Map (higher-order function)


In many programming languages, map is the name of a higher-order function that applies a given function to each element of a functor, e.g. a list, returning a list of results in the same order. It is often called apply-to-all when considered in functional form.
The concept of a map is not limited to lists: it works for sequential containers, tree-like containers, or even abstract containers such as futures and promises.

Examples: mapping a list

Suppose we have a list of integers and would like to calculate the square of each integer. To do this, we first define a function to square a single number :

square x = x * x

Afterwards we may call

>>> map square

which yields , demonstrating that map has gone through the entire list and applied the function square to each element.

Visual example

Below, you can see a view of each step of the mapping process for a list of integers X = that we want to map into a new list X' according to the function :
The map is provided as part of the Haskell's base prelude and is implemented as:

map :: -> ->
map _ =
map f = f x : map f xs

Generalization

In Haskell, the polymorphic function map :: -> -> is generalized to a polytypic function fmap :: Functor f => -> f a -> f b, which applies to any type belonging the Functor type class.
The type constructor of lists can be defined as an instance of the Functor type class using the map function from the previous example:

instance Functor where
fmap = map

Other examples of Functor instances include trees:

-- a simple binary tree
data Tree a = Leaf a | Fork
instance Functor Tree where
fmap f = Leaf
fmap f = Fork

Mapping over a tree yields:

>>> fmap square ) ))
Fork ) )

For every instance of the Functor type class, fmap is contractually obliged to obey the functor laws:

fmap id ≡ id -- identity law
fmap ≡ fmap f. fmap g -- composition law

where . denotes function composition in Haskell.
Among other uses, this allows defining element-wise operations for various kinds of collections.
Moreover, if and are two functors, a natural transformation is a function of polymorphic type which respects :
If the function is defined by parametric polymorphism as in the type definition above, this specification is always satisfied.

Optimizations

The mathematical basis of maps allow for a number of optimizations. The composition law ensures that both
lead to the same result; that is,. However, the second form is more efficient to compute than the first form, because each map requires rebuilding an entire list from scratch. Therefore, compilers will attempt to transform the first form into the second; this type of optimization is known as map fusion and is the functional analog of loop fusion.
Map functions can be and often are defined in terms of a fold such as foldr, which means one can do a map-fold fusion: foldr f z. map g is equivalent to foldr z.
The implementation of map above on singly linked lists is not tail-recursive, so it may build up a lot of frames on the stack when called with a large list. Many languages alternately provide a "reverse map" function, which is equivalent to reversing a mapped list, but is tail-recursive. Here is an implementation which utilizes the fold-left function.

reverseMap f = foldl

Since reversing a singly linked list is also tail-recursive, reverse and reverse-map can be composed to perform normal map in a tail-recursive way, though it requires performing two passes over the list.

Language comparison

The map function originated in functional programming languages.
The language Lisp introduced a map function called maplist in 1959, with slightly different versions already appearing in 1958. This is the original definition for maplist, mapping a function over successive rest lists:

maplist = -> NIL;T -> cons;maplist;f]

The function maplist is still available in newer Lisps like Common Lisp, though functions like mapcar or the more generic map would be preferred.
Squaring the elements of a list using maplist would be written in S-expression notation like this:

')

Using the function mapcar, above example would be written like this:

')

Today mapping functions are supported in many procedural, object-oriented, and multi-paradigm languages as well: In C++'s Standard Template Library, it is called std::transform, in C# 's LINQ library, it is provided as an extension method called Select. Map is also a frequently used operation in high level languages such as ColdFusion Markup Language, Perl, Python, and Ruby; the operation is called map in all four of these languages. A collect alias for map is also provided in Ruby. Common Lisp provides a family of map-like functions; the one corresponding to the behavior described here is called mapcar. There are also languages with syntactic constructs providing the same functionality as the map function.
Map is sometimes generalized to accept dyadic functions that can apply a user-supplied function to corresponding elements from two lists. Some languages use special names for this, such as map2 or zipWith. Languages using explicit variadic functions may have versions of map with variable arity to support variable-arity functions. Map with 2 or more lists encounters the issue of handling when the lists are of different lengths. Various languages differ on this. Some raise an exception. Some stop after the length of the shortest list and ignore extra items on the other lists. Some continue on to the length of the longest list, and for the lists that have already ended, pass some placeholder value to the function indicating no value.
In languages which support first-class functions, map may be partially applied to lift a function that works on only one value to an element-wise equivalent that works on an entire container; for example, map square is a Haskell function which squares each element of a list.
LanguageMapMap 2 listsMap n listsNotesHandling lists of different lengths
APLfunc listlist1 func list2func/ list1 list2 list3 list4APL's array processing abilities make operations like map implicitlength error if list lengths not equal or 1
Common Lispstops after the length of the shortest list
C++std::transformstd::transformin header
begin, end, and result are iterators
result is written starting at result
C#ienum.Select
or
ienum1.ZipSelect is an extension method
ienum is an IEnumerable
Zip is introduced in.NET 4.0
Similarly in all.NET languages		stops after the shortest list ends
CFMLobj.mapWhere obj is an array or a structure. func receives as arguments each item's value, its index or key, and a reference to the original object.
Clojurestops after the shortest list ends
Dlist.map!funczip.map!funczip.map!funcSpecified to zip by StoppingPolicy: shortest, longest, or requireSameLength
Erlanglists:maplists:zipwithzipwith3 also availableLists must be equal length
ElixirEnum.mapEnum.zip |> Enum.mapList.zip |> Enum.mapstops after the shortest list ends
F#List.map func listList.map2 func list1 list2Functions exist for other types Throws exception
Groovylist.collect.transpose.collect.transpose.collect
Haskellmap func listzipWith func list1 list2zipWithn func list1 list2...n corresponds to the number of lists; predefined up to zipWith7stops after the shortest list ends
Haxearray.map

list.map

Lambda.map
Jfunc listlist1 func list2func/ list1, list2, list3,: list4J's array processing abilities make operations like map implicitlength error if list lengths not equal
Java 8+stream.map
JavaScript 1.6
ECMAScript 5		List1.mapList1.mapArray#map passes 3 arguments to func: the element, the index of the element, and the array. Unused arguments can be omitted.Stops at the end of List1, extending the shorter arrays with undefined items if needed.
JuliamapmapmapERROR: DimensionMismatch
Logtalkmapmapmap Only the Closure argument must be instantiated.Failure
Mathematicafunc /@ list
Map		MapThreadMapThreadLists must be same length
Maximamap
maplist		map returns an expression which leading operator is the same as that of the expressions;
maplist returns a list
OCamlList.map func list
Array.map func array		List.map2 func list1 list2raises Invalid_argument exception
PARI/GPapply
Perlmap block list
map expr, list		In block or expr special variable $_ holds each value from list in turn.Helper List::MoreUtils::each_array combines more than one list until the longest one is exhausted, filling the others with undef.
PHParray_maparray_maparray_mapThe number of parameters for callable
should match the number of arrays.		extends the shorter lists with NULL items
Prologmaplist.maplist.maplist.List arguments are input, output or both. Subsumes also zipWith, unzip, allSilent failure
PythonmapmapmapReturns a list in Python 2 and an iterator in Python 3.zip and map stops after the shortest list ends, whereas map and itertools.zip_longest extends the shorter lists with None items
Rubyenum.collect
enum.map 		enum1.zip.map enum1.zip.map
.transpose.map 		enum is an Enumerationstops at the end of the object it is called on ; if any other list is shorter, it is extended with nil items
Rust
list1.iter.map
list1.iter.zip).map		map returns an iterator.stops after the shorter list ends
S-RlapplymapplymapplyShorter lists are cycled
Scalalist.map.zipped.map.zipped.mapnote: more than 3 not possible.stops after the shorter list ends
Scheme lists must all have same length
SmalltalkaCollection collect: aBlockaCollection1 with: aCollection2 collect: aBlockFails
Standard MLmap func listListPair.map func
ListPair.mapEq func 		For 2-argument map, func takes its arguments in a tupleListPair.map stops after the shortest list ends, whereas ListPair.mapEq raises UnequalLengths exception
Swiftsequence.mapzip.mapstops after the shortest list ends
XPath 3
XQuery 3		list ! block
for-each		for-each-pairIn block the context item . holds the current valuestops after the shortest list ends