# [Haskell-cafe] Memoization in Haskell?

Edward Kmett ekmett at gmail.com
Fri Jul 9 04:39:23 EDT 2010

```On Thu, Jul 8, 2010 at 5:30 PM, Angel de Vicente <angelv at iac.es> wrote:

> Hi,
>
> I'm going through the first chapters of the Real World Haskell book,
> so I'm still a complete newbie, but today I was hoping I could solve
> the following function in Haskell, for large numbers (n > 108)
>
> f(n) = max(n,f(n/2)+f(n/3)+f(n/4))
>
> I've seen examples of memoization in Haskell to solve fibonacci
> numbers, which involved computing (lazily) all the fibonacci numbers
> up to the required n. But in this case, for a given n, we only need to
> compute very few intermediate results.
>
> How could one go about solving this efficiently with Haskell?
>
> We can do this very efficiently by making a structure that we can index in
sub-linear time.

But first,

>    {-# LANGUAGE BangPatterns #-}

>    import Data.Function (fix)

Lets define f, but make it use 'open recursion' rather than call itself
directly.

>    f :: (Int -> Int) -> Int -> Int
>    f mf 0 = 0
>    f mf n = max n \$ mf (div n 2) +
>                     mf (div n 3) +
>                     mf (div n 4)

You can get an unmemoized f by using `fix f`

This will let you test that f does what you mean for small values of f by
calling, for example: `fix f 123` = 144

We could memoize this by defining:

>    f_list :: [Int]
>    f_list = map (f faster_f) [0..]

>    faster_f :: Int -> Int
>    faster_f n = f_list !! n

That performs passably well, and replaces what was going to take O(n^3) time
with something that memoizes the intermediate results.

But it still takes linear time just to index to find the memoized answer for
`mf`. This means that results like:

*Main Data.List> faster_f 123801
248604

are tolerable, but the result doesn't scale much better than that. We can do
better!

First lets define an infinite tree:

>    data Tree a = Tree (Tree a) a (Tree a)
>    instance Functor Tree where
>        fmap f (Tree l m r) = Tree (fmap f l) (f m) (fmap f r)

And then we'll define a way to index into it, so we can find a node with
index n in O(log n) time instead:

>    index :: Tree a -> Int -> a
>    index (Tree _ m _) 0 = m
>    index (Tree l _ r) n = case (n - 1) `divMod` 2 of
>        (q,0) -> index l q
>        (q,1) -> index r q

... and we may find a tree full of natural numbers to be convenient so we
don't have to fiddle around with those indices:

>    nats :: Tree Int
>    nats = go 0 1
>        where
>            go !n !s = Tree (go l s') n (go r s')
>                where
>                    l = n + s
>                    r = l + s
>                    s' = s * 2

Since we can index, you can just convert a tree into a list:

>    toList :: Tree a -> [a]
>    toList as = map (index as) [0..]

You can check the work so far by verifying that `toList nats` gives you
[0..]

Now,

>    f_tree :: Tree Int
>    f_tree = fmap (f fastest_f) nats

>    fastest_f :: Int -> Int
>    fastest_f = index f_tree

works just like with list above, but instead of taking linear time to find
each node, can chase it down in logarithmic time.

The result is considerably faster:

*Main> fastest_f 12380192300
67652175206

*Main> fastest_f 12793129379123
120695231674999

In fact it is so much faster that you can go through and replace Int with
Integer above and get ridiculously large answers almost instantaneously

*Main> fastest_f' 1230891823091823018203123
93721573993600178112200489

*Main> fastest_f' 12308918230918230182031231231293810923
11097012733777002208302545289166620866358

-Edward Kmett
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