Map (higher-order function)

Computer programming function

For the abstract data type of the same name, see Map (data structure).

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In many programming languages, map is a higher-order function that applies a given function to each element of a collection, e.g. a list or set, returning the results in a collection of the same type. 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

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Suppose there is list of integers `[1, 2, 3, 4, 5]` and would like to calculate the square of each integer. To do this, first define a function to `square` a single number (shown here in Haskell):

square x = x * x

Afterwards, call:

>>> map square [1, 2, 3, 4, 5]

which yields `[1, 4, 9, 16, 25]`, demonstrating that `map` has gone through the entire list and applied the function `square` to each element.

### Visual example

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Below, there is view of each step of the mapping process for a list of integers `X = [0, 5, 8, 3, 2, 1]` mapping into a new list `X'` according to the function  f ( x ) = x + 1 {\displaystyle f(x)=x+1} :

View of processing steps when applying map function on a list

The `map` is provided as part of the Haskell's base prelude (i.e. "standard library") and is implemented as:

map :: (a -> b) -> [a] -> [b]
map _ []       = []
map f (x : xs) = f x : map f xs

## Generalization

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See also: Functor and Category theory

In Haskell, the polymorphic function `map :: (a -> b) -> [a] -> [b]` is generalized to a polytypic function `fmap :: Functor f => (a -> b) -> 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 (Tree a) (Tree a)

instance Functor Tree where
fmap f (Leaf x) = Leaf (f x)
fmap f (Fork l r) = Fork (fmap f l) (fmap f r)

Mapping over a tree yields:

>>> fmap square (Fork (Fork (Leaf 1) (Leaf 2)) (Fork (Leaf 3) (Leaf 4)))
Fork (Fork (Leaf 1) (Leaf 4)) (Fork (Leaf 9) (Leaf 16))

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

fmap id      ≡ id              -- identity law
fmap (f . g) ≡ 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.

### Category-theoretic background

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In category theory, a functor F : C -> D {\displaystyle F:C\rightarrow D} consists of two maps: one that sends each object A of the category to another object F A, and one that sends each morphism  f : A -> B {\displaystyle f:A\rightarrow B} to another morphism  F f : F A -> F B {\displaystyle Ff:FA\rightarrow FB} , which acts as a homomorphism on categories (i.e. it respects the category axioms). Interpreting the universe of data types as a category Type, with morphisms being functions, then a type constructor `F` that is a member of the `Functor` type class is the object part of such a functor, and `fmap :: (a -> b) -> F a -> F b` is the morphism part. The functor laws described above are precisely the category-theoretic functor axioms for this functor.

Functors can also be objects in categories, with "morphisms" called natural transformations. Given two functors  F , G : C -> D {\displaystyle F,G:C\rightarrow D} , a natural transformation  η : F -> G {\displaystyle \eta :F\rightarrow G} consists of a collection of morphisms  η A : F A -> G A {\displaystyle \eta _{A}:FA\rightarrow GA} , one for each object A of the category D, which are 'natural' in the sense that they act as a 'conversion' between the two functors, taking no account of the objects that the functors are applied to. Natural transformations correspond to functions of the form `eta :: F a -> G a`, where `a` is a universally quantified type variable – `eta` knows nothing about the type which inhabits `a`. The naturality axiom of such functions is automatically satisfied because it is a so-called free theorem, depending on the fact that it is parametrically polymorphic.[1] For example, `reverse :: List a -> List a`, which reverses a list, is a natural transformation, as is `flattenInorder :: Tree a -> List a`, which flattens a tree from left to right, and even `sortBy :: (a -> a -> Bool) -> List a -> List a`, which sorts a list based on a provided comparison function.

## Optimizations

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The mathematical basis of maps allow for a number of optimizations. The composition law ensures that both

* `(map f . map g) list` and
* `map (f . g) list`

lead to the same result; that is,  map ⁡ ( f ) ∘ map ⁡ ( g ) = map ⁡ ( f ∘ g ) {\displaystyle \operatorname {map} (f)\circ \operatorname {map} (g)=\operatorname {map} (f\circ g)} . 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.[2]

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 (f . g) 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 (\ys x -> f x : ys) []

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

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The map function originated in functional programming languages.

The language Lisp introduced a map function called `maplist`[3] in 1959, with slightly different versions already appearing in 1958.[4] This is the original definition for `maplist`, mapping a function over successive rest lists:

maplist[x;f] = [null[x] -> NIL;T -> cons[f[x];maplist[cdr[x];f]]]

The function `maplist` is still available in newer Lisps like Common Lisp,[5] 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:

(maplist (lambda (l) (sqr (car l))) '(1 2 3 4 5))

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

(mapcar (function sqr) '(1 2 3 4 5))

Today mapping functions are supported (or may be defined) in many procedural, object-oriented, and multi-paradigm languages as well: In C++'s Standard Library, it is called `std::transform`, in C# (3.0)'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 (CFML), 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 (from Smalltalk). Common Lisp provides a family of map-like functions; the one corresponding to the behavior described here is called `mapcar` (`-car` indicating access using the CAR operation). There are also languages with syntactic constructs providing the same functionality as the map function.

Map is sometimes generalized to accept dyadic (2-argument) 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 and currying, `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.

Map in various languages  Language  | Map  | Map 2 lists  | Map n lists  | Notes  | Handling lists of different lengths

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APL | `func list` | `list1 func list2` | `func/ list1 list2 list3 list4` | APL's array processing abilities make operations like map implicit  | length error if list lengths not equal or 1
Common Lisp | `(mapcar func list)` | `(mapcar func list1 list2)` | `(mapcar func list1 list2 ...)` |  | stops after the length of the shortest list
C++ | `std::transform(begin, end, result, func)` | `std::transform(begin1, end1, begin2, result, func)` |  | in header <algorithm>
begin, end, and result are iterators
result is written starting at result |
C# | `ienum.Select(func)`

The `select` clause | `ienum1.Zip(ienum2, func)` |  | `Select` 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
CFML | `obj.map(func)` |  |  | Where `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.  |
Clojure | `(map func list)` | `(map func list1 list2)` | `(map func list1 list2 ...)` |  | stops after the shortest list ends
D | `list.map!func` | `zip(list1, list2).map!func` | `zip(list1, list2, ...).map!func` |  | Specified to zip by StoppingPolicy: shortest, longest, or requireSameLength
Erlang | `lists:map(Fun, List)` | `lists:zipwith(Fun, List1, List2)` | `zipwith3` also available  |  | Lists must be equal length
Elixir | `Enum.map(list, fun)` | `Enum.zip(list1, list2) |> Enum.map(fun)` | `List.zip([list1, list2, ...]) |> Enum.map(fun)` |  | stops after the shortest list ends
F# | `List.map func list` | `List.map2 func list1 list2` |  | Functions exist for other types (Seq and Array)  | Throws exception
Gleam | `list.map(list, func)`
`yielder.map(yielder, func)` | `list.map2(list1, list2, func)`
`yielder.map2(yielder1, yielder2, func)` |  |  | drops the extra elements of the longer list
Groovy | `list.collect(func)` | `[list1 list2].transpose().collect(func)` | `[list1 list2 ...].transpose().collect(func)` |  |
Haskell | `map func list` | `zipWith func list1 list2` | `zipWithn func list1 list2 ...` | `n` corresponds to the number of lists; predefined up to `zipWith7` | stops after the shortest list ends
Haxe | `array.map(func)
list.map(func)
Lambda.map(iterable, func)` |  |  |  |
J | `func list` | `list1 func list2` | `func/ list1, list2, list3 ,: list4` | J's array processing abilities make operations like map implicit  | length error if list lengths not equal
Java 8+  | `stream.map(func)` |  |  |  |
JavaScript 1.6
ECMAScript 5  | `array#map(func)` | `List1.map(function (elem1, i) {
return func(elem1, List2[i]); })` | `List1.map(function (elem1, i) {
return func(elem1, List2[i], List3[i], ...); })` | Array#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.
Julia | `map(func, list)` | `map(func, list1, list2)` | `map(func, list1, list2, ..., listN)` |  | ERROR: DimensionMismatch
Logtalk | `map(Closure, List)` | `map(Closure, List1, List2)` | `map(Closure, List1, List2, List3, ...) (up to seven lists)` | Only the Closure argument must be instantiated.  | Failure
Mathematica | `func /@ list
Map[func, list]` | `MapThread[func, {list1, list2}]` | `MapThread[func, {list1, list2, ...}]` |  | Lists must be same length
Maxima | `map(f, expr1, ..., exprn)
maplist(f, expr1, ..., exprn)` |  |  | map returns an expression which leading operator is the same as that of the expressions;
maplist returns a list  |
OCaml | `List.map func list
Array.map func array` | `List.map2 func list1 list2` |  |  | raises Invalid_argument exception
PARI/GP | `apply(func, list)` |  |  |  | —
Perl | `map 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.`
PHP | `array_map(callable, array)` | `array_map(callable, array1,array2)` | `array_map(callable, array1,array2, ...)` | The number of parameters for callable
should match the number of arrays.  | extends the shorter lists with NULL items
Prolog | `maplist(Cont, List1, List2).` | `maplist(Cont, List1, List2, List3).` | `maplist(Cont, List1, ...).` | List arguments are input, output or both. Subsumes also zipWith, unzip, all  | Silent failure (not an error)
Python | `map(func, list)` | `map(func, list1, list2)` | `map(func, list1, list2, ...)` | Returns a list in Python 2 and an iterator in Python 3.  | `zip()` and `map()` (3.x) stops after the shortest list ends, whereas `map()` (2.x) and `itertools.zip_longest()` (3.x) extends the shorter lists with `None` items
Ruby | `enum.collect {block}
enum.map {block}` | `enum1.zip(enum2).map {block}` | `enum1.zip(enum2, ...).map {block}
[enum1, enum2, ...].transpose.map {block}` | `enum is an Enumeration` | stops at the end of the object it is called on (the first list); if any other list is shorter, it is extended with nil items
Rust | `list1.into_iter().map(func)` | `list1.into_iter().zip(list2).map(func)` |  | the `Iterator::map` and `Iterator::zip` methods both take ownership of the original iterator and return a new one; the `Iterator::zip` method internally calls the `IntoIterator::into_iter` method on `list2` | stops after the shorter list ends
S-R | `lapply(list, func)` | `mapply(func, list1, list2)` | `mapply(func, list1, list2, ...)` |  | Shorter lists are cycled
Scala | `list.map(func)` | `(list1, list2).zipped.map(func)` | `(list1, list2, list3).zipped.map(func)` | note: more than 3 not possible.  | stops after the shorter list ends
Scheme (including Guile and Racket)  | `(map func list)` | `(map func list1 list2)` | `(map func list1 list2 ...)` |  | lists must all have same length (SRFI-1 extends to take lists of different length)
Smalltalk | `aCollection collect: aBlock` | `aCollection1 with: aCollection2 collect: aBlock` |  |  | Fails
Standard ML | `map func list` | `ListPair.map func (list1, list2)
ListPair.mapEq func (list1, list2)` |  | For 2-argument map, func takes its arguments in a tuple  | `ListPair.map` stops after the shortest list ends, whereas `ListPair.mapEq` raises `UnequalLengths` exception
Swift | `sequence.map(func)` | `zip(sequence1, sequence2).map(func)` |  |  | stops after the shortest list ends
XPath 3
XQuery 3  | `list ! block`
`for-each(list, func)` | `for-each-pair(list1, list2, func)` |  | In `block` the context item `.` holds the current value  | stops after the shortest list ends

## See also

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* Functor (functional programming)
* Zipping (computer science) or zip, mapping 'list' over multiple lists
* Filter (higher-order function)
* Fold (higher-order function)
* foreach loop
* Free monoid
* Functional programming
* Higher-order function
* List comprehension
* Map (parallel pattern)

## References

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1. ^ In a non-strict language that permits general recursion, such as Haskell, this is only true if the first argument to `fmap` is strict. Wadler, Philip (September 1989). Theorems for free! (PDF). 4th International Symposium on Functional Programming Languages and Computer Architecture. London: Association for Computing Machinery.
2. ^ "Map fusion: Making Haskell 225% faster"
3. ^ J. McCarthy, K. Maling, S. Russell, N. Rochester, S. Goldberg, J. Slagle. LISP Programmer's Manual. March-April, 1959
4. ^ J. McCarthy: Symbol Manipulating Language - Revisions of the Language. AI Memo No. 4, October 1958
5. ^ Function MAPC, MAPCAR, MAPCAN, MAPL, MAPLIST, MAPCON in ANSI Common Lisp