(link to The Fibonacci sequence)
(Memoizing fix point operator by Jon Fairbairn)
Revision as of 21:12, 5 August 2007
Memoization is a technique for storing values of a function instead of recomputing them each time the function is called.
A classic example is the recursive computation of Fibonacci numbers.
The naive implementation of Fibonacci numbers without memoization is horribly slow.Try
slow_fib :: Int -> Integer slow_fib 0 = 0 slow_fib 1 = 1 slow_fib n = slow_fib (n-2) + slow_fib (n-1)
The memoized version is much faster.Try
memoized_fib :: Int -> Integer memoized_fib = let fib 0 = 0 fib 1 = 1 fib n = memoized_fib (n-2) + memoized_fib (n-1) in (map fib [0 ..] !!)
1 Memoizing fix point operator
You can factor out the memoizing trick to a function, the memoizing fix point operator,which we will call
fib :: (Int -> Integer) -> Int -> Integer fib f 0 = 1 fib f 1 = 1 fib f n = f (n-1) + f (n-2) fibonacci :: Int -> Integer fibonacci = memoFix fib
I suppose if you want to "put it in a library",you should just put
This allows the user e.g. to define the data structure used for memoization.
The memoising fixpoint operator works by putting the result of the first call of the function for each natural number into a data structure and using that value for subsequent calls ;-)
memoFix :: ((Int -> Integer) -> (Int -> Integer)) -> (Int -> Integer) memoFix f = mf where memo = fmap (f mf) (naturals 1 0) mf = (memo !!!)
A data structure with a node corresponding to each natural number to use as a memo.
data NaturalTree a = Node a (NaturalTree a) (NaturalTree a)
Map the nodes to the naturals in this order:
0 1 2 3 5 4 6 7 ...
Look up the node for a particular number
Node a tl tr !!! 0 = a Node a tl tr !!! n = if odd n then tl !!! top else tr !!! (top-1) where top = n `div` 2
We surely want to be able to map on these things...
instance Functor NaturalTree where fmap f (Node a tl tr) = Node (f a) (fmap f tl) (fmap f tr)
If only so that we can write cute, but inefficient things like the below,which is just a
naturals = Node 0 (fmap ((+1).(*2)) naturals) (fmap ((*2).(+1)) naturals)
The following is probably more efficient (and, having arguments won't hang around at top level, I think)-- have I put more
naturals r n = Node n ((naturals $! r2) $! (n+r)) ((naturals $! r2) $! (n+r2)) where r2 = 2*r