List (abstract data type)


In computer science, a list or sequence is an abstract data type that represents a countable number of ordered values, where the same value may occur more than once. An instance of a list is a computer representation of the mathematical concept of a tuple or finite sequence; the infinite analog of a list is a stream. Lists are a basic example of containers, as they contain other values. If the same value occurs multiple times, each occurrence is considered a distinct item.
The name list is also used for several concrete data structures that can be used to implement abstract lists, especially linked lists and arrays. In some contexts, such as in Lisp programming, the term list may refer specifically to a linked list rather than an array. In class-based programming, lists are usually provided as instances of subclasses of a generic "list" class, and traversed via separate iterators.
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 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.

Operations

Implementation of the list data structure may provide some of the following operations:
Lists are typically implemented either as linked lists 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 also supported "compressed lists" which had a special internal representation. 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, 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.

Programming language support

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.
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 . 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, making a singly linked list.

Applications

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

The abstract list type L with elements of some type E is defined by the following functions:
with the axioms
for any element e and any list l. It is implicit that
Note that first ) and rest ) 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

The list type forms a monad with the following functions :
where append is defined as:
Alternatively, the monad may be defined in terms of operations return, fmap and join, with:
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.