Map (higher-order function): Difference between revisions

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{{one source|date=November 2012}}

{{Short description|Computer programming function}}

{{for|the abstract data type of the same name|Map (data structure)}}

{{one source|date=November 2012}}

In many [[programming language]]s, '''map''' is the name of a [[higher-order function]] that applies a [[procedural parameter|given function]] to each element of a [[functorCollection (abstract data type)|collection]], e.g. a [[list (computing)|list]], returningor a[[set list(abstract ofdata type)|set]], returning the results in a collection of the same ordertype. 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 [[Container (abstract data type)|containers]], tree-like containers, or even abstract containers such as [[futures and promises]].

==Example Examples: mapping a list ==

Suppose wethere have ais list of integers <code>[1, 2, 3, 4, 5]</code> and would like to calculate the [[Square (algebra)|square]] of each integer. To do this, we first define a function to <code>square</code> a single number (shown here in [[Haskell (programming language)|Haskell]]):

<syntaxhighlight lang="haskell">

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</syntaxhighlight>

Afterwards we may, call:

<syntaxhighlight lang="haskell">

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</syntaxhighlight>

which yields <code>[1, 4, 9, 16, 25]</code>, demonstrating that <code>map</code> has gone through the entire list and applied the function <code>square</code> to each element. The <code>map</code> is provided as part of the Haskell's base prelude (i.e. "standard library") and is implemented as:

=== Visual example ===

Below, there is view of each step of the mapping process for a list of integers <code>X = [0, 5, 8, 3, 2, 1]</code> mapping into a new list <code>X'</code> according to the function <math>f(x) = x + 1</math> :

[[File:Mapping-steps-loillibe-new.gif|alt=applying map function processing steps|none|thumb|480x480px|View of processing steps when applying map function on a list]]

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

<syntaxhighlight lang="haskell">

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map _ [] = []

map f (x : xs) = f x : map f xs

</syntaxhighlight>

</syntaxhighlight>Now, suppose we have a list of integers X <code>[0, 5, 8, 3, 2, 1]</code> and we want to map it into a new list X' with the function <math>f(x) = x + 1</math>, you can see below a view of each step of the mapping process :

[[File:Mapping-steps-loillibe-new.gif|alt=applying map function processing steps|none|thumb|480x480px|View of processing steps when applying map function on a list]]

== Generalization ==

{{see also|Functor|Category theory}}

In Haskell, the [[polymorphic function]] <{{code>|2=haskell|1=map :: (a -> b) -> [a] -> [b]</code>}} is generalized to a [[polytypic function]] <{{code>|2=haskell|1=fmap :: Functor f => (a -> b) -> f a -> f b</code>}}, which applies to any type belonging the [[functor (category theory)|<code>Functor</code>]] [[type class]].

The [[type constructor]] of lists <code>[]</code> can be defined as an instance of the <code>Functor</code> type class using the <code>map</code> function from the previous example:

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Among other uses, this allows defining element-wise operations for various kinds of [[collection (computer science)|collections]].

=== Category-theoretic background ===

Moreover, if {{math|''F''}} and {{math|''G''}} are two functors, a [[natural transformation]] is a function of polymorphic type <math>h : \forall T . F(T) \to G(T)</math> which respects {{math|fmap}}:

: <math>h_Y \circ \operatorname{fmap}(f) = \operatorname{fmap}(f) \circ h_X</math> for any function <math>f : X \to Y</math>.

In [[category theory]], a [[functor]] <math>F : C \rarr D</math> consists of two maps: one that sends each object {{mvar|A}} of the category to another object {{mvar|F A}}, and one that sends each morphism <math>f : A \rarr B</math> to another morphism <math>Ff : FA \rarr FB</math>, which acts as a [[homomorphism]] on categories (i.e. it respects the category axioms). Interpreting the universe of data types as a category {{mvar|Type}}, with morphisms being functions, then a type constructor <code>F</code> that is a member of the <code>Functor</code> type class is the object part of such a functor, and {{code|2=haskell|1=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.

If the {{math|''h''}} function is defined by [[parametric polymorphism]] as in the type definition above, this specification is always satisfied.

Functors can also be objects in categories, with "morphisms" called [[natural transformation]]s. Given two functors <math>F, G : C \rarr D</math>, a natural transformation <math>\eta : F \rarr G</math> consists of a collection of morphisms <math>\eta_A : FA \rarr GA</math>, one for each object {{mvar|A}} of the category {{mvar|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 {{code|2=haskell|1=eta :: F a -> G a}}, where <code>a</code> is a universally quantified type variable – <code>eta</code> knows nothing about the type which inhabits <code>a</code>. The naturality axiom of such functions is automatically satisfied because it is a so-called free theorem, depending on the fact that it is [[parametric polymorphism|parametrically polymorphic]].<ref>In a [[non-strict language]] that permits general recursion, such as Haskell, this is only true if the first argument to <code>fmap</code> is strict. {{cite conference |title=Theorems for free! |first=Philip |last=Wadler |author-link=Philip Wadler |conference=4th International Symposium on Functional Programming Languages and Computer Architecture |___location=London |date=September 1989 |url=https://people.mpi-sws.org/~dreyer/tor/papers/wadler.pdf |publisher=[[Association for Computing Machinery]]}}</ref> For example, {{code|2=haskell|1=reverse :: List a -> List a}}, which reverses a list, is a natural transformation, as is {{code|2=haskell|1=flattenInorder :: Tree a -> List a}}, which flattens a tree from left to right, and even {{code|2=haskell|1=sortBy :: (a -> a -> Bool) -> List a -> List a}}, which sorts a list based on a provided comparison function.

==Optimizations==

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* <code>(map f . map g) list</code> and

* <code>map (f . g) list</code>

lead to the same result; that is, <math>\operatorname{map}(f) \circ \operatorname{map}(g) = \operatorname{map}(f \circ g)</math>.{{citation needed|date=October 2016}} However, the second form is more efficient to compute than the first form, because each <code>map</code> 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 programming|functional]] analog of [[loop fusion]].<ref>[http://www.randomhacks.net/articles/2007/02/10/map-fusion-and-haskell-performance "Map fusion: Making Haskell 225% faster"]</ref>

Map functions can be and often are defined in terms of a [[Fold (higher-order function)|fold]] such as <code>foldr</code>, which means one can do a ''map-fold fusion'': <code>foldr f z . map g</code> is equivalent to <code>foldr (f . g) z</code>.

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</syntaxhighlight>

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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</syntaxhighlight>

Today mapping functions are supported (or may be defined) in many [[procedural programming|procedural]], [[Object-oriented programming|object-oriented]], and [[multi-paradigm]] languages as well: In [[C++]]'s [[C++ Standard TemplateLibrary|Standard Library]], it is called <code>std::transform</code>, in [[C Sharp (programming language)|C#]] (3.0)'s LINQ library, it is provided as an extension method called <code>Select</code>. Map is also a frequently used operation in high level languages such as [[ColdFusion Markup Language]] (CFML), [[Perl]], [[Python (programming language)|Python]], and [[Ruby (programming language)|Ruby]]; the operation is called <code>map</code> in all four of these languages. A <code>collect</code> alias for <code>map</code> 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 <code>mapcar</code> (<code>-car</code> indicating access using the [[CAR and CDR|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 function]]s 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 function]]s and [[currying]], <code>map</code> may be [[partial application|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, <code>map square</code> is a Haskell function which squares each element of a list.

{| class="wikitable" style="font-size: 85%"

|+ Map in various languages

! scope="col" | Language

! Language !! Map !! Map 2 lists !! Map n lists !! Notes !! Handling lists of different lengths

! scope="col" | Map

! scope="col" | Map 2 lists

! scope="col" | Map n lists

! scope="col" | Notes

! scope="col" | Handling lists of different lengths

|- valign="top"

| [[APL (programming language)|APL]]

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| <code>std::transform(<wbr/>''begin1'', ''end1'', ''begin2'', ''result'', ''func'')</code>

|

| in header {{mono|<algorithm>}}<br /> ''begin'', ''end'', and ''result'' are iterators<br /> result is written starting at ''result''

|

|- valign="top"

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| Functions exist for other types (''Seq'' and ''Array'')

| Throws exception

|- valign="top"

| [[Gleam (programming language)|Gleam]]

| <code>list.map(''list'', ''func'')</code><br><code>yielder.map(''yielder'', ''func'')</code>

| <code>list.map2(''list1'', ''list2'', ''func'')</code><br><code>yielder.map2(''yielder1'', ''yielder2'', ''func'')</code>

|

|

| drops the extra elements of the longer list

|- valign="top"

| [[Groovy (programming language)|Groovy]]

| <{{code>|list.collect(func)</code>|groovy}}

| <{{code>|[list1 list2]<wbr>.transpose()<wbr>.collect(func)</code>|groovy}}

| <{{code>|[list1 list2 ...]<wbr>.transpose()<wbr>.collect(func)</code>|groovy}}

|

|

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|- valign="top"

| [[Haxe]]

| <code>''array''.map(''func'')<br />''list''.map(''func'')<br />Lambda.map(''iterable'', ''func'')</code>

''list''.map(''func'')<br />

Lambda.map(''iterable'', ''func'')

</code>

|

|

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|-

|[[Rust (programming language)|Rust]]

| <code>''list1''.into_iter().map(''func'')</code>

| <code>''list1''.iterinto_iter().zip(''list2.iter()'').map(''func'')</code>

|

| the <code>Iterator::map</code> and <code>Iterator::zip</code> methods both take ownership of the original iterator and return a new one; the <code>Iterator::zip</code> method internally calls the <code>IntoIterator::into_iter</code> method on <code>''list2''</code>

list1.iter().map(func)

| stops after the shorter list ends

|

list1.iter().zip(list2.iter()).map(func)

|

|map() returns an iterator.

|stops after the shorter list ends

|- valign="top"

| [[S (programming language)|S]]-[[R (programming language)|R]]

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| stops after the shorter list ends

|- valign="top"

| [[Scheme (programming language)|Scheme]], (including [[Guile (programming language)|Guile]] and [[Racket (programming language)|Racket]])

| <code>(map ''func'' ''list'')</code>

| <code>(map ''func'' ''list1'' ''list2'')</code>

| <code>(map ''func'' ''list1'' ''list2'' ...)</code>

|

| lists must all have same length (SRFI-1 extends to take lists of different length)

|- valign="top"

| [[Smalltalk]]

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|- valign="top"

| [[XPath 3]]<br />[[XQuery]] 3

| <{{code>|list ! block</code>|xquery}}<br /> <{{code>|for-each(list, func)</code>|xquery}}

| <{{code>|for-each-pair(list1, list2, func)</code>|xquery}}

|

| In <code>block</code> the context item <code>.</code> holds the current value

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==See also==

* [[Functor (functional programming)]]

{{Portal|Computer science}}

* [[ConvolutionZipping (computer science)]] or zip, also termedmapping ''conv'list' orover ''zip''multiple lists

* [[Filter (higher-order function)]]

* [[Fold (higher-order function)]]

* [[foreach loop]]

* [[Free monoid]]

* [[Functional programming]]