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From: Jon Harrop <jon@ffconsultancy.com>
Organization: Flying Frog Consultancy Ltd.
To: caml-list@yquem.inria.fr
Subject: Re: [Caml-list] More problems with memoization
Date: Sun, 1 Oct 2006 01:23:48 +0100
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X-Spam: no; 0.00; memoization:01 pons:01 memoized:01 fib:01 fib:01 val:01 val:01 recursion:01 higher-order:01 recursive:01 combinator:01 higher-order:01 memoization:01 memoize:01 hashtbl:01 

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