 Functional Thursday #33
 2015.12.03
 Yen3 (yen3rc@gmail.com)
About the slide

這份投影片是 Parallel and Concurrent Programming in Haskell  Chatper 1 ~ 5 的筆記

主題包含
 Eval monad
 Par monad
 Repa

今天的內容皆與 datalevel parallelism 相關
 在 Haskell 中，一個較容易平行化 function 是
map
，這份投影片會很常 討論到它。
Definition of Parallel
 A parallel program is one that uses a multiplicity of computational hardware (e.g., several processor cores) to perform a computation more quickly.
 Parallel programming in Haskell is deterministic: The parallel program always produces the same answer, regardless of how many processors are used to run it.
The status of value in ghc (1/)
 There are three conditions of a value.
 Unevaluated
 WeakHead Normal Form (WHNF)  evaluated with first constructor
 Normal Form (NF)  fully evaluated
The status of value in ghc (2/)

sprint
 prints a value without forcing its evaluation 
seq
: only far as the first constructor and doesn't evaluate any more of the structure. It evaluates first argument to WHNF.
seq :: a > b > b
The status of value in ghc (3/)
 Example
Prelude> let x = 1 + 2 :: Int Prelude> let y = x + 1 Prelude> :sprint x x = _ Prelude> :sprint y y = _ Prelude> seq y () () Prelude> :sprint x x = 3 Prelude> :sprint y y = 4
The status of value in ghc (4/)
Prelude> let xs = map (+1) [1..10] :: [Int] Prelude> :sprint xs xs = _ Prelude> seq xs () () Prelude> :sprint xs xs = _ : _ Prelude> length xs 10 Prelude> :sprint xs xs = [_,_,_,_,_,_,_,_,_,_] Prelude> sum xs 65 Prelude> :sprint xs xs = [2,3,4,5,6,7,8,9,10,11]
force
function

force
 fully evaluated it's argument and returns it. (WHNF into NF)
import Control.DeepSeq class NFData a where rnf :: a > ()  reduce to normalform rnf a = a `seq` () deepseq :: NFData a => a > b > b deepseq a b = rnf a `seq` b force :: NFData a => a > a force x = x `deepseq` x

seq
: only far as the first constructor and doesn't evaluate any more of the structure. It evaluates first argument to WHNF.
Eval monad
 Decoupling of the algorithm from the parallelism
 The type declaration for eval monad
data Eval a instance Monad Eval runEval :: Eval a > a rpar :: a > Eval a  rpar :: Strategy a rseq :: a > Eval a  rseq :: Strategy a

rpar
 evaluate its argument in parallel. 
rseq
 Evaluate the argument and wait for the result. evaluates its argument to WHNF.
Eval monad  simple example
 Example
runEval $ do a < rpar (f x) b < rseq (f y) rseq a return (a, b)
Eval monad  Strategy
 Strategy  modularize parllel code by separating the algorithm from the
parallelism
 use
using
function to add parallelism with the existing codes 
withStrategy
 a another version ofusing
with the arguments flipped
 use
type Strategy a = a > Eval a using :: a > Strategy a > a x `using` s = runEval (s x) withStrategy :: Strategy a > a > a withStrategy s x = runEval (s x)
Eval monad  Strategy
 The
rpar
,rseq
are also Strategies.
rpar :: Strategy a rseq :: Strategy a
 You could write the algorithm first and add the parallelism code later ideally.
Eval monad  example for pair
 Example
import Control.Parallel.Strategies import Control.DeepSeq evalPair :: Strategy a > Strategy b > Strategy (a,b) evalPair sa sb (a,b) = do a' < sa a b' < sb b return (a',b')
Eval monad  example for pair
rparWith :: Strategy a > Strategy a rparWith s a = do ra < rpar a sa < s ra return sa (+) 1 2  (11) ((+) 1 2, (+) 3 4)  (12) (+) 1 2 `using` rpar  (21) <!((+) 1 2, (+) 3 4) `using` evalPair (rparWith rseq) (rparWith rseq)  (22)> ((+) 1 2, (+) 3 4) `using` evalPair (rparWith rseq) (rparWith rseq)  (22)
 (11), (12)  sequential version
 (21), (22)  parallel version and reduce the value to WHNF
Eval monad  some help functions (1/)
 About some helper function

rdeepseq
 evaluates the argument to NF

rdeepseq :: NFData a => Strategy a rdeepseq x = rseq (force x)
 `rparWith`  wraps the Strategy s in an `rpar`
rparWith :: Strategy a > Strategy a
Eval monad  some help functions (2/)
 The code reduced to NF in previous slide could also be written as
 NF (+) 1 2 `using` rparWith rdeepseq ((+) 1 2, (+) 3 4) `using` evalPair (rparWith rdeepseq) (rparWith rdeepseq)
Eval monad  parallelize map
parMap :: (a > b) > [a] > [b] parMap f xs = map f x `using` parList rseq evalList :: Strategy a > Strategy [a] evalList start [] = return [] evalList start (x:xs) = do x' < start x xs' < evalList start xs return (x': xs') parList :: Strategy a > Strategy [a] parList start = evalList (rparWith start)

parMap
will calculate its list to WHNF 
parList
 evaluate the list element in parallel
Eval monad
 Example
import Control.Parallel.Strategies import Control.DeepSeq map (+1) [1..100]  (1) map (+1) [1..100] `using` parList rseq  (2) map (+1) [1..100] `using` parList rdeepseq  (3)
 (1) sequential version
 (2) parallelize version reduce value to WHNF
 (3) parallelize version reduce value to NF
Example  Mandelbrot set

You could get more details from my blog post.

some type define
type Point = (Double, Double) type Range = (Double, Double) type Plane = (Range, Range)
 sequential version
planePoints :: Plane > V.Vector Point mandelSet :: Plane > V.Vector Point mandelSet = planeToMandelPoints
Example  Mandelbrot set
 basic parallel version with
parList
splitPlane :: Integer > Plane > [Plane] mandelSetStart :: Integer > Plane > V.Vector Point mandelSetStart size p = V.concat (map planeToMandelPoints (splitPlane size p) `using` parList rseq)
 In 2010 late MBP15 (Intel Core i5 2.4 Ghz, 8Gb)
 sequential  about 45 secs
 run in 2 cores  about 25 secs (
./Mandelbrot par 100 +RTS N2 s
)
Par Monad

Goal
 be more explicit about granularity and data dependences
 Avoid the reliance on lazy evaluation, but without sacrificing the determinism that we value for parallel programming.
 The parallel computations are pure (and deterministic)
 The library is implemented entirely as a Haskell library
 You can accommodate alternative scheduling strategies.
Par Monad
 Par monad  a monad for parallel computation
newtype Par a instance Applicative Par instance Monad Par runPar :: Par a > a  produce a pure result. fork :: Par () > Par ()  the way to create parallel tasks
 IVar  results are communicated through IVars
data IVar a new :: Par (IVar a) put :: NFData a => IVar a > a > Par () get :: IVar a > Par a
Par Monad
 IVar
data IVar a new :: Par (IVar a) put :: NFData a => IVar a > a > Par () get :: IVar a > Par a

IVar
 as a box that stars empty 
put
 store a value in the box All values communicated through IVars are fully evaluated. There is a headstrict version
put_
 All values communicated through IVars are fully evaluated. There is a headstrict version

get
 read the value. If the box is empty, it waits until the box is filled by put. Theget
operation does not remove the value from the box. Once the box is full. It stays the state constantly.
Par Monad
 Example
runPar $ do i < new j < new fork (put i (fib n)) fork (put j (fib m)) a < get i b < get j return (a+b)
Par Monad

spawn
 Like fork, but returns a IVar that can be used to query the result of the forked computation. Therefore spawn provides futures or promises. 
parMap
 parallel version map implemented with par monad
spawn :: NFData a => Par a > Par (IVar a) spawn p = do i < new fork (do x < p; put i x) return i parMap :: NFData b => (a > b) > [a] > Par [b] parMap f as = do ibs < mapM (spawn . return . f) as mapM get ibs
Example  prime number
 Example

primeIntVector
 Eval monad 
primeIntVector'
 Par monad

primeIntVector :: Int > VU.Vector Int primeIntVector n = let ls = genNumberRange 0 n 100 in VU.concat (map (uncurry primes) ls `using` parList rseq) primeIntVector' :: Int > VU.Vector Int primeIntVector' n = let ls = genNumberRange 0 n 100 in VU.concat $ Par.runPar $ Par.parMap (uncurry primes) ls
Difference between Par
and Eval
 Par Monad
 Always evaluate its value to normal form. It avoids the problem about the weaknormal form
 The cost of calling
runPar
function is bigger thenrunEval
 Easy to redefine the scheduling strategy
Difference between Par
and Eval

Eval Monad
 Need use
force
function to evaluate its value from weakhead normal form to normal form. It’s suitable for lazy data structure  The cost of calling
runEval
function is free  Provide the speculative parallelism
 Eval Monad has more diagnostics in ThreadScope compared Par Monad.
 Need use
Repa
 Repa  REgular PArallel arrays

Goal
 efficient numerical array computations in Haskell and run them in parallel
 It could provides efficient unboxed data computation, but not Par monad and
Strategy monad
 Repa also support boxed data.
Repa  type
 The array type
data Array r sh e

r
 representation type 
e
 element type 
sh
 the shape of array (the dimension(s) of array)
data Z = Z  scalar data data tail :. head = tail :. head type DIM0 = Z type DIM1 = DIM0 :. Int type DIM2 = DIM1 :. Int
Repa  array
 how to create an array with
Array
type ?
fromListUnboxed
 from list of unboxed type 
fromUnboxed
 from the vector withData.Vector.Unboxed
type 
fromFunction
 from the shape information to generate the array  ... etc

fromListUnboxed :: (Shape sh, Unbox a) => sh > [a] > Array U sh a fromFunction :: sh > (sh > a) > Array D sh a fromUnboxed :: (Shape sh, Data.Vector.Unboxed e) :: sh > e > Array U sh e
Repa  create array example
 Example  create an array
Prelude> import Data.Array.Repa as R Prelude R> let a = R.fromListUnboxed (Z :. 10) [1..10] :: Array U DIM1 Int Prelude R> :t a a :: Array U DIM1 Int Prelude R> let b = R.fromFunction (Z :. 10) (\(Z :. i) > i + 1 :: Int) Prelude R> :t b b :: Array D (Z :. Int) Int Prelude R > import qualified Data.Vector.Unboxed as VU Prelude R VU> let v = VU.enumFromN 1 10 :: VU.Vector Int Prelude R VU> let c = R.fromUnboxed (Z :. (VU.length v)) v Prelude R VU> :t c c :: Array U (Z :. Int) Int
Repa  array computation
 All array will transfer to a delayed array type (ex:
Array D sh e
) after array computations (ex:Repa.map
)
Repa.map :: (Shape sh, Source r a) => (a > b) > Array r sh a > Array D sh b (+^) :: (Num c, Shape sh, Source r1 c, Source r2 c) => Array r1 sh c > Array r2 sh c > Array D sh c
Repa  compute

computeS
 calculate the array operations in sequentially. 
computeP
 the same ascomputeS
but in parallel. the purpose of the monad is only to ensure that
computeP
operations are performed in sequence and not nested. the simplest way to reduce the monad effect 
runIdentity
 See page p.94 to get more information
 the simplest way to reduce the monad effect 
 the purpose of the monad is only to ensure that
computeS :: (Load r1 sh e, Target r2 e) => Array r1 sh e > Array r2 sh e computeP :: (Monad m, Source r2 e, Target r2 e, Load r1 sh e) => Array r1 sh e > m (Array r2 sh e)
Repa  array computation example
 calculate $e^x = \sum^{\infty}_{n=0}\frac{x^n}{n!} \forall x$
import Data.Array.Repa as R import Control.Monad.Identity fact x = foldr (*) 1 [1..x] enumN :: Int > Array D DIM1 Double enumN n = R.fromFunction (Z :. n) (\(Z :. i) > fromIntegral i) exp' :: Int > Double exp' x = let ns = enumN 100 ys = R.map (\n > (((fromIntegral x)**n) / (fact n))) ns in runIdentity . R.sumAllP $ yswzxhzdk:32 >
Repa  example
 Example  prime numbers
primeArray :: Int > VU.Vector Int primeArray n = let a = genArray n ps = runIdentity . Repa.computeUnboxedP . primeArrayCheck $ a :: Array U DIM1 Int in VU.filter (/=0) (Repa.toUnboxed ps)
Conclusion

The simplest parallel method  parallel map
 use
parMap
orparList
 use

Repa is useful especially for numeric calculation.

The remaining part of the book is about.
 Parallel programming with GPU (
Data.Accelerate
)  Concurrent programming
 Parallel programming with GPU (

Bool unbxoed type ?
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