In many programming languages, map is the name of a higher-order function that applies a given function to a sequence of elements (such as a list) and returns a sequence of results. They are examples of both catamorphisms and anamorphisms.
For example, if we define a function square as follows:
square x = x * x
Then calling (map square [1,2,3,4,5]) will return [1,4,9,16,25].
Map itself may be defined recursively, like this code in Haskell:
map f [] = [] map f (x:xs) = f x : map f xs
Language comparison
The map function is especially common in functional programming languages, but some high-level procedural languages support it as well, and in others it may be defined. Common Lisp provides a whole family of map-like functions. The one that corresponds to the behavior described here is called mapcar. In C++'s Standard Template Library, the map function is called transform.
Optimizations
The mathematical basis of maps allow for a number of optimizations. If one has map f . map g ('.' is function composition) then it is the same as the simpler map (f . g) xs; that is,
. This particular optimization eliminates an expensive second map by fusing it with the first map; thus it is a "map fusion"[1].
Map functions can and often are defined in terms of a fold such as foldr, which means one can do a "map-fold fusion": 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 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.
rev_map f l = foldl (\ys x -> f x : ys) [] l
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.
Haskell's Functor class
In the Haskell programming language, map is generalized to a polymorphic function called fmap using the Functor type class. For every Functor instance, fmap must be defined such that it obeys the Functor laws:
fmap id = id -- identityfmap (f . g) = fmap f . fmap g -- composition