Knapsack Problem in Haskell
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I recently described two versions of the Knapsack problem written in Ruby and Python and one common thing is that I used a global cache to store the results of previous calculations.
From my experience of coding in Haskell it’s not considered very idiomatic to write code like that and although I haven’t actually tried it, potentially more tricky to achieve.
I thought it’d be interesting to try and write the algorithm in Haskell with that constraint in mind and my first version looked like this:
ref :: a -> IORef a ref x = unsafePerformIO (newIORef x) knapsackCached1 :: [[Int]] -> Int -> Int -> IORef (Map.Map (Int, Int) Int) -> Int knapsackCached1 rows knapsackWeight index cacheContainer = unsafePerformIO $ do cache <- readIORef cacheContainer if index == 0 || knapsackWeight == 0 then do return 0 else let (value:weight:_) = rows !! index best = knapsackCached1 rows knapsackWeight prevIndex cacheContainer in if weight > knapsackWeight && lookupPreviousIn cache == Nothing then do let updatedCache = Map.insert (prevIndex, knapsackWeight) best cache writeIORef cacheContainer updatedCache return $ fromJust $ lookupPreviousIn updatedCache else if lookupPreviousIn cache == Nothing then do let newBest = maximum [best, value + knapsackCached1 rows (knapsackWeight-weight) prevIndex cacheContainer] updatedCache = Map.insert (prevIndex, knapsackWeight) newBest cache writeIORef cacheContainer updatedCache return $ fromJust $ lookupPreviousIn updatedCache else return $ fromJust $ lookupPreviousIn cache where lookupPreviousIn cache = Map.lookup (prevIndex,knapsackWeight) cache prevIndex = index-1
We then call it like this:
let (knapsackWeight, numberOfItems, rows) = process contents cache = ref (Map.empty :: Map.Map (Int, Int) Int) knapsackCached1 rows knapsackWeight (numberOfItems-1) cache
As you can see, we’re passing around the cache as a parameter where the cache is a Map wrapped inside an IORef – a data type which allows us to pass around a mutable variable in the IO monad.
We write our new value into the cache on lines 11 and 17 so that our updates to the map will be picked up in the other recursive steps.
Apart from that the shape of the code is the same as the Ruby and Python versions except I’m now only using a map with a pair as the key instead of an array + map as in the other versions.
The annoying thing about this solution is that we have to pass the cache around as a parameter when it’s just a means of optimisation and not part of the actual problem.
An alternative solution could be the following where we abstract the writing/reading of the map into a memoize function which we wrap our function in:
memoize :: ((Int, Int) -> Int) -> (Int, Int) -> Int memoize fn mapKey = unsafePerformIO $ do let cache = ref (Map.empty :: Map.Map (Int, Int) Int) items <- readIORef cache if Map.lookup mapKey items == Nothing then do let result = fn mapKey writeIORef cache $ Map.insert mapKey result items return result else return (fromJust $ Map.lookup mapKey items) knapsackCached :: [[Int]] -> Int -> Int -> Int knapsackCached rows weight numberOfItems = inner (numberOfItems-1, weight) where inner = memoize (\(i,w) -> if i < 0 || w == 0 then 0 else let best = inner (i-1,w) (vi:wi:_) = rows !! i in if wi > w then best else maximum [best, vi + inner (i-1, w-wi)])
We can call that function like this:
let (knapsackWeight, numberOfItems, rows) = process contents cache = ref (Map.empty :: Map.Map (Int, Int) Int) knapsackCached rows knapsackWeight numberOfItems
Here we define an inner function inside knapsackCached which is a partial application of the memoize function. We then pass our cache key to that function on the previous line.
One thing which I noticed while writing this code is that there is some strangeness around the use of ‘in’ after let statements. It seems like if you’re inside an if/else block you need to use ‘in’ unless you’re in the context of a Monad (do statement) in which case you don’t need to.
I was staring a screen of compilation errors for about an hour until I realised this!
These are the timings for the two versions of the algorithm:
# First one $ time ./k knapsack2.txt real 0m14.993s user 0m14.646s sys 0m0.320s # Second one $ time ./k knapsack2.txt real 0m12.594s user 0m12.259s sys 0m0.284s
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