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Today's Topics:
1. Re: parallelizing a function for generating prime numbers
(Mateusz Kowalczyk)
2. Re: parallelizing a function for generating prime numbers
(mukesh tiwari)
----------------------------------------------------------------------
Message: 1
Date: Sat, 17 May 2014 14:09:59 +0200
From: Mateusz Kowalczyk <[email protected]>
To: [email protected]
Subject: Re: [Haskell-beginners] parallelizing a function for
generating prime numbers
Message-ID: <[email protected]>
Content-Type: text/plain; charset=windows-1252
On 05/16/2014 10:13 PM, Norbert Melzer wrote:
> I had some fears, that there will be answers like this ;)
>
> The problem with improving the generation itself is, that I don't
> understand the faster implementations that I found (namely the
> implementation of `Data.Numbers.Primes` in the `primes`-package and some
> other Wheel-Sieves).
>
> And for the Project-Euler-Problems I only use code that I have created
> myself or at least I have a small idea how it works if it is from a package?
>
Euler is a mathematics (mostly number theory) exercise, not a
programming one, so it's understandable that in some cases the problems
and some of their solutions are not suited for parallelism. It's not of
much benefit to try and parallelise an inherently linear solution.
> 2014-05-16 18:33 GMT+02:00 Benjamin Edwards <[email protected]>:
>
>> Given the linear dependencies in prime number generation, shy of using a
>> probabilistic sieving method, I'm not sure that it's possible to hope for
>> any kind of parallel number generation. All you are going to do is for
>> yourself to eat the cost of synchronisation for no gain.
>>
>> Ben
>>
>> On Fri May 16 2014 at 16:53:38, Norbert Melzer <[email protected]>
>> wrote:
>>
>>> Hi there!
>>>
>>> I am trying to enhence the speed of my Project Euler solutions?
>>>
>>> My original function is this:
>>>
>>> ```haskell
>>> problem10' :: Integer
>>> problem10' = sum $ takeWhile (<=2000000) primes
>>> where
>>> primes = filter isPrime possiblePrimes
>>> isPrime n = n == head (primeFactors n)
>>> possiblePrimes = (2:3:concat [ [6*pp-1, 6*pp+1] | pp <-
>>> [1..] ])
>>> primeFactors m = pf 2 m
>>> pf n m | n*n > m = [m]
>>> | n*n == m = [n,n]
>>> | m `mod` n == 0 = n:pf n (m `div` n)
>>> | otherwise = pf (n+1) m
>>> ```
>>>
>>> Even if the generation of primes is relatively slow and could be much
>>> better, I want to focus on parallelization, so I tried the following:
>>>
>>> ```haskell
>>> parFilter :: (a -> Bool) -> [a] -> [a]
>>> parFilter _ [] = []
>>> parFilter f (x:xs) =
>>> let px = f x
>>> pxs = parFilter f xs
>>> in par px $ par pxs $ case px of True -> x : pxs
>>> False -> pxs
>>>
>>> problem10' :: Integer
>>> problem10' = sum $ takeWhile (<=2000000) primes
>>> where
>>> primes = parFilter isPrime possiblePrimes
>>> isPrime n = n == head (primeFactors n)
>>> possiblePrimes = (2:3:concat [ [6*pp-1, 6*pp+1] | pp <-
>>> [1..] ])
>>> primeFactors m = pf 2 m
>>> pf n m | n*n > m = [m]
>>> | n*n == m = [n,n]
>>> | m `mod` n == 0 = n:pf n (m `div` n)
>>> | otherwise = pf (n+1) m
>>> ```
>>>
>>> This approach was about half as slow as the first solution (~15 seconds
>>> old, ~30 the new one!).
>>>
>>> Trying to use `Control.Parallel.Strategies.evalList` for `possiblePrimes`
>>> resulted in a huge waste of memory, since it forced to generate an endless
>>> list, and does not stop?
>>>
>>> Trying the same for `primeFactors` did not gain any speed, but was not
>>> much slower at least, but I did not expect much, since I look at its head
>>> only?
>>>
>>> Only thing I could imagine to parallelize any further would be the
>>> takeWhile, but then I don't get how I should do it?
>>>
>>> Any ideas how to do it?
>>>
>>> TIA
>>> Norbert
>>>
>>> _______________________________________________
>>> Beginners mailing list
>>> [email protected]
>>> http://www.haskell.org/mailman/listinfo/beginners
>>>
>>
>> _______________________________________________
>> Beginners mailing list
>> [email protected]
>> http://www.haskell.org/mailman/listinfo/beginners
>>
>>
>
>
>
> _______________________________________________
> Beginners mailing list
> [email protected]
> http://www.haskell.org/mailman/listinfo/beginners
>
--
Mateusz K.
------------------------------
Message: 2
Date: Sun, 18 May 2014 12:21:52 +0530
From: mukesh tiwari <[email protected]>
To: The Haskell-Beginners Mailing List - Discussion of primarily
beginner-level topics related to Haskell <[email protected]>
Subject: Re: [Haskell-beginners] parallelizing a function for
generating prime numbers
Message-ID:
<CAFHZvE_q9pGWGhEg8tan4VzxN3zKHBkGpB4i=7mb4hpj12n...@mail.gmail.com>
Content-Type: text/plain; charset="utf-8"
You can use Rabin-Miller[1] primality testing. The idea is, divide the
range 1 to 2000000 in chunk of 10000 numbers and evaluate the all the
chunks in parallel.
import Data.Bits
import Control.Parallel.Strategies
powM :: Integer -> Integer -> Integer -> Integer
powM a d n
| d == 0 = 1
| d == 1 = mod a n
| otherwise = mod q n where
p = powM ( mod ( a^2 ) n ) ( shiftR d 1 ) n
q = if (.&.) d 1 == 1 then mod ( a * p ) n else p
calSd :: Integer -> ( Integer , Integer )
calSd n = ( s , d ) where
s = until ( \x -> testBit ( n - 1) ( fromIntegral x ) ) ( +1 ) 0
d = div ( n - 1 ) ( shiftL 1 ( fromIntegral s ) )
rabinMiller::Integer->Integer->Integer->Integer-> Bool
rabinMiller n s d a
| n == a = True
| otherwise = case powM a d n of
1 -> True
x -> any ( == pred n ) . take ( fromIntegral s )
. iterate (\e -> mod ( e^2 ) n ) $ x
isPrime::Integer-> Bool
isPrime n
| n <= 1 = False
| n == 2 = True
| even n = False
| otherwise = all ( == True ) . map ( rabinMiller n s d ) $ [ 2 , 3 ,
5 , 7 , 11 , 13 , 17 ] where
( s , d ) = calSd n
primeRange :: Integer -> Integer -> [ Bool ]
primeRange m n = ( map isPrime [ m .. n ] ) `using` parListChunk 10000
rdeepseq
sum' :: Integer -> Integer -> Integer
sum' m n = sum . map ( \( x, y ) -> if y then x else 0 ) . zip [ m .. n ]
. primeRange m $ n
main = print ( sum' 1 2000000 )
Mukeshs-MacBook-Pro:Puzzles mukeshtiwari$ time ./Proj10 +RTS -N2
142913828922
real 0m6.301s
user 0m11.937s
sys 0m0.609s
Mukeshs-MacBook-Pro:Puzzles mukeshtiwari$ time ./Proj10 +RTS -N1
142913828922
real 0m8.202s
user 0m8.026s
sys 0m0.174s
-Mukesh
[1] http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test
On Fri, May 16, 2014 at 9:23 PM, Norbert Melzer <[email protected]> wrote:
> Hi there!
>
> I am trying to enhence the speed of my Project Euler solutions?
>
> My original function is this:
>
> ```haskell
> problem10' :: Integer
> problem10' = sum $ takeWhile (<=2000000) primes
> where
> primes = filter isPrime possiblePrimes
> isPrime n = n == head (primeFactors n)
> possiblePrimes = (2:3:concat [ [6*pp-1, 6*pp+1] | pp <- [1..]
> ])
> primeFactors m = pf 2 m
> pf n m | n*n > m = [m]
> | n*n == m = [n,n]
> | m `mod` n == 0 = n:pf n (m `div` n)
> | otherwise = pf (n+1) m
> ```
>
> Even if the generation of primes is relatively slow and could be much
> better, I want to focus on parallelization, so I tried the following:
>
> ```haskell
> parFilter :: (a -> Bool) -> [a] -> [a]
> parFilter _ [] = []
> parFilter f (x:xs) =
> let px = f x
> pxs = parFilter f xs
> in par px $ par pxs $ case px of True -> x : pxs
> False -> pxs
>
> problem10' :: Integer
> problem10' = sum $ takeWhile (<=2000000) primes
> where
> primes = parFilter isPrime possiblePrimes
> isPrime n = n == head (primeFactors n)
> possiblePrimes = (2:3:concat [ [6*pp-1, 6*pp+1] | pp <- [1..]
> ])
> primeFactors m = pf 2 m
> pf n m | n*n > m = [m]
> | n*n == m = [n,n]
> | m `mod` n == 0 = n:pf n (m `div` n)
> | otherwise = pf (n+1) m
> ```
>
> This approach was about half as slow as the first solution (~15 seconds
> old, ~30 the new one!).
>
> Trying to use `Control.Parallel.Strategies.evalList` for `possiblePrimes`
> resulted in a huge waste of memory, since it forced to generate an endless
> list, and does not stop?
>
> Trying the same for `primeFactors` did not gain any speed, but was not
> much slower at least, but I did not expect much, since I look at its head
> only?
>
> Only thing I could imagine to parallelize any further would be the
> takeWhile, but then I don't get how I should do it?
>
> Any ideas how to do it?
>
> TIA
> Norbert
>
>
> _______________________________________________
> Beginners mailing list
> [email protected]
> http://www.haskell.org/mailman/listinfo/beginners
>
>
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