Editing List (abstract data type)

{{short description|Abstract data type used in computer science}}
{{about|sequential data structures|random-access data structures|Array data type}}
In [[computer science]], a '''list''' or '''sequence''' is an [[abstract data type]] that represents a countable number of ordered [[value (computer science)|values]], where the same value may occur more than once. An instance of a list is a computer representation of the [[mathematics|mathematical]] concept of a finite [[sequence (mathematics)|sequence]]; the (potentially) infinite analog of a list is a [[Stream (computing)|stream]].<ref>{{cite book |title=[[Structure and Interpretation of Computer Programs]] |first1=Harold |last1=Abelson |first2=Gerald Jay |last2=Sussman |year=1996 |publisher=MIT Press}}</ref>{{rp|§3.5}} Lists are a basic example of [[Container (abstract data type)|containers]], as they contain other values. If the same value occurs multiple times, each occurrence is considered a distinct item.

[[Image:Singly linked list.png|thumb|right|A singly linked list structure, implementing a list with 3 integer elements.]]
The name '''list''' is also used for several concrete [[data structures]] that can be used to implement abstract lists, especially [[linked list]]s.

Many [[programming language]]s provide support for '''list data types''', and have special syntax and semantics for lists and list operations. A list can often be constructed by writing the items in sequence, separated by [[comma]]s, [[semicolon]]s, and/or [[Space (punctuation)|spaces]], within a pair of delimiters such as [[parentheses]] '()', [[brackets]] '[]', [[Braces (punctuation)|braces]] '{}', or [[angle brackets]] '<>'. Some languages may allow list types to be [[Array index|indexed]] or [[array slicing|sliced]] like [[array data type|array type]]s, in which case the data type is more accurately described as an array. In [[object-oriented programming language]]s, lists are usually provided as [[instance (computer science)|instance]]s of subclasses of a generic "list" class, and traversed via separate [[iterator]]s. List data types are often implemented using [[array data structure]]s or linked lists of some sort, but other [[data structures]] may be more appropriate for some applications. In some contexts, such as in [[Lisp programming language|Lisp]] programming, the term list may refer specifically to a linked list rather than an array.

In [[type theory]] and [[functional programming]], abstract lists are usually defined [[inductive type|inductively]] by two operations: ''nil'' that yields the empty list, and ''cons'', which adds an item at the beginning of a list.<ref>{{cite book|last1=Reingold|first1=Edward|last2=Nievergelt|first2=Jurg|last3=Narsingh|first3=Deo|title=Combinatorial Algorithms: Theory and Practice|date=1977|publisher=Prentice Hall|location=Englewood Cliffs, New Jersey|isbn=0-13-152447-X|pages=38–41}}</ref>

==Operations==
Implementation of the list data structure may provide some of the following [[operation (mathematics)|operations]]:
* a [[constructor (computer science)|constructor]] for creating an empty list;
* an operation for testing whether or not a list is empty;
* an operation for prepending an entity to a list
* an operation for appending an entity to a list
* an operation for determining the first component (or the "head") of a list
* an operation for referring to the list consisting of all the components of a list except for its first (this is called the "tail" of the list.)
* an operation for accessing the element at a given index.

==Implementations==
Lists are typically implemented either as [[linked list]]s (either singly or doubly linked) or as [[Array data structure|arrays]], usually variable length or [[dynamic array]]s.

The standard way of implementing lists, originating with the programming language [[Lisp (programming language)|Lisp]], is to have each element of the list contain both its value and a pointer indicating the location of the next element in the list. This results in either a [[linked list]] or a [[tree data structure|tree]], depending on whether the list has nested sublists. Some older Lisp implementations (such as the Lisp implementation of the [[Symbolics]] 3600) also supported "compressed lists" (using [[CDR coding]]) which had a special internal representation (invisible to the user). Lists can be manipulated using [[iteration]] or [[recursion]]. The former is often preferred in [[imperative programming|imperative programming languages]], while the latter is the norm in [[functional language]]s.

Lists can be implemented as [[self-balancing binary search tree]]s holding index-value pairs, providing equal-time access to any element (e.g. all residing in the fringe, and internal nodes storing the right-most child's index, used to guide the search), taking the time logarithmic in the list's size, but as long as it doesn't change much will provide the illusion of [[random access]] and enable swap, prefix and append operations in logarithmic time as well.<ref>{{cite web|last1=Barnett|first1=Granville|last2=Del tonga|first2=Luca|title=Data Structures and Algorithms|url=http://www.mta.ca/~rrosebru/oldcourse/263114/Dsa.pdf|website=mta.ca|accessdate=12 November 2014|date=2008}}</ref>

==Programming language support==
Some languages do not offer a list [[data structure]], but offer the use of [[associative array]]s or some kind of table to emulate lists. For example, [[Lua programming language|Lua]] provides tables. Although Lua stores lists that have numerical indices as arrays internally, they still appear as dictionaries.<ref>{{cite book|last1=Lerusalimschy|first1=Roberto|title=Programming in Lua (first edition)|date=December 2003|publisher=Lua.org|isbn=8590379817|edition=First|url=http://www.lua.org/pil/11.3.html|accessdate=12 November 2014}}</ref>

In [[Lisp programming language|Lisp]], lists are the fundamental data type and can represent both program code and data. In most dialects, the list of the first three prime numbers could be written as <code>(list 2 3 5)</code>. In several dialects of Lisp, including [[Scheme (programming language)|Scheme]], a list is a collection of pairs, consisting of a value and a pointer to the next pair (or null value), making a singly linked list.<ref>{{cite book|last1=Steele|first1=Guy|title=Common Lisp|date=1990|publisher=Digital Press|isbn=1-55558-041-6|pages=29–31|edition=Second}}</ref>

==Applications==
As the name implies, lists can be used to store a list of elements. However, unlike in traditional [[array data structure|arrays]], lists can expand and shrink, and are stored dynamically in memory.

In computing, lists are easier to implement than sets. A finite [[set (mathematics)|set]] in the mathematical sense can be realized as a list with additional restrictions; that is, duplicate elements are disallowed and order is irrelevant. Sorting the list speeds up determining if a given item is already in the set, but in order to ensure the order, it requires more time to add new entry to the list. In efficient implementations, however, sets are implemented using [[self-balancing binary search tree]]s or [[hash table]]s, rather than a list.

Lists also form the basis for other [[abstract data types]] including the [[Queue (abstract data type)|queue]], the [[Stack (abstract data type)|stack]], and their variations.

==Abstract definition==
The abstract list type ''L'' with elements of some type ''E'' (a [[Type polymorphism|monomorphic]] list) is defined by the following functions:

:nil: () → ''L''
:cons: ''E'' × ''L'' → ''L''
:first: ''L'' → ''E''
:rest: ''L'' → ''L''

with the axioms

:first (cons (''e'', ''l'')) = ''e''
:rest (cons (''e'', ''l'')) = ''l''

for any element ''e'' and any list ''l''. It is implicit that

:cons (''e'', ''l'') ≠ ''l''
:cons (''e'', ''l'') ≠ ''e''
:cons (''e''<sub>1</sub>, ''l''<sub>1</sub>) = cons (''e''<sub>2</sub>, ''l''<sub>2</sub>) if ''e''<sub>1</sub> = ''e''<sub>2</sub> and ''l''<sub>1</sub> = ''l''<sub>2</sub>

Note that first (nil ()) and rest (nil ()) are not defined.

These axioms are equivalent to those of the abstract [[Stack (abstract data type)|stack]] data type.

In [[type theory]], the above definition is more simply regarded as an [[inductive type]] defined in terms of constructors: ''nil'' and ''cons''. In algebraic terms, this can be represented as the transformation 1 + ''E'' &times; ''L'' → ''L''.  ''first'' and ''rest'' are then obtained by [[pattern matching]] on the ''cons'' constructor and separately handling the ''nil'' case.

===The list monad===
The list type forms a [[monad (functional programming)|monad]] with the following functions (using ''E''<sup>*</sup> rather than ''L'' to represent monomorphic lists with elements of type ''E''):

:<math>\text{return}\colon A \to A^{*} = a \mapsto \text{cons} \, a \, \text{nil}</math>
:<math>\text{bind}\colon A^{*} \to (A \to B^{*}) \to B^{*} = l \mapsto f \mapsto \begin{cases} \text{nil} & \text{if} \ l = \text{nil}\\ \text{append} \, (f \, a) \, (\text{bind} \, l' \, f) & \text{if} \ l = \text{cons} \, a \, l' \end{cases}</math>
where ''append'' is defined as:
:<math>\text{append}\colon A^{*} \to A^{*} \to A^{*} = l_1 \mapsto l_2 \mapsto \begin{cases} l_2 & \text{if} \ l_1 = \text{nil} \\ \text{cons} \, a \, (\text{append} \, l_1' \, l_2) & \text{if} \ l_1 = \text{cons} \, a \, l_1' \end{cases}</math>

Alternatively, the monad may be defined in terms of operations ''return'', ''fmap'' and ''join'', with:
:<math>\text{fmap} \colon (A \to B) \to (A^{*} \to B^{*}) = f \mapsto l \mapsto \begin{cases} \text{nil} & \text{if} \ l = \text{nil}\\ \text{cons} \, (f \, a) (\text{fmap} f \, l') & \text{if} \ l = \text{cons} \, a \, l' \end{cases}</math>
:<math>\text{join} \colon {A^{*}}^{*} \to A^{*} = l \mapsto \begin{cases} \text{nil} & \text{if} \ l = \text{nil}\\ \text{append} \, a \, (\text{join} \, l') & \text{if} \ l = \text{cons} \, a \, l' \end{cases}</math>

Note that ''fmap'', ''join'', ''append'' and ''bind'' are well-defined, since they're applied to progressively deeper arguments at each recursive call.

The list type is an additive monad, with ''nil'' as the monadic zero and ''append'' as monadic sum.

Lists form a [[monoid]] under the ''append'' operation. The identity element of the monoid is the empty list, ''nil''. In fact, this is the [[free monoid]] over the set of list elements.

==References==
{{reflist}}

==See also==
{{Wiktionary|list}}
*[[Array data type|Array]]
*[[Queue (abstract data type)|Queue]]
*[[Set (abstract data type)|Set]]
*[[Stack (abstract data type)|Stack]]
*[[Stream (computing)|Stream]]
{{Data structures}}
{{Data types}}

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