GHC Optimization: Collatz conjecture

haskell, optimization

Solution

Some problems with your (mutable array) code:

- You use a fold to find the maximal chain length, for that the array has to be converted to an association list, that takes time and allocation the C++ version doesn't need.

- You use `even` and `div` for testing resp dividing by 2. These are slow. g++ optimises both operations to the faster bit operations (on platforms where that is supposedly faster, at least), but GHC doesn't do these low-level optimisations (yet), so for the time being, they have to be done by hand.

- You use `readArray` and `writeArray`. The extra bounds-checking that isn't done in the C++ code also takes time, once the other problems are dealt with, that amounts to a significant portion of the running time (ca. 25% on my box), since there are done a lot of reads and writes in the algorithm.

Incorporating that into the implementation, I get

import Data.Array.ST
import Data.Array.Base
import Control.Monad.ST
import Data.Bits

collatz_array :: ST s (STUArray s Int Int)
collatz_array = do
    let upper = 10000000
    arr <- newArray (0,upper) 0
    unsafeWrite arr 2 1
    let check i
            | upper < i = return arr
            | i .&. 1 == 0 = do
                l <- unsafeRead arr (i `shiftR` 1)
                unsafeWrite arr i (l+1)
                check (i+1)
            | otherwise = do
                let j = (3*i+1) `shiftR` 1
                    find k l
                        | upper < k = find (next k) $! l+1
                        | k < i     = do
                            m <- unsafeRead arr k
                            return (m+l)
                        | otherwise = do
                            m <- unsafeRead arr k
                            if m == 0
                              then do
                                  n <- find (next k) 1
                                  unsafeWrite arr k n
                                  return (n+l)
                              else return (m+l)
                          where
                            next h
                                | h .&. 1 == 0 = h `shiftR` 1
                                | otherwise = (3*h+1) `shiftR` 1
                l <- find j 1
                unsafeWrite arr i l
                check (i+1)
    check 3

collatz_max :: ST s (Int,Int)
collatz_max = do
    car <- collatz_array
    (_,upper) <- getBounds car
    let find w m i
            | upper < i = return (w,m)
            | otherwise = do
                l <- unsafeRead car i
                if m < l
                  then find i l (i+1)
                  else find w m (i+1)
    find 1 0 2

main :: IO ()
main = print (runST collatz_max)

And the timings (both for 10 million):

$ time ./cccoll
8400511 429

real    0m0.210s
user    0m0.200s
sys     0m0.009s
$ time ./stcoll
(8400511,429)

real    0m0.341s
user    0m0.307s
sys     0m0.033s

which doesn't look too bad.

Important note: That code only works on 64-bit GHC (so, in particular, on Windows, you need ghc-7.6.1 or later, previous GHCs were 32-bit even on 64-bit Windows) since intermediate chain elements exceed 32-bit range. On 32-bit systems, one would have to use `Integer` or a 64-bit integer type (`Int64` or `Word64`) for following the chains, at a drastic performance cost, since the primitive 64-bit operations (arithmetic and shifts) are implemented as foreign calls to C functions in 32-bit GHCs (fast foreign calls, but still much slower than direct machine ops).

Problem

Original source