Two parameter memoization in Haskell

Question

I\'m trying to memoize the following function:

gridwalk x y
    | x == 0 = 1
    | y == 0 = 1
    | otherwise = (gridwalk (x - 1) y) + (gridwalk x (y - 1))
<         


        

Answer 1

Here is a version using Data.MemoTrie from the MemoTrie package to memoize the function:

import Data.MemoTrie(memo2)

gridwalk :: Int -> Int -> Int
gridwalk = memo2 gw
  where
    gw 0 _ = 1
    gw _ 0 = 1
    gw x y = gridwalk (x - 1) y + gridwalk x (y - 1)

Answer 2

Use the data-memocombinators package from hackage. It provides easy to use memorization techniques and provides an easy and breve way to use them:

import Data.MemoCombinators (memo2,integral)

gridwalk = memo2 integral integral gridwalk' where
  gridwalk' x y
    | x == 0 = 1
    | y == 0 = 1
    | otherwise = (gridwalk (x - 1) y) + (gridwalk x (y - 1))

Answer 3

You can use a list of lists to memoize the function result for both parameters:

memo :: (Int -> Int -> a) -> [[a]]
memo f = map (\x -> map (f x) [0..]) [0..]


gw :: Int -> Int -> Int
gw 0 _ = 1
gw _ 0 = 1
gw x y = (fastgw (x - 1) y) + (fastgw x (y - 1))

gwstore :: [[Int]]
gwstore = memo gw

fastgw :: Int -> Int -> Int
fastgw x y = gwstore !! x !! y

Answer 4

If you want maximum generality, you can memoize a memoizing function.

memo :: (Num a, Enum a) => (a -> b) -> [b]
memo f = map f (enumFrom 0)

gwvals = fmap memo (memo gw)

fastgw :: Int -> Int -> Int
fastgw x y = gwvals !! x !! y

This technique will work with functions that have any number of arguments.

Edit: thanks to Philip K. for pointing out a bug in the original code. Originally memo had a "Bounded" constraint instead of "Num" and began the enumeration at minBound, which would only be valid for natural numbers.

Lists aren't a good data structure for memoizing, though, because they have linear lookup complexity. You might be better off with a Map or IntMap. Or look on Hackage.

Note that this particular code does rely on laziness, so if you wanted to switch to using a Map you would need to take a bounded amount of elements from the list, as in:

gwByMap :: Int -> Int -> Int -> Int -> Int
gwByMap maxX maxY x y = fromMaybe (gw x y) $ M.lookup (x,y) memomap
 where
  memomap = M.fromList $ concat [[((x',y'),z) | (y',z) <- zip [0..maxY] ys]
                                              | (x',ys) <- zip [0..maxX] gwvals]

fastgw2 :: Int -> Int -> Int
fastgw2 = gwByMap 20 20

I think ghc may be stupid about sharing in this case, you may need to lift out the x and y parameters, like this:

gwByMap maxX maxY = \x y -> fromMaybe (gw x y) $ M.lookup (x,y) memomap