Solving knapsack prob in F#: performance - performance

Question

I found an article:
Solving the 0-1 knapsack problem using continuation-passing style with memoization in F#
about knapsack problem implemented in F#. As I'm learning this language, I found this really interesting and tried to investigate this a bit. Here's the code I crafted:
open System
open System.IO
open System.Collections.Generic
let parseToTuple (line : string) =
let parsedLine = line.Split(' ') |> Array.filter(not << String.IsNullOrWhiteSpace) |> Array.map Int32.Parse
(parsedLine.[0], parsedLine.[1])
let memoize f =
let cache = Dictionary<_, _>()
fun x ->
if cache.ContainsKey(x)
then cache.[x]
else
let res = f x
cache.[x] <- res
res
type Item =
{
Value : int
Size : int
}
type ContinuationBuilder() =
member b.Bind(x, f) = fun k -> x (fun x -> f x k)
member b.Return x = fun k -> k x
member b.ReturnFrom x = x
let cont = ContinuationBuilder()
let set1 =
[
(4, 11)
(8, 4)
(10, 5)
(15, 8)
(4, 3)
]
let set2 =
[
(50, 341045); (1906, 4912); (41516, 99732); (23527, 56554); (559, 1818); (45136, 108372); (2625, 6750); (492, 1484)
(1086, 3072); (5516, 13532); (4875, 12050); (7570, 18440); (4436, 10972); (620, 1940); (50897, 122094); (2129, 5558)
(4265, 10630); (706, 2112); (2721, 6942); (16494, 39888); (29688, 71276); (3383, 8466); (2181, 5662); (96601, 231302)
(1795, 4690); (7512, 18324); (1242, 3384); (2889, 7278); (2133, 5566); (103, 706); (4446, 10992); (11326, 27552)
(3024, 7548); (217, 934); (13269, 32038); (281, 1062); (77174, 184848); (952, 2604); (15572, 37644); (566, 1832)
(4103, 10306); (313, 1126); (14393, 34886); (1313, 3526); (348, 1196); (419, 1338); (246, 992); (445, 1390)
(23552, 56804); (23552, 56804); (67, 634)
]
[<EntryPoint>]
let main args =
// prepare list of items from a file args.[0]
let header, items = set1
|> function
| h::t -> h, t
| _ -> raise (Exception("Wrong data format"))
let N, K = header
printfn "N = %d, K = %d" N K
let items = List.map (fun x -> {Value = fst x ; Size = snd x}) items |> Array.ofList
let rec combinations =
let innerSolver key =
cont
{
match key with
| (i, k) when i = 0 || k = 0 -> return 0
| (i, k) when items.[i-1].Size > k -> return! combinations (i-1, k)
| (i, k) -> let item = items.[i-1]
let! v1 = combinations (i-1, k)
let! beforeItem = combinations (i-1, k-item.Size)
let v2 = beforeItem + item.Value
return max v1 v2
}
memoize innerSolver
let res = combinations (N, K) id
printfn "%d" res
0
However, the problem with this implementation is that it's veeeery slow (in practice I'm unable to solve problem with 50 items and capacity of ~300000, which gets solved by my naive implementation in C# in less than 1s).
Could you tell me if I made a mistake somewhere? Or maybe the implementation is correct and this is simply the inefficient way of solving this problem.

Answer 1

When you naively apply a generic memoizer like this, and use continuation passing, the values in your memoization cache are continuations, not regular "final" results. Thus, when you get a cache hit, you aren't getting back a finalized result, you are getting back some function which promises to compute a result when you invoke it. This invocation might be expensive, might invoke various other continuations, might ultimately hit the memoization cache again itself, etc.
Effectively memoizing continuation-passing functions such that a) the caching works to full effect and b) the function remains tail-recursive is quite difficult. Read this discussion and come back when you fully understand it all. ;-)
The author of the blog post you linked is using a more sophisticated, less generic memoizer which is specially fitted to the problem. Admittedly, I don't fully grok it yet (code on the blog is incomplete/broken, so hard to try it out), but I think the gist of it is that it "forces" the chain of continuations before caching the final integer result.
To illustrate the point, here's a quick refactor of your code which is fully self-contained and traces out relevant info:
open System
open System.Collections.Generic
let mutable cacheHits = 0
let mutable cacheMisses = 0
let memoize f =
let cache = Dictionary<_, _>()
fun x ->
match cache.TryGetValue(x) with
| (true, v) ->
cacheHits <- cacheHits + 1
printfn "Hit for %A - Result is %A" x v
v
| _ ->
cacheMisses <- cacheMisses + 1
printfn "Miss for %A" x
let res = f x
cache.[x] <- res
res
type Item = { Value : int; Size : int }
type ContinuationBuilder() =
member b.Bind(x, f) = fun k -> x (fun x -> f x k)
member b.Return x = fun k -> k x
member b.ReturnFrom x = x
let cont = ContinuationBuilder()
let genItems n =
[| for i = 1 to n do
let size = i % 5
let value = (size * i)
yield { Value = value; Size = size }
|]
let N, K = (5, 100)
printfn "N = %d, K = %d" N K
let items = genItems N
let rec combinations_cont =
memoize (
fun key ->
cont {
match key with
| (0, _) | (_, 0) -> return 0
| (i, k) when items.[i-1].Size > k -> return! combinations_cont (i - 1, k)
| (i, k) -> let item = items.[i-1]
let! v1 = combinations_cont (i-1, k)
let! beforeItem = combinations_cont (i-1, k - item.Size)
let v2 = beforeItem + item.Value
return max v1 v2
}
)
let res = combinations_cont (N, K) id
printfn "Answer: %d" res
printfn "Memo hits: %d" cacheHits
printfn "Memo misses: %d" cacheMisses
printfn ""
let rec combinations_plain =
memoize (
fun key ->
match key with
| (i, k) when i = 0 || k = 0 -> 0
| (i, k) when items.[i-1].Size > k -> combinations_plain (i-1, k)
| (i, k) -> let item = items.[i-1]
let v1 = combinations_plain (i-1, k)
let beforeItem = combinations_plain (i-1, k-item.Size)
let v2 = beforeItem + item.Value
max v1 v2
)
cacheHits <- 0
cacheMisses <- 0
let res2 = combinations_plain (N, K)
printfn "Answer: %d" res2
printfn "Memo hits: %d" cacheHits
printfn "Memo misses: %d" cacheMisses
As you can see, the CPS version is caching continuations (not integers), and there are is a lot of extra activity going on toward the end as the continuations are invoked.
If you boost the problem size to let (N, K) = (20, 100) (and remove the printfn statements in the memoizer), you will see that the CPS version ends up doing over 1 million cache lookups, compared to plain version doing only a few hundred.

Answer 2

From running this code in FSI:
open System
open System.Diagnostics
open System.Collections.Generic
let time f =
System.GC.Collect()
let sw = Stopwatch.StartNew()
let r = f()
sw.Stop()
printfn "Took: %f" sw.Elapsed.TotalMilliseconds
r
let mutable cacheHits = 0
let mutable cacheMisses = 0
let memoize f =
let cache = Dictionary<_, _>()
fun x ->
match cache.TryGetValue(x) with
| (true, v) ->
cacheHits <- cacheHits + 1
//printfn "Hit for %A - Result is %A" x v
v
| _ ->
cacheMisses <- cacheMisses + 1
//printfn "Miss for %A" x
let res = f x
cache.[x] <- res
res
type Item = { Value : int; Size : int }
type ContinuationBuilder() =
member b.Bind(x, f) = fun k -> x (fun x -> f x k)
member b.Return x = fun k -> k x
member b.ReturnFrom x = x
let cont = ContinuationBuilder()
let genItems n =
[| for i = 1 to n do
let size = i % 5
let value = (size * i)
yield { Value = value; Size = size }
|]
let N, K = (80, 400)
printfn "N = %d, K = %d" N K
let items = genItems N
//let rec combinations_cont =
// memoize (
// fun key ->
// cont {
// match key with
// | (0, _) | (_, 0) -> return 0
// | (i, k) when items.[i-1].Size > k -> return! combinations_cont (i - 1, k)
// | (i, k) -> let item = items.[i-1]
// let! v1 = combinations_cont (i-1, k)
// let! beforeItem = combinations_cont (i-1, k - item.Size)
// let v2 = beforeItem + item.Value
// return max v1 v2
// }
// )
//
//
//cacheHits <- 0
//cacheMisses <- 0
//let res = time(fun () -> combinations_cont (N, K) id)
//printfn "Answer: %d" res
//printfn "Memo hits: %d" cacheHits
//printfn "Memo misses: %d" cacheMisses
//printfn ""
let rec combinations_plain =
memoize (
fun key ->
match key with
| (i, k) when i = 0 || k = 0 -> 0
| (i, k) when items.[i-1].Size > k -> combinations_plain (i-1, k)
| (i, k) -> let item = items.[i-1]
let v1 = combinations_plain (i-1, k)
let beforeItem = combinations_plain (i-1, k-item.Size)
let v2 = beforeItem + item.Value
max v1 v2
)
cacheHits <- 0
cacheMisses <- 0
printfn "combinations_plain"
let res2 = time (fun () -> combinations_plain (N, K))
printfn "Answer: %d" res2
printfn "Memo hits: %d" cacheHits
printfn "Memo misses: %d" cacheMisses
printfn ""
let recursivelyMemoize f =
let cache = Dictionary<_, _>()
let rec memoizeAux x =
match cache.TryGetValue(x) with
| (true, v) ->
cacheHits <- cacheHits + 1
//printfn "Hit for %A - Result is %A" x v
v
| _ ->
cacheMisses <- cacheMisses + 1
//printfn "Miss for %A" x
let res = f memoizeAux x
cache.[x] <- res
res
memoizeAux
let combinations_plain2 =
let combinations_plain2Aux combinations_plain2Aux key =
match key with
| (i, k) when i = 0 || k = 0 -> 0
| (i, k) when items.[i-1].Size > k -> combinations_plain2Aux (i-1, k)
| (i, k) -> let item = items.[i-1]
let v1 = combinations_plain2Aux (i-1, k)
let beforeItem = combinations_plain2Aux (i-1, k-item.Size)
let v2 = beforeItem + item.Value
max v1 v2
let memoized = recursivelyMemoize combinations_plain2Aux
fun x -> memoized x
cacheHits <- 0
cacheMisses <- 0
printfn "combinations_plain2"
let res3 = time (fun () -> combinations_plain2 (N, K))
printfn "Answer: %d" res3
printfn "Memo hits: %d" cacheHits
printfn "Memo misses: %d" cacheMisses
printfn ""
let recursivelyMemoizeCont f =
let cache = Dictionary HashIdentity.Structural
let rec memoizeAux x k =
match cache.TryGetValue(x) with
| (true, v) ->
cacheHits <- cacheHits + 1
//printfn "Hit for %A - Result is %A" x v
k v
| _ ->
cacheMisses <- cacheMisses + 1
//printfn "Miss for %A" x
f memoizeAux x (fun y ->
cache.[x] <- y
k y)
memoizeAux
let combinations_cont2 =
let combinations_cont2Aux combinations_cont2Aux key =
cont {
match key with
| (0, _) | (_, 0) -> return 0
| (i, k) when items.[i-1].Size > k -> return! combinations_cont2Aux (i - 1, k)
| (i, k) -> let item = items.[i-1]
let! v1 = combinations_cont2Aux (i-1, k)
let! beforeItem = combinations_cont2Aux (i-1, k - item.Size)
let v2 = beforeItem + item.Value
return max v1 v2
}
let memoized = recursivelyMemoizeCont combinations_cont2Aux
fun x -> memoized x id
cacheHits <- 0
cacheMisses <- 0
printfn "combinations_cont2"
let res4 = time (fun () -> combinations_cont2 (N, K))
printfn "Answer: %d" res4
printfn "Memo hits: %d" cacheHits
printfn "Memo misses: %d" cacheMisses
printfn ""
I get these results:
N = 80, K = 400
combinations_plain
Took: 7.191000
Answer: 6480
Memo hits: 6231
Memo misses: 6552
combinations_plain2
Took: 6.310800
Answer: 6480
Memo hits: 6231
Memo misses: 6552
combinations_cont2
Took: 17.021200
Answer: 6480
Memo hits: 6231
Memo misses: 6552
combinations_plain is from latkin's answer.
combinations_plain2 exposes the recursive memoization step explicitly.
combinations_cont2 adapts the recursive memoization function into one that memoizes the continuation results.
combinations_cont2 works by intercepting the result in the continuation before passing it on to the actual continuation. Subsequent calls on the same key provide a continuation and this continuation is fed the answer we intercepted originally.
This demonstrates that we are able to:
Memoize using continuation passing style.
Achieve similar (ish) performance characteristics to the vanilla memoized version.
I hope this clears things up a little. Sorry, my blog code snippet was incomplete (I think I might have lost it when reformatting recently).