How to optimize this Haskell code summing up the primes in sublinear time?

Question

Problem 10 from Project Euler is to find the sum of all the primes below given n.

I solved it simply by summing up the primes generated by the sieve of Eratosth

Answer 1

First as a baseline, the timings of the existing approaches on my machine:

1. Original program posted in the question:

   time stack exec primorig
   95673602693282040
   
   real    0m4.601s
   user    0m4.387s
   sys     0m0.251s
   

2. Second the version using Data.IntMap.Strict from here

   time stack exec primIntMapStrict
   95673602693282040
   
   real    0m2.775s
   user    0m2.753s
   sys     0m0.052s
   

3. Shershs code with Data.Judy dropped in here

   time stack exec prim-hash2
   95673602693282040
   
   real    0m0.945s
   user    0m0.955s
   sys     0m0.028s
   

4. Your python solution.

   I compiled it with

   python -O -m py_compile problem10.py
   

   and the timing:

   time python __pycache__/problem10.cpython-36.opt-1.pyc
   95673602693282040
   
   real    0m1.163s
   user    0m1.160s
   sys     0m0.003s
   

5. Your C++ version:

   $ g++ -O2 --std=c++11 p10.cpp -o p10
   $ time ./p10
   sum(2000000000) = 95673602693282040
   
   real    0m0.314s
   user    0m0.310s
   sys     0m0.003s
   

I didn't bother to provide a baseline for slow.hs, as I didn't want to wait for it to complete when run with an argument of 2*10^9.

Subsecond performance

The following program runs in under a second on my machine.

It uses a hand rolled hashmap, which uses closed hashing with linear probing and uses some variant of knuths hashfunction, see here.

Certainly it is somewhat tailored to the case, as the lookup function for example expects the searched keys to be present.

Timings:

time stack exec prim
95673602693282040

real    0m0.725s
user    0m0.714s
sys     0m0.047s

First I implemented my hand rolled hashmap simply to hash the keys with

key `mod` size

and selected a size multiple times higher than the expected input, but the program took 22s or more to complete.

Finally it was a matter of choosing a hash function which was good for the workload.

Here is the program:

import Data.Maybe
import Control.Monad
import Data.Array.IO
import Data.Array.Base (unsafeRead)

type Number = Int

data Map = Map { keys :: IOUArray Int Number
               , values :: IOUArray Int Number
               , size :: !Int 
               , factor :: !Int
               }

newMap :: Int -> Int -> IO Map
newMap s f = do
  k <- newArray (0, s-1) 0
  v <- newArray (0, s-1) 0
  return $ Map k v s f 

storeKey :: IOUArray Int Number -> Int -> Int -> Number -> IO Int
storeKey arr s f key = go ((key * f) `mod` s)
  where
    go :: Int -> IO Int
    go ind = do
      v <- readArray arr ind
      go2 v ind
    go2 v ind
      | v == 0    = do { writeArray arr ind key; return ind; }
      | v == key  = return ind
      | otherwise = go ((ind + 1) `mod` s)

loadKey :: IOUArray Int Number -> Int -> Int -> Number -> IO Int
loadKey arr s f key = s `seq` key `seq` go ((key *f) `mod` s)
  where
    go :: Int -> IO Int
    go ix = do
      v <- unsafeRead arr ix
      if v == key then return ix else go ((ix + 1) `mod` s)

insertIntoMap :: Map -> (Number, Number) -> IO Map
insertIntoMap m@(Map ks vs s f) (k, v) = do
  ix <- storeKey ks s f k
  writeArray vs ix v
  return m

fromList :: Int -> Int -> [(Number, Number)] -> IO Map
fromList s f xs = do
  m <- newMap s f
  foldM insertIntoMap m xs

(!) :: Map -> Number -> IO Number
(!) (Map ks vs s f) k = do
  ix <- loadKey ks s f k
  readArray vs ix

mupdate :: Map -> Number -> (Number -> Number) -> IO ()
mupdate (Map ks vs s fac) i f = do
  ix <- loadKey ks s fac i
  old <- readArray vs ix
  let x' = f old
  x' `seq` writeArray vs ix x'

r' :: Number -> Number
r'  = floor . sqrt . fromIntegral

vs' :: Integral a => a -> a -> [a]
vs' n r = [n `div` i | i <- [1..r]] ++ reverse [1..n `div` r - 1]  

vss' n r = r + n `div` r -1

list' :: Int -> Int -> [Number] -> IO Map
list' s f vs = fromList s f [(i, i * (i + 1) `div` 2 - 1) | i <- vs]

problem10 :: Number -> IO Number
problem10 n = do
      m <- list' (19*vss) (19*vss+7) vs
      nm <- sieve m 2 r vs
      nm ! n
    where vs = vs' n r
          vss = vss' n r
          r  = r' n

sieve :: Map -> Number -> Number -> [Number] -> IO Map
sieve m p r vs | p > r     = return m
               | otherwise = do
                   v1 <- m ! p
                   v2 <- m ! (p - 1)
                   nm <- if v1 > v2 then update m vs p else return m
                   sieve nm (p + 1) r vs

update :: Map -> [Number] -> Number -> IO Map
update m vs p = foldM (decrease p) m $ takeWhile (>= p*p) vs

decrease :: Number -> Map -> Number -> IO Map
decrease p m k = do
  v <- sumOfSieved m k p
  mupdate m k (subtract v)
  return m

sumOfSieved :: Map -> Number -> Number -> IO Number
sumOfSieved m v p = do
  v1 <- m ! (v `div` p)
  v2 <- m ! (p - 1)
  return $ p * (v1 - v2)

main = do { n <- problem10 (2*10^9) ; print n; } -- 2*10^9

I am not a professional with hashing and that sort of stuff, so this can certainly be improved a lot. Maybe we Haskellers should improve the of the shelf hash maps or provide some simpler ones.

My hashmap, Shershs code

If I plug my hashmap in Shershs (see answer below) code, see here we are even down to

time stack exec prim-hash2
95673602693282040

real    0m0.601s
user    0m0.604s
sys     0m0.034s

Why is slow.hs slow?

If you read through the source for the function insert in Data.HashTable.ST.Basic, you will see that it deletes the old key value pair and inserts a new one. It doesn't look up the "place" for the value and mutate it, as one might imagine, if one reads that it is a "mutable" hashtable. Here the hashtable itself is mutable, so you don't need to copy the whole hashtable for insertion of a new key value pair, but the value places for the pairs are not. I don't know if that is the whole story of slow.hs being slow, but my guess is, it is a pretty big part of it.

A few minor improvements

So that's the idea I followed while trying to improve your program the first time.

See, you don't need a mutable mapping from keys to values. Your key set is fixed. You want a mapping from keys to mutable places. (Which is, by the way, what you get from C++ by default.)

And so I tried to come up with that. I used IntMap IORef from Data.IntMap.Strict and Data.IORef first and got a timing of

tack exec prim
95673602693282040

real    0m2.134s
user    0m2.141s
sys     0m0.028s

I thought maybe it would help to work with unboxed values and to get that, I used IOUArray Int Int with 1 element each instead of IORef and got those timings:

time stack exec prim
95673602693282040

real    0m2.015s
user    0m2.018s
sys     0m0.038s

Not much of a difference and so I tried to get rid of bounds checking in the 1 element arrays by using unsafeRead and unsafeWrite and got a timing of

time stack exec prim
95673602693282040

real    0m1.845s
user    0m1.850s
sys     0m0.030s

which was the best I got using Data.IntMap.Strict.

Of course I ran each program multiple times to see if the times are stable and the differences in run time aren't just noise.

It looks like these are all just micro-optimizations.

And here is the program that ran fastest for me without using a hand rolled data structure:

import qualified Data.IntMap.Strict as M
import Control.Monad
import Data.Array.IO
import Data.Array.Base (unsafeRead, unsafeWrite)

type Number = Int
type Place = IOUArray Number Number
type Map = M.IntMap Place

tupleToRef :: (Number, Number) -> IO (Number, Place)
tupleToRef = traverse (newArray (0,0))

insertRefs :: [(Number, Number)] -> IO [(Number, Place)]
insertRefs = traverse tupleToRef

fromList :: [(Number, Number)] -> IO Map 
fromList xs = M.fromList <$> insertRefs xs

(!) :: Map -> Number -> IO Number
(!) m i = unsafeRead (m M.! i) 0

mupdate :: Map -> Number -> (Number -> Number) -> IO ()
mupdate m i f = do
  let place = m M.! i
  old <- unsafeRead place 0
  let x' = f old
  -- make the application of f strict
  x' `seq` unsafeWrite place 0 x'

r' :: Number -> Number
r'  = floor . sqrt . fromIntegral

vs' :: Integral a => a -> a -> [a]
vs' n r = [n `div` i | i <- [1..r]] ++ reverse [1..n `div` r - 1]  

list' :: [Number] -> IO Map
list' vs = fromList [(i, i * (i + 1) `div` 2 - 1) | i <- vs]

problem10 :: Number -> IO Number
problem10 n = do
      m <- list' vs
      nm <- sieve m 2 r vs
      nm ! n
    where vs = vs' n r
          r  = r' n

sieve :: Map -> Number -> Number -> [Number] -> IO Map
sieve m p r vs | p > r     = return m
               | otherwise = do
                   v1 <- m ! p
                   v2 <- m ! (p - 1)
                   nm <- if v1 > v2 then update m vs p else return m
                   sieve nm (p + 1) r vs

update :: Map -> [Number] -> Number -> IO Map
update m vs p = foldM (decrease p) m $ takeWhile (>= p*p) vs

decrease :: Number -> Map -> Number -> IO Map
decrease p m k = do
  v <- sumOfSieved m k p
  mupdate m k (subtract v)
  return m

sumOfSieved :: Map -> Number -> Number -> IO Number
sumOfSieved m v p = do
  v1 <- m ! (v `div` p)
  v2 <- m ! (p - 1)
  return $ p * (v1 - v2)

main = do { n <- problem10 (2*10^9) ; print n; } -- 2*10^9

If you profile that, you see that it spends most of the time in the custom lookup function (!), don't know how to improve that further. Trying to inline (!) with {-# INLINE (!) #-} didn't yield better results; maybe ghc already did this.