In many programming languages, map is the name of a higher-order function that applies a given function to each element of a list, returning a list of results in the same order. It is often called apply-to-all when considered in functional form.
Contents
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.
Example: mapping a list
Suppose we have a list of integers [1, 2, 3, 4, 5] and would like to calculate the square of each integer. To do this, we first define a function to square a single number (shown here in Haskell):
Afterwards we may call
which yields [1, 4, 9, 16, 25], demonstrating that map has gone through the entire list and applied the function square to each element. The map is provided as part of the Haskell's base prelude (i.e. "standard library") and is implemented as:
Generalization
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:
Other examples of Functor instances include trees:
Mapping over a tree yields:
For every instance of the Functor type class, fmap is contractually obliged to obey the functor laws:
where . denotes function composition in Haskell.
Among other uses, this allows defining element-wise operations for various kinds of collections.
Moreover, if F and G are two functors, a natural transformation is a function of polymorphic type
If the h 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
(map f . map g) list andmap (f . g) listlead to the same result; that is,
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 (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.
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.
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:
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 (or may be defined) in many procedural, object-oriented, and multi-paradigm languages as well: In C++'s Standard Template 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, 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.