Question

I am dealing with the computation which has as an intermediate result a list A=[B], which is a list of K lists of the length L. The time-complexity to compute an element of B is controlled by the parameter M and is theoretically linear in M. Theoretically I would expect the time-complexity for the computation of A to be O(K*L*M). However, this is not the case and I don’t understand why?

Here is the simple complete sketch program which exhibits the problem I have explained

import System.Random (randoms, mkStdGen)
import Control.Parallel.Strategies (parMap, rdeepseq)
import Control.DeepSeq (NFData)
import Data.List (transpose)

type Point = (Double, Double)

fmod :: Double -> Double -> Double
fmod a b | a < 0     = b - fmod (abs a) b 
         | otherwise = if a < b then a 
                       else let q = a / b in b * (q - fromIntegral (floor q))

standardMap :: Double -> Point -> Point
standardMap k (q, p) = (fmod (q + p) (2 * pi), fmod (p + k * sin(q)) (2 * pi))

trajectory :: (Point -> Point) -> Point -> [Point] 
trajectory map initial = initial : (trajectory map $ map initial)

justEvery :: Int -> [a] -> [a]
justEvery n (x:xs) = x : (justEvery n $ drop (n-1) xs)
justEvery _ []     = []

subTrace :: Int -> Int -> [a] -> [a]
subTrace n m = take (n + 1) . justEvery m

ensemble :: Int -> [Point]
ensemble n = let qs = randoms (mkStdGen 42)
                 ps = randoms (mkStdGen 21)
             in take n $ zip qs ps 

ensembleTrace :: NFData a => (Point -> [Point]) -> (Point -> a) -> 
                              Int -> Int -> [Point] -> [[a]]
ensembleTrace orbitGen observable n m = 
    parMap rdeepseq ((map observable . subTrace n m) . orbitGen)

main = let k = 100
           l = 100
           m = 100
           orbitGen = trajectory (standardMap 7)
           observable (p, q) = p^2 - q^2
           initials = ensemble k
           mean xs = (sum xs) / (fromIntegral $ length xs)
           result =   (map mean) 
                    $ transpose 
                    $ ensembleTrace orbitGen observable l m 
                    $ initials
       in mapM_ print result

I am compiling with

$ ghc -O2 stdmap.hs -threaded

and runing with

$ ./stdmap +RTS -N4 > /dev/null

on the intel Q6600, Linux 3.6.3-1-ARCH, with GHC 7.6.1 and get the following results
for the different sets of the parameters K, L, M (k, l, m in the code of the program)

(K=200,L=200,N=200)   -> real    0m0.774s
                         user    0m2.856s
                         sys     0m0.147s

(K=2000,L=200,M=200)  -> real    0m7.409s
                         user    0m28.102s
                         sys     0m1.080s

(K=200,L=2000,M=200)  -> real    0m7.326s
                         user    0m27.932s
                         sys     0m1.020s

(K=200,L=200,M=2000)  -> real    0m10.581s
                         user    0m38.564s
                         sys     0m3.376s

(K=20000,L=200,M=200) -> real    4m22.156s
                         user    7m30.007s
                         sys     0m40.321s

(K=200,L=20000,M=200) -> real    1m16.222s
                         user    4m45.891s
                         sys     0m15.812s

(K=200,L=200,M=20000) -> real    8m15.060s
                         user    23m10.909s
                         sys     9m24.450s

I don’t quite understand where the problem of such a pure scaling might be. If I understand correctly the lists are lazy and should not be constructed, since they are consumed in the head-tail direction? As could be observed from the measurements there is a correlation between the excessive real-time consumption and the excessive system-time consumption as the excess would be on the system account. But if there is some memory management wasting time, this should still scale linearly in K, L, M.

Help!

EDIT

I made changes in the code according to the suggestions given by Daniel Fisher, which indeed solved the bad scaling with respect to M. As pointed out, by forcing the strict evaluation in the trajectory, we avoid the construction of large thunks. I understand the performance improvement behind that, but I still don’t understand the bad scaling of the original code, because (if I understand correctly) the space-time-complexity of the construction of the thunk should be linear in M?

Additionally, I still have problems understanding the bad scaling with respect to K (the size of the ensemble). I performed two additional measurements with the improved code for K=8000 and K=16000, keeping L=200, M=200. Scaling up to K=8000 is as expected but for K=16000 it is already abnormal. The problem seems to be in the number of overflowed SPARKS, which is 0 for K=8000 and 7802 for K=16000. This probably reflects in the bad concurrency which I quantify as a quotient Q = (MUT cpu time) / (MUT real time) which would be ideally equal to the number of CPU-s. However, Q ~ 4 for K = 8000 and Q ~ 2 for K = 16000.
Please help me understand the origin of this problem and the possible solutions.

K = 8000:

$ ghc -O2 stmap.hs -threaded -XBangPatterns
$ ./stmap +RTS -s -N4 > /dev/null

56,905,405,184 bytes allocated in the heap
 503,501,680 bytes copied during GC
  53,781,168 bytes maximum residency (15 sample(s))
   6,289,112 bytes maximum slop
         151 MB total memory in use (0 MB lost due to fragmentation)

                                Tot time (elapsed)  Avg pause  Max pause
Gen  0     27893 colls, 27893 par    7.85s    1.99s     0.0001s    0.0089s
Gen  1        15 colls,    14 par    1.20s    0.30s     0.0202s    0.0558s

Parallel GC work balance: 23.49% (serial 0%, perfect 100%)

TASKS: 6 (1 bound, 5 peak workers (5 total), using -N4)

SPARKS: 8000 (8000 converted, 0 overflowed, 0 dud, 0 GC'd, 0 fizzled)

INIT    time    0.00s  (  0.00s elapsed)
MUT     time   95.90s  ( 24.28s elapsed)
GC      time    9.04s  (  2.29s elapsed)
EXIT    time    0.00s  (  0.00s elapsed)
Total   time  104.95s  ( 26.58s elapsed)

Alloc rate    593,366,811 bytes per MUT second

Productivity  91.4% of total user, 360.9% of total elapsed

gc_alloc_block_sync: 315819

and

K = 16000:

$ ghc -O2 stmap.hs -threaded -XBangPatterns
$ ./stmap +RTS -s -N4 > /dev/null

113,809,786,848 bytes allocated in the heap
 1,156,991,152 bytes copied during GC
  114,778,896 bytes maximum residency (18 sample(s))
    11,124,592 bytes maximum slop
      300 MB total memory in use (0 MB lost due to fragmentation)

                                Tot time (elapsed)  Avg pause  Max pause
Gen  0     135521 colls, 135521 par   22.83s    6.59s     0.0000s    0.0190s
Gen  1        18 colls,    17 par    2.72s    0.73s     0.0405s    0.1692s

Parallel GC work balance: 18.05% (serial 0%, perfect 100%)

TASKS: 6 (1 bound, 5 peak workers (5 total), using -N4)

SPARKS: 16000 (8198 converted, 7802 overflowed, 0 dud, 0 GC'd, 0 fizzled)

INIT    time    0.00s  (  0.00s elapsed)
MUT     time  221.77s  (139.78s elapsed)
GC      time   25.56s  (  7.32s elapsed)
EXIT    time    0.00s  (  0.00s elapsed)
Total   time  247.34s  (147.10s elapsed)

Alloc rate    513,176,874 bytes per MUT second

Productivity  89.7% of total user, 150.8% of total elapsed

gc_alloc_block_sync: 814824

Answer 1

M. A. D.‘s point about fmod is a good one, but it is not necessary to call out to C, and we can do better staying in Haskell land (the ticket the linked thread was about is meanwhile fixed). The trouble in

fmod :: Double -> Double -> Double
fmod a b | a < 0     = b - fmod (abs a) b 
         | otherwise = if a < b then a 
                       else let q = a / b in b * (q - fromIntegral (floor q))

is that type defaulting leads to floor :: Double -> Integer (and consequently fromIntegral :: Integer -> Double) being called. Now, Integer is a comparatively complicated type, with slow operations, and the conversion from Integer to Double is also relatively complicated. The original code (with parameters k = l = 200 and m = 5000) produced the stats

./nstdmap +RTS -s -N2 > /dev/null
  60,601,075,392 bytes allocated in the heap
  36,832,004,184 bytes copied during GC
       2,435,272 bytes maximum residency (13741 sample(s))
         887,768 bytes maximum slop
               9 MB total memory in use (0 MB lost due to fragmentation)

                                    Tot time (elapsed)  Avg pause  Max pause
  Gen  0     46734 colls, 46734 par   41.66s   20.87s     0.0004s    0.0058s
  Gen  1     13741 colls, 13740 par   23.18s   11.62s     0.0008s    0.0041s

  Parallel GC work balance: 60.58% (serial 0%, perfect 100%)

  TASKS: 4 (1 bound, 3 peak workers (3 total), using -N2)

  SPARKS: 200 (200 converted, 0 overflowed, 0 dud, 0 GC'd, 0 fizzled)

  INIT    time    0.00s  (  0.00s elapsed)
  MUT     time   34.99s  ( 17.60s elapsed)
  GC      time   64.85s  ( 32.49s elapsed)
  EXIT    time    0.00s  (  0.00s elapsed)
  Total   time   99.84s  ( 50.08s elapsed)

  Alloc rate    1,732,048,869 bytes per MUT second

  Productivity  35.0% of total user, 69.9% of total elapsed

on my machine (-N2 because I have only two physical cores). Simply changing the code to use a type signature floor q :: Int brings that down to

./nstdmap +RTS -s -N2 > /dev/null
  52,105,495,488 bytes allocated in the heap
  29,957,007,208 bytes copied during GC
       2,440,568 bytes maximum residency (10481 sample(s))
         893,224 bytes maximum slop
               8 MB total memory in use (0 MB lost due to fragmentation)

                                    Tot time (elapsed)  Avg pause  Max pause
  Gen  0     36979 colls, 36979 par   32.96s   16.51s     0.0004s    0.0066s
  Gen  1     10481 colls, 10480 par   16.65s    8.34s     0.0008s    0.0018s

  Parallel GC work balance: 68.64% (serial 0%, perfect 100%)

  TASKS: 4 (1 bound, 3 peak workers (3 total), using -N2)

  SPARKS: 200 (200 converted, 0 overflowed, 0 dud, 0 GC'd, 0 fizzled)

  INIT    time    0.01s  (  0.01s elapsed)
  MUT     time   29.78s  ( 14.94s elapsed)
  GC      time   49.61s  ( 24.85s elapsed)
  EXIT    time    0.00s  (  0.00s elapsed)
  Total   time   79.40s  ( 39.80s elapsed)

  Alloc rate    1,749,864,775 bytes per MUT second

  Productivity  37.5% of total user, 74.8% of total elapsed

a reduction of about 20% in elapsed time, 13% in MUT time. Not bad. If we look at the code for floor that you get with optimisations, we can see why:

floorDoubleInt :: Double -> Int
floorDoubleInt (D# x) =
    case double2Int# x of
      n | x <## int2Double# n   -> I# (n -# 1#)
        | otherwise             -> I# n

floorDoubleInteger :: Double -> Integer
floorDoubleInteger (D# x) =
    case decodeDoubleInteger x of
      (# m, e #)
        | e <# 0#   ->
          case negateInt# e of
            s | s ># 52#    -> if m < 0 then (-1) else 0
              | otherwise   ->
                case TO64 m of
                  n -> FROM64 (n `uncheckedIShiftRA64#` s)
        | otherwise -> shiftLInteger m e

floor :: Double -> Int just uses the machine conversion, while floor :: Double -> Integer needs an expensive decodeDoubleInteger and more branches. But where floor is called here, we know that all involved Doubles are nonnegative, hence floor is the same as truncate, which maps directly to the machine conversion double2Int#, so let’s try that instead of floor:

  INIT    time    0.00s  (  0.00s elapsed)
  MUT     time   29.29s  ( 14.70s elapsed)
  GC      time   49.17s  ( 24.62s elapsed)
  EXIT    time    0.00s  (  0.00s elapsed)
  Total   time   78.45s  ( 39.32s elapsed)

a really small reduction (to be expected, the fmod isn’t really the bottleneck). For comparison, calling out to C:

  INIT    time    0.01s  (  0.01s elapsed)
  MUT     time   31.46s  ( 15.78s elapsed)
  GC      time   54.05s  ( 27.06s elapsed)
  EXIT    time    0.00s  (  0.00s elapsed)
  Total   time   85.52s  ( 42.85s elapsed)

is a bit slower (unsurprisingly, you can execute a number of primops in the time calling out to C takes).

But that’s not where the big fish swim. The bad thing is that picking only every m-th element of the trajectories leads to large thunks that cause a lot of allocation and take long to evaluate when the time comes. So let’s eliminate that leak and make the trajectories strict:

{-# LANGUAGE BangPatterns #-}

trajectory :: (Point -> Point) -> Point -> [Point] 
trajectory map !initial@(!a,!b) = initial : (trajectory map $ map initial)

That reduces the allocations and GC time drastically, and as a consequence also the MUT time:

  INIT    time    0.00s  (  0.00s elapsed)
  MUT     time   21.83s  ( 10.95s elapsed)
  GC      time    0.72s  (  0.36s elapsed)
  EXIT    time    0.00s  (  0.00s elapsed)
  Total   time   22.55s  ( 11.31s elapsed)

with the original fmod,

  INIT    time    0.00s  (  0.00s elapsed)
  MUT     time   18.26s  (  9.18s elapsed)
  GC      time    0.58s  (  0.29s elapsed)
  EXIT    time    0.00s  (  0.00s elapsed)
  Total   time   18.84s  (  9.47s elapsed)

with floor q :: Int, and within measuring accuracy the same times for truncate q :: Int (the allocation figures are a bit lower for truncate).

---

The problem seems to be in the number of overflowed SPARKS, which is 0 for K=8000 and 7802 for K=16000. This probably reflects in the bad concurrency

Yes (though as far as I know the more correct term here would be parallelism instead of concurrency), there is a spark pool, and when that’s full, any further sparks are not scheduled for being evaluated in whatever thread next has time when its turn comes, the computation is then done without parallelism, from the parent thread. In this case that means after an initial parallel phase, the computation falls back to sequential.

The size of the spark pool is apparently about 8K (2^13).

If you watch the CPU load via top, you will see that it drops from (close to 100%)*(number of cores) to a much lower value after a while (for me, it was ~100% with -N2 and ~130% with -N4).

The cure is to avoid sparking too much, and letting each spark do some more work. With the quick-and-dirty modification

ensembleTrace orbitGen observable n m =
    withStrategy (parListChunk 25 rdeepseq) . map ((map observable . subTrace n m) . orbitGen)

I’m back to 200% with -N2 for practically the entire run and a good productivity,

  INIT    time    0.00s  (  0.00s elapsed)
  MUT     time   57.42s  ( 29.02s elapsed)
  GC      time    5.34s  (  2.69s elapsed)
  EXIT    time    0.00s  (  0.00s elapsed)
  Total   time   62.76s  ( 31.71s elapsed)

  Alloc rate    1,982,155,167 bytes per MUT second

  Productivity  91.5% of total user, 181.1% of total elapsed

and with -N4 it’s also fine (even a wee bit faster on the wall-clock – not much because all threads do basically the same, and I have only 2 physical cores),

  INIT    time    0.00s  (  0.00s elapsed)
  MUT     time   99.17s  ( 26.31s elapsed)
  GC      time   16.18s  (  4.80s elapsed)
  EXIT    time    0.00s  (  0.00s elapsed)
  Total   time  115.36s  ( 31.12s elapsed)

  Alloc rate    1,147,619,609 bytes per MUT second

  Productivity  86.0% of total user, 318.7% of total elapsed

since now the spark pool doesn’t overflow.

The proper fix is to make the size of the chunks a parameter that is computed from the number of trajectories and available cores so that the number of sparks doesn’t exceed the pool size.