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CAFs and arrays can be used to memoize functions
eg. if we had a function which typically took a long time to calulate (i.e. not like the example that's coming up) we could memoise this using an array. This would incur some extra time the first time the function was evaluated but afterwards would return the result in constant time.
import Array isUppercase :: Char -> Bool isUppercase x = let ox = ord x in (ox >= ord 'A' && ox <= ord 'Z') isUC x = uc!x uc:: Array Char Bool uc = array bnds [ (x,isUppercase x) | x <- range bnds] where bnds = ((chr 0,chr 255) :: (Char,Char))
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 | x `mod` 2 == 0 = 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 | x `mod` 2 == 0 = 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 LazyEvaluation, 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 (i-1)) (buildMemoTable' (base + midSize) (i-1)) where midSize = 2 ^ (i-1) 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 | x `mod` 2 == 0 = 1 + wonderous3 (x `div` 2) | otherwise = 1 + wonderous3 (3*x+1)