Memoization
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(→Memoization with recursion: the etaexpanded version doesn't actually memoize properly) 

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Memoization is a technique for storing values of a function instead of recomputing them each time the function is called.
Contents 
1 Memoization without recursion
You can just write a memoization function using a data structure that is suitable for your application. We don't go into the details of this case. If you want a general solution for several types,
you need a type class, saymemoize :: Memoizable a => (a>b) > (a>b)
Now, how to implement something like this? Of course, one needs a finite
map that stores valuesMap () b := b Map (Either a a') b := (Map a b, Map a' b) Map (a,a') b := Map a (Map a' b)
Its construction is based on the following laws for functions
() > b =~= b (a + a') > b =~= (a > b) x (a' > b)  = case analysis (a x a') > b =~= a > (a' > b)  = currying
For further and detailed explanations, see
 Ralf Hinze: Memo functions, polytypically !
 Ralf Hinze: Generalizing generalized tries
 Conal Elliott: Elegant memoization with functional memo tries and other posts on memoization.
 Conal Elliott Denotational design with type class morphisms, section 9 (Memo tries).
2 Memoization with recursion
Things become more complicated if the function is recursively defined and it should use memoized calls to itself. A classic example is the recursive computation of Fibonacci numbers.
The naive implementation of Fibonacci numbers without memoization is horribly slow.
Tryslow_fib :: Int > Integer slow_fib 0 = 0 slow_fib 1 = 1 slow_fib n = slow_fib (n2) + slow_fib (n1)
The memoized version is much faster.
Trymemoized_fib :: Int > Integer memoized_fib = (map fib [0 ..] !!) where fib 0 = 0 fib 1 = 1 fib n = memoized_fib (n2) + memoized_fib (n1)
2.1 Memoizing fix point operator
You can factor out the memoizing trick to a function, the memoizing fix point operator,
which we will callfib :: (Int > Integer) > Int > Integer fib f 0 = 1 fib f 1 = 1 fib f n = f (n1) + f (n2) fibonacci :: Int > Integer fibonacci = memoFix fib
I suppose if you want to "put it in a library",
you should just putThis 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 ;)
In general it is
memoFix :: ((a > b) > (a > b)) > a > b memoFix f = let mf = memoize (f mf) in mf
3 Efficient tree data structure for maps from Int to somewhere
Here we present a special tree data type (datainttrie) which is useful as memoizing data structure e.g. for the Fibonacci function.
memoizeInt :: (Int > a) > (Int > a) memoizeInt f = (fmap f (naturals 1 0) !!!)
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 !!! (top1) 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 anaturals = 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 morenaturals r n = Node n ((naturals $! r2) $! (n+r)) ((naturals $! r2) $! (n+r2)) where r2 = 2*r
4 Memoising CAFS
Note: This is migrated from the old wiki.
Memoising constructor functions gives you HashConsing, and you can sometimes use MemoisingCafs to implement that.
The MemoisingCafs idiom also supports recursion.
Consider, for example:
wonderous :: Integer > Integer wonderous 1 = 0 wonderous x  even x = 1 + wonderous (x `div` 2)  otherwise = 1 + wonderous (3*x+1)
This function is not at all understood by mathematicians and has a surprisingly complex recursion pattern, so if you need to call it many times with different values, optimising it would not be easy.
However, we can memoise some of the domain using an array CAF:
wonderous2 :: Integer > Integer wonderous2 x  x <= maxMemo = memoArray ! x  otherwise = wonderous2' x where maxMemo = 100 memoArray = array (1,maxMemo) [ (x, wonderous2' x)  x < [1..maxMemo] ] wonderous2' 1 = 0 wonderous2' x  even x = 1 + wonderous2 (x `div` 2)  otherwise = 1 + wonderous2' (3*x+1)
When using this pattern in your own code, note carefully when to call the memoised version (wonderous2 in the above example) and when not to. In general, the partially memoised version (wonderous2' in the above example) should call the memoised version if it needs to perform a recursive call. However, in this instance, we only memoize for small values of x, so the branch of the recursion that passes a larger argument need not bother checking the memo table. (This does slow the array initialization, however.) Thanks to lazy evaluation, we can even memoise an infinite domain, though we lose constant time lookup. This data structure is O(log N):
type MemoTable a = [(Integer, BinTree a)] data BinTree a = Leaf a  Node Integer (BinTree a) (BinTree a) wonderous3 :: Integer > Integer wonderous3 x = searchMemoTable x memoTable where memoTable :: MemoTable Integer memoTable = buildMemoTable 1 5 buildMemoTable n i = (nextn, buildMemoTable' n i) : buildMemoTable nextn (i+1) where nextn = n + 2^i buildMemoTable' base 0 = Leaf (wonderous3' base) buildMemoTable' base i = Node (base + midSize) (buildMemoTable' base (i1)) (buildMemoTable' (base + midSize) (i1)) where midSize = 2 ^ (i1) searchMemoTable x ((x',tree):ms)  x < x' = searchMemoTree x tree  otherwise = searchMemoTable x ms searchMemoTree x (Leaf y) = y searchMemoTree x (Node mid l r)  x < mid = searchMemoTree x l  otherwise = searchMemoTree x r wonderous3' 1 = 0 wonderous3' x  even x = 1 + wonderous3 (x `div` 2)  otherwise = 1 + wonderous3 (3*x+1)
Naturally, these techniques can be combined, say, by using a fast CAF data structure for the most common part of the domain and an infinite CAF data structure for the rest.
5 Memoizing polymorphic functions
What about memoizing polymorphic functions defined with polymorphic recursion? How can such functions be memoized? The caching data structures used in memoization typically handle only one type of argument at a time. For instance, one can have finite maps of differing types, but each concrete finite map holds just one type of key and one type of value.
See the discussion on *Memoizing polymorphic functions*, part one and part two.
6 See also
 HaskellCafe "speeding up fibonacci with memoizing"
 HaskellCafe about memoization utility function
 HaskellCafe "memoisation"
 HaskellCafe about Memoization and Data.Map
 http://programming.reddit.com/info/16ofr/comments
 Monadic Memoization Mixins by Daniel Brown and William R. Cook
 datamemocombinators: Combinators for building memo tables.
 MemoTrie: Triebased memo functions