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More problems with memoization
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 Date: 2006-10-01 (00:23) From: Jon Harrop Subject: Re: [Caml-list] More problems with memoization
```On Saturday 30 September 2006 19:01, Diego Olivier FERNANDEZ PONS wrote:
> I wrote the following (classical) memoized code for the fibonacci
> function and I have been unsuccessfully trying to generalize it with a
> higher order function.
>
> let rec fib = function
>
>    | 0 -> 0
>    | 1 -> 1
>    | n -> fib_mem (n - 1) + fib_mem (n - 2)
>
> and fib_mem =
>    let table = ref [] in
>      function n ->
>        try
> 	List.assoc n !table
>        with Not_found ->
> 	let f_n = fib n in
> 	  table := (n, f_n) :: !table;
> 	  f_n
>
> # val fib : int -> int = <fun>
> # val fib_mem : int -> int = <fun>
>
> It works: fib 35 answers instantaneously.
>
> Now I want to achieve the same result with a higher order function
> [make_memo] and apply it to fib

I believe you want to "untie the knot" of recursion, creating an higher-order,
auxiliary fibonacci function fib_aux that accepts the recursive call as an
argument:

# let rec fib_aux fib = function
| 0 | 1 as n -> n
| n -> fib(n - 1) + fib(n - 2);;
val fib_aux : (int -> int) -> int -> int = <fun>

You can recover the ordinary fibonacci function using the Y combinator:

# let rec fib n = fib_aux fib n;;
val fib : int -> int = <fun>

You can write a higher-order memoization function that accepts an argument
with the type of fib_aux:

# let memoize f =
let m = Hashtbl.create 0 in
let rec f' n =
try Hashtbl.find m n with Not_found ->
let x = f f' n in
Hashtbl.replace m n x;
x in
f';;
val memoize : (('a -> 'b) -> 'a -> 'b) -> 'a -> 'b = <fun>

Now you can memoize recursive functions easily:

# memoize fib_aux 35;;
- : int = 9227465

--
Dr Jon D Harrop, Flying Frog Consultancy Ltd.
Objective CAML for Scientists
http://www.ffconsultancy.com/products/ocaml_for_scientists

```