## OCaml and sequences from scratch, part 2

This is the second part of my series about simple sequences in OCaml from scratch, the first part is here.

We are going to explore a few more complex but not difficult list (or our own definition, seq) functions, they will be our building blocks for (hopefully) future blog posts. This is a very common pattern in functional programming algorithms, base in smaller functions doing a simple operation. Isn’t it beautiful?!

Now it is time for a little more complicated functions, we will start with one of the jewels of the OCaml standard library, rev:

let sample = Next(1, Next(2, Next(3, End))) ;;

let seq_rev l =
let rec rev' acc = function
| End -> acc
| Next (x, tl) -> rev' (seq_cons x acc) tl
in
rev' End l
;;

seq_rev sample

Out[*]:
val seq_rev : 'a seq -> 'a seq = <fun>
- : int seq = Next (3, Next (2, Next (1, End)))


I said before this is one of the jewels of OCaml, while implementing sequence operations you will use it many times.

Now let’s try something a little more complex, the init function, the idea is that you provide a function f and a number n and it will return a sequence with n number of elements when applying the function f n number of times. The signature of the function is int -> 'a

let seq_init n f =
let rec init' n' acc =
if n' = 0 then acc else init' (n' - 1) (cons_seq (f n') acc)
in
init' n End
;;

seq_init 3 (fun x -> x)

Out[*]:
val seq_init : int -> (int -> 'a) -> 'a seq = <fun>
- : int seq = Next (1, Next (2, Next (3, End)))


As you see, the inner accumulator pattern is used here as well, get used to it because it is very common in OCaml, oh, and the function is tail recursive!.

#### Concatenate lists

We already saw an operation to append an element to the beginning of the list. As you may remember, cons (expressed as well in OCaml as the operator ::) is a time constant operation, it doesn’t matter how big is the list we are appending the element to, it will always take the same constant time.

This is not the case with appending an element at the end, which is the same case as appending two lists (after all, appending another list is just adding its first element as the last element of the first list).

Let’s give our first try:

let rec seq_append a b =
match a with
| End -> b
| Next (x, tl) -> Next(x, ((append_seq [@tailcall]) tl b))
;;
seq_append sample (Next(4, End))

Out[*]:
val seq_append : 'a seq -> 'a seq -> 'a seq
- : int seq = Next (1, Next (2, Next (3, Next (4, End))))


But wait, the OCaml compiler emits a warning! (yeah, that is why I actually added the [@tailcall] notation there)

Warning 51: expected tailcall

Yes, that is because our implementation is not tail recursive!, it means it cannot work in very long sequences, I did that on purpose because the standard implementation of append in the OCaml List module it is not tail recursive neither!.

We can do better than this, we can write a tail recursive function!. It is not as efficient though, the operation will roughly be $$O(n^2)$$

let seq_append a b =
let rec append' acc = function
| End -> acc
| Next (x, t) -> append' (Next(x, acc)) t
in
append' End (seq_rev a)
;;

seq_append sample (Next(4, End))

Out[*]:
val seq_append : 'a seq -> 'a seq -> 'a seq
- : int seq = Next (1, Next (2, Next (3, Next (4, End))))


Did you notice the usage of rev here? this is another patter I like to call reverse inner accumulator loop (again, I made up that name, I am pretty sure it is known with another sexier name). That pattern is another of the things you will see very often in OCaml list functions, it looks weird but you will see it everywhere with list functions. In the standard Caml list module we have the function rev_append and that does exactly the same.

That is all for today, as usual you can find the original Jupyter Notebook in my GitHub repository.