Knapsack Problem in Haskell
Date: 2013-01-09 00:12:25

I recently described two versions of the http://www.markhneedham.com/blog/2013/01/07/knapsack-problem-python-vs-ruby/[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:

[source,haskell]
----

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:

[source,haskell]
----

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 http://www.haskell.org/ghc/docs/latest/html/libraries/base/Data-IORef.html[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 +++<cite>+++memoize+++</cite>+++ function which we wrap our function in:

[source,haskell]
----

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:

[source,haskell]
----

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 +++<cite>+++knapsackCached+++</cite>+++ 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:

[source,text]
----

# 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
----

I'm still trying to understand exactly how to http://book.realworldhaskell.org/read/profiling-and-optimization.html[profile] http://www.haskell.org/haskellwiki/Performance/GHC[and] http://stackoverflow.com/questions/3276240/tools-for-analyzing-performance-of-a-haskell-program[then] http://www.haskell.org/ghc/docs/7.4.1/html/users_guide/runtime-control.html[optimise] the program so any tips are always welcome.