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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 lists. Many programming languages 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 commas, semicolons, and/or spaces, within a pair of delimiters such as parentheses '()', brackets '[]', braces ' ', or angle brackets '<>'. Some languages may allow list types to be indexed or sliced like array types, in which case the data type is more accurately described as an array. In object-oriented programming languages, lists are usually provided as instances of subclasses of a generic "list" class, and traversed via separate iterators. List data types are often implemented using array data structures 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, 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 inductively by two operations: nil that yields the empty list, and cons, which adds an item at the beginning of a list.[2]

Contents

1 Operations 2 Implementations 3 Programming language support 4 Applications 5 Abstract definition

5.1 The list monad

6 References 7 See also

Operations[edit] Implementation of the list data structure may provide some of the following operations:

a 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.)

Implementations[edit] Lists are typically implemented either as linked lists (either singly or doubly linked) or as arrays, usually variable length or dynamic arrays. The standard way of implementing lists, originating with the 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, 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 languages, while the latter is the norm in functional languages. Lists can be implemented as self-balancing binary search trees 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.[3] Programming language support[edit] Some languages do not offer a list data structure, but offer the use of associative arrays or some kind of table to emulate lists. For example, Lua provides tables. Although Lua stores lists that have numerical indices as arrays internally, they still appear as dictionaries.[4] In 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 (list 2 3 5). In several dialects of Lisp, including 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.[5] Applications[edit] As the name implies, lists can be used to store a list of elements. However, unlike in traditional arrays, lists can expand and shrink, and are stored dynamically in memory. In computing, lists are easier to implement than sets. A finite 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 trees or hash tables, rather than a list. Lists also form the basis for other abstract data types including the queue, the stack, and their variations. Abstract definition[edit] The abstract list type L with elements of some type E (a 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 (e1, l1) = cons (e2, l2) if e1 = e2 and l1 = l2

Note that first (nil ()) and rest (nil ()) are not defined. These axioms are equivalent to those of the abstract 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 × L → L. first and rest are then obtained by pattern matching on the cons constructor and separately handling the nil case. The list monad[edit] The list type forms a monad with the following functions (using E* rather than L to represent monomorphic lists with elements of type E):

return

: A →

A

∗

= a ↦

cons

a

nil

displaystyle text return colon Ato A^ * =amapsto text cons ,a, text nil

bind

:

A

∗

→ ( A →

B

∗

) →

B

∗

= l ↦ f ↦

nil

if

l =

nil

append

( f

a )

(

bind

l ′

f )

if

l =

cons

a

l ′

displaystyle text bind colon A^ * to (Ato B^ * )to B^ * =lmapsto fmapsto 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

where append is defined as:

append

:

A

∗

→

A

∗

→

A

∗

=

l

1

↦

l

2

↦

l

2

if

l

1

=

nil

cons

a

(

append

l

1

′

l

2

)

if

l

1

=

cons

a

l

1

′

displaystyle 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

Alternatively, the monad may be defined in terms of operations return, fmap and join, with:

fmap

: ( A → B ) → (

A

∗

→

B

∗

) = f ↦ l ↦

nil

if

l =

nil

cons

( f

a ) (

fmap

f

l ′

)

if

l =

cons

a

l ′

displaystyle text fmap colon (Ato B)to (A^ * to B^ * )=fmapsto lmapsto 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

join

:

A

∗

∗

→

A

∗

= l ↦

nil

if

l =

nil

append

a

(

join

l ′

)

if

l =

cons

a

l ′

displaystyle text join colon A^ * ^ * to A^ * =lmapsto begin cases text nil & text if l= text nil \ text append ,a,( text join ,l')& text if l= text cons ,a,l'end cases

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[edit]

^ Abelson, Harold; Sussman, Gerald Jay (1996). Structure and Interpretation of Computer Programs. MIT Press. ^ Reingold, Edward; Nievergelt, Jurg; Narsingh, Deo (1977). Combinatorial Algorithms: Theory and Practice. Englewood Cliffs, New Jersey: Prentice Hall. pp. 38–41. ISBN 0-13-152447-X. ^ Barnett, Granville; Del tonga, Luca (2008). "Data Structures and Algorithms" (PDF). mta.ca. Retrieved 12 November 2014. ^ Lerusalimschy, Roberto (December 2003). Programming in Lua (first edition) (First ed.). Lua.org. ISBN 8590379817. Retrieved 12 November 2014. ^ Steele, Guy (1990). Common Lisp (Second ed.). Digital Press. pp. 29–31. ISBN 1-55558-041-6.

See also[edit]

Look up list in Wiktionary, the free dictionary.

Array Queue Set Stream

v t e

Data structures

Types

Collection Container

Abstract

Associative array

Multimap

List Stack Queue

Double-ended queue

Priority queue

Double-ended priority queue

Set

Multiset Disjoint-set

Arrays

Bit array Circular buffer Dynamic array Hash table Hashed array tree Sparse matrix

Linked

Association list Linked list Skip list Unrolled linked list XOR linked list

Trees

B-tree Binary search tree

AA tree AVL tree Red–black tree Self-balancing tree Splay tree

Heap

Binary heap Binomial heap Fibonacci heap

R-tree

R* tree R+ tree Hilbert R-tree

Trie

Hash tree

Graphs

Binary decision diagram Directed acyclic graph Directed acyclic word graph

List of data structures

v t e

Data types

Uninterpreted

Bit Byte Trit Tryte Word Bit array

Numeric

Arbitrary-precision or bignum Complex Decimal Fixed point Floating point

Double precision Extended precision Half precision Long double Minifloat Octuple precision Quadruple precision Single precision

Integer

signedness

Interval Rational

Pointer

Address

physical virtual

Reference

Text

Character String

null-terminated

Composite

Algebraic data type

generalized

Array Associative array Class Dependent Equality Inductive List Object

metaobject

Option type Product Record Set Union

tagged

Other

Boolean Bottom type Collection Enumerated type Exception Function type Opaque data type Recursive data type Semaphore Stream Top type Type class Unit type Void

Related topics

Abstract data type Data structure Generic Kind

metaclass

Parametric polymorphism Primitive data type Protocol

interface

Subtyping Type constructor Type conversion Type system Type theory

See also platform-dependent and independent units of