Wednesday, April 02, 2008

Memoization, etc.

I've been sweating over some problems at Project Euler, and I've verified that I'm no mathematician. Some enterprising Haskell programmers got there way before I did, and if I were inclined to cheat, I'd simply translate their solutions into F#. However, in the effort or out of the necessity to remain pure, they don't use for-loops over arrays, or their own memoization, which I'd almost certainly do for any production code that had to calculate the Fibonacci series...well, actually, I don't do that kind of work. Anyway...

I did find the Haskell samples instructive, though I don't want to get sidetracked from my project of learning F#. To this end I've once again resorted to Don Syme's thoughts on memoization: here on the CAML list, and here more recently, where he avails himself of some popular .NET collections. So a memoized Fibonacci function in Dr. Syme's older formulation looks like this:

> #light;;

> let cache f =

- let table = ref [] in

- fun n ->

- try

- List.assoc n !table

- with Not_found ->

- let f_n = f n in

- table := (n, f_n) :: !table;

- f_n



- let rec fib_mem =

- cache (function

- // You'd better return a bigint, not an int!

- ¦ 0 -> 0I

- ¦ 1 -> 1I

- ¦ n -> fib_mem (n - 1) + fib_mem (n - 2))



- ;;

val cache : ('a -> 'b) -> ('a -> 'b)

val fib_mem : (int -> bigint)

> fib_mem(1000);;

val it : bigint

= 434665576869374564356885276750406258025646605173717804024817290895365554179490



val ns : seq
Several problems at Project Euler ask you to do something with the first 10 or last 10 digits of some big integer; for anyone who was wondering how to do that with a number such as the one above, here's one way:

> fib_mem(1000) |> Seq.unfold (fun n -> if n > 0I
- then
- let quotient,remainder = BigInt.divmod n 10I
- Some (remainder, quotient)
- else
- None);;
val it : seq = seq [5I; 7I; 8I; 8I; ...]
You'll notice that the least significant digits appear first, which is of course by design, since dividing by 10 is probably cheap. The last thing I'd want to do is reverse this sequence, i.e.

- |> List.of_seq
- |> List.rev;;
I could clean this up further by applying BigInt.to_Int32 to the generated elements, etc. To sum the first 10 digits (note that this is *not* the Project Euler problem; I'm not about to give away any answers) I could go further, like so:

- |> Seq.take 10
- |> Seq.fold1 (+);;
You get the idea.

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