| publisher = Cambridge University Press

| location = Cambridge

| isbn = 0-521-59924-5

}}

}}

Category:String (computer science)

Category:Formal languages