4.2. Map

Here are two functions we might want to write:

(** [add1 lst] adds 1 to each element of [lst] *)
let rec add1 = function
  | [] -> []
  | h :: t -> (h + 1) :: add1 t

let lst1 = add1 [1; 2; 3]
val add1 : int list -> int list = <fun>
val lst1 : int list = [2; 3; 4]
(** [concat_bang lst] concatenates "!" to each element of [lst] *)
let rec concat_bang = function
  | [] -> []
  | h :: t -> (h ^ "!") :: concat_bang t

let lst2 = concat_bang ["sweet"; "salty"]
val concat_bang : string list -> string list = <fun>
val lst2 : string list = ["sweet!"; "salty!"]

There’s a lot of similarity between those two functions:

  • They both pattern match against a list.

  • They both return the same value for the base case of the empty list.

  • They both recurse on the tail in the case of a non-empty list.

In fact the only difference (other than their names) is what they do for the head element: add versus concatenate. Let’s rewrite the two functions to make that difference even more explicit:

(** [add1 lst] adds 1 to each element of [lst] *)
let rec add1 = function
  | [] -> []
  | h :: t ->
    let f = fun x -> x + 1 in
    f h :: add1 t

(** [concat_bang lst] concatenates "!" to each element of [lst] *)
let rec concat_bang = function
  | [] -> []
  | h :: t ->
    let f = fun x -> x ^ "!" in
    f h :: concat_bang t
val add1 : int list -> int list = <fun>
val concat_bang : string list -> string list = <fun>

Now the only difference between the two functions (again, other than their names) is the body of helper function f. Why repeat all that code when there’s such a small difference between the functions? We might as well abstract that one helper function out from each main function and make it an argument:

let rec add1' f = function
  | [] -> []
  | h :: t -> f h :: add1' f t

(** [add1 lst] adds 1 to each element of [lst] *)
let add1 = add1' (fun x -> x + 1)

let rec concat_bang' f = function
  | [] -> []
  | h :: t -> f h :: concat_bang' f t

(** [concat_bang lst] concatenates "!" to each element of [lst] *)
let concat_bang = concat_bang' (fun x -> x ^ "!")
val add1' : ('a -> 'b) -> 'a list -> 'b list = <fun>
val add1 : int list -> int list = <fun>
val concat_bang' : ('a -> 'b) -> 'a list -> 'b list = <fun>
val concat_bang : string list -> string list = <fun>

But now there really is no difference at all between add1' and concat_bang' except for their names. They are totally duplicated code. Even their types are now the same, because nothing about them mentions integers or strings. We might as well just keep only one of them and come up with a good new name for it. One possibility could be transform, because they tranform a list by applying a function to each element of the list:

let rec transform f = function
  | [] -> []
  | h :: t -> f h :: transform f t

(** [add1 lst] adds 1 to each element of [lst] *)
let add1 = transform (fun x -> x + 1)

(** [concat_bang lst] concatenates "!" to each element of [lst] *)
let concat_bang = transform (fun x -> x ^ "!")
val transform : ('a -> 'b) -> 'a list -> 'b list = <fun>
val add1 : int list -> int list = <fun>
val concat_bang : string list -> string list = <fun>

Note

Instead of

let add1 lst = transform (fun x -> x + 1) lst

above we wrote

let add1 = transform (fun x -> x + 1)

This is another way of being higher order, but it’s one we already learned about under the guise of partial application. The latter way of writing the function partially applies transform to just one of its two arguments, thus returning a function. That function is bound to the name add1.

Indeed, the C++ library does call the equivalent function transform. But OCaml and many other languages (including Java and Python) use the shorter word map, in the mathematical sense of how a function maps an input to an output. So let’s make one final change to that name:

let rec map f = function
  | [] -> []
  | h :: t -> f h :: map f t

(** [add1 lst] adds 1 to each element of [lst] *)
let add1 = map (fun x -> x + 1)

(** [concat_bang lst] concatenates "!" to each element of [lst] *)
let concat_bang = map (fun x -> x ^ "!")
val map : ('a -> 'b) -> 'a list -> 'b list = <fun>
val add1 : int list -> int list = <fun>
val concat_bang : string list -> string list = <fun>

We have now successfully applied the Abstraction Principle: the common structure has been factored out. What’s left clearly expresses the computation, at least to the reader who is familiar with map, in a way that the original versions do not as quickly make apparent.

4.2.1. Side Effects

The map function exists already in OCaml’s standard library as List.map, but with one small difference from the implementation we discovered above. First, let’s see what’s potentially wrong with our own implementation, then we’ll look at the standard library’s implementation.

We’ve seen before in our discussion of exceptions that the OCaml language specification does not generally specify evaluation order of subexpressions, and that the current language implementation generally evaluates right-to-left. Because of that, the following (rather contrived) code actually causes the list elements to be printed in what might seem like reverse order:

let p x = print_int x; print_newline(); x + 1

let lst = map p [1; 2]
val p : int -> int = <fun>
2
1
val lst : int list = [2; 3]

Here’s why:

  • Expression map p [1; 2] evaluates to p 1 :: map p [2].

  • The right-hand side of that expression is then evaluated to p 1 :: (p 2 :: map p []). The application of p to 1 has not yet occurred.

  • The right-hand side of :: is again evaluated next, yielding p 1 :: (p 2 :: []).

  • Then p is applied to 2, and finally to 1.

That is likely surprising to anyone who is predisposed to thinking that evaluation would occur left-to-right. The solution is to use a let expression to cause the evaluation of the function application to occur before the recursive call:

let rec map f = function
  | [] -> []
  | h :: t -> let h' = f h in h' :: map f t

let lst2 = map p [1; 2]
val map : ('a -> 'b) -> 'a list -> 'b list = <fun>
1
2
val lst2 : int list = [2; 3]

Here’s why that works:

  • Expression map p [1; 2] evaluates to let h' = p 1 in h'' :: map p [2].

  • The binding expression p 1 is evaluated, causing 1 to be printed and h' to be bound to 2.

  • The body expression h'' :: map p [2] is then evaluated, which leads to 2 being printed next.

So that’s how the standard library defines List.map. We should use it instead of re-defining the function ourselves from now on. But it’s good that we have discovered the function “from scratch” as it were, and that if needed we could quickly re-code it.

The bigger lesson to take away from this discussion is that when evaluation order matters, we need to use let to ensure it. When does it matter? Only when there are side effects. Printing and exceptions are the two we’ve seen so far. Later we’ll add mutability.

4.2.2. Map and Tail Recursion

Astute readers will have noticed that the implementation of map is not tail recursive. That is to some extent unavoidable. Here’s a tempting but awful way to create a tail-recursive version of it:

let rec map_tr_aux f acc = function
  | [] -> acc
  | h :: t -> map_tr_aux f (acc @ [f h]) t

let map_tr f = map_tr_aux f []

let lst = map_tr (fun x -> x + 1) [1; 2; 3]
val map_tr_aux : ('a -> 'b) -> 'b list -> 'a list -> 'b list = <fun>
val map_tr : ('a -> 'b) -> 'a list -> 'b list = <fun>
val lst : int list = [2; 3; 4]

To some extent that works: the output is correct, and map_tr_aux is tail recursive. The subtle flaw is the subexpression acc @ [f h]. Recall that append is a linear-time operation on singly-linked lists. That is, if there are \(n\) list elements then append takes time \(O(n)\). So at each recursive call we perform a \(O(n)\) operation. And there will be \(n\) recursive calls, one for each element of the list. That’s a total of \(n \cdot O(n)\) work, which is \(O(n^2)\). So we achieved tail recursion, but at a high cost: what ought to be a linear-time operation became quadratic time.

In an attempt to fix that, we could use the constant-time cons operation instead of the linear-time append operation:

let rec map_tr_aux f acc = function
  | [] -> acc
  | h :: t -> map_tr_aux f (f h :: acc) t

let map_tr f = map_tr_aux f []

let lst = map_tr (fun x -> x + 1) [1; 2; 3]
val map_tr_aux : ('a -> 'b) -> 'b list -> 'a list -> 'b list = <fun>
val map_tr : ('a -> 'b) -> 'a list -> 'b list = <fun>
val lst : int list = [4; 3; 2]

And to some extent that works: it’s tail recursive and linear time. The not-so-subtle flaw this time is that the output is backwards. As we take each element off the front of the input list, we put it on the front of the output list, but that reverses their order.

Note

To understand why the reversal occurs, it might help to think of the input and output lists as people standing in a queue:

  • Input: Alice, Bob.

  • Output: empty.

Then we remove Alice from the input and add her to the output:

  • Input: Bob.

  • Output: Alice.

Then we remove Bob from the input and add him to the output:

  • Input: empty.

  • Output: Bob, Alice.

The point is that with singly-linked lists, we can only operate on the head of the list and still be constant time. We can’t move Bob to the back of the output without making him walk past Alice—and anyone else who might be standing in the output.

For that reason, the standard library calls this function List.rev_map, that is, a (tail-recursive) map function that returns its output in reverse order.

let rec rev_map_aux f acc = function
  | [] -> acc
  | h :: t -> rev_map_aux f (f h :: acc) t

let rev_map f = rev_map_aux f []

let lst = rev_map (fun x -> x + 1) [1; 2; 3]
val rev_map_aux : ('a -> 'b) -> 'b list -> 'a list -> 'b list = <fun>
val rev_map : ('a -> 'b) -> 'a list -> 'b list = <fun>
val lst : int list = [4; 3; 2]

If you want the output in the “right” order, that’s easy: just apply List.rev to it:

let lst = List.rev (List.rev_map (fun x -> x + 1) [1; 2; 3])
val lst : int list = [2; 3; 4]

Since List.rev is both linear time and tail recursive, that yields a complete solution. We get a linear-time and tail-recursive map computation. The expense is that it requires two passes through the list: one to transform, the other to reverse. We’re not going to do better than this efficiency with a singly-linked list. Of course, there are other data structures that implement lists, and we’ll come to those eventually. Meanwhile, recall that we generally don’t have to worry about tail recursion (which is to say, about stack space) until lists have 10,000 or more elements.

Why doesn’t the standard library provide this all-in-one function? Maybe it will someday if there’s good enough reason. But you might discover in your own programming there’s not a lot of need for it. In many cases, we can either do without the tail recursion, or be content with a reversed list.

The bigger lesson to take away from this discussion is that there can be a tradeoff between time and space efficiency for recursive functions. By attempting to make a function more space efficient (i.e., tail recursive), we can accidentally make it asympotically less time efficient (i.e., quadratic instead of linear), or if we’re clever keep the asymptotic time efficiency the same (i.e., linear) at the cost of a constant factor (i.e., processing twice).

4.2.3. Map in Other Languages

We mentioned above that the idea of map exists in many programming languages. Here’s an example from Python:

>>> print(list(map(lambda x: x + 1, [1, 2, 3])))
[2, 3, 4]

We have to use the list function to convert the result of the map back to a list, because Python for sake of efficiency produces each element of the map output as needed. Here again we see the theme of “when does it get evaluated?” returning.

In Java, map is part of the Stream abstraction that was added in Java 8. Since there isn’t a built-in Java syntax for lists or streams, it’s a little more verbose to give an example. Here we use a factory method Stream.of to create a stream:

jshell> Stream.of(1, 2, 3).map(x -> x + 1).collect(Collectors.toList())
$1 ==> [2, 3, 4]

Like in the Python example, we have to use something to convert the stream back into a list. In this case it’s the collect method.