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Efficient Haskell Arrays featuring Parallel computation
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array arrays convolution delayed-arrays haskell haskell-arrays massiv massiv-io multidimensional-arrays parallel-computing parallel-processing stencil

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massiv

massiv is a Haskell library for array manipulation. Performance is one of its main goals, thus it is capable of seamless parallelization of most of the operations provided by the library

The name for this library comes from the Russian word Massiv (Масси́в), which means an Array.

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Package Hackage Nightly LTS
massiv Hackage Nightly LTS
massiv-test Hackage Nightly LTS
haskell-scheduler Hackage Nightly LTS

Introduction

Everything in the library revolves around an Array r ix e - a data family for anything that can be thought of as an array. The type variables, from the end, are:

Manifest

These are your classical arrays that are located in memory and allow constant time lookup of elements. Another main property they share is that they have a mutable interface. An Array with manifest representation can be thawed into a mutable MArray and then frozen back into its immutable counterpart after some destructive operation is applied to the mutable copy. The differences among representations below is in the way that elements are being accessed in memory:

Delayed

Main trait of delayed arrays is that they do not exist in memory and instead describe the contents of an array as a function or a composition of functions. In fact all of the fusion capabilities in massiv can be attributed to delayed arrays.

Construct

Creating a delayed type of array allows us to fuse any future operations we decide to perform on it. Let's look at this example:

λ> import Data.Massiv.Array as A
λ> makeVectorR D Seq 10 id
Array D Seq (Sz1 10)
  [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ]

Here we created a delayed vector of size 10, which is in reality just an id function from its index to an element (see the Computation section for the meaning of Seq). So let's go ahead and square its elements

λ> vec = makeVectorR D Seq 10 id
λ> evaluateM vec 4
4
λ> vec2 = A.map (^ (2 :: Int)) vec
λ> evaluateM vec2 4
16

It's not that exciting, since every time we call evaluateM it will recompute the element, every time, therefore this function should be avoided at all costs! Instead we can use all of the functions that take Source like arrays and then fuse that computation together by calling compute, or a handy computeAs function and only afterwards apply an indexM function or its partial synonym: (!). Any delayed array can also be reduced using one of the folding functions, thus completely avoiding any memory allocation, or converted to a list, if that's what you need:

λ> vec2U = computeAs U vec2
λ> vec2U
Array U Seq (Sz1 10)
  [ 0, 1, 4, 9, 16, 25, 36, 49, 64, 81 ]
λ> vec2U ! 4
16
λ> toList vec2U
[0,1,4,9,16,25,36,49,64,81]
λ> A.sum vec2U
285

There is a whole multitude of ways to construct arrays:

It's worth noting that, in the next example, nested lists will be loaded into an unboxed manifest array and the sum of its elements will be computed in parallel on all available cores.

λ> A.sum (fromLists' Par [[0,0,0,0,0],[0,1,2,3,4],[0,2,4,6,8]] :: Array U Ix2 Double)
30.0

The above wouldn't run in parallel in ghci of course, as the program would have to be compiled with ghc using -threaded -with-rtsopts=-N flags in order to use all available cores. Alternatively we could compile with the -threaded flag and then pass the number of capabilities directly to the runtime with +RTS -N<n>, where <n> is the number of cores you'd like to utilize.

Index

The main Ix n closed type family can be somewhat confusing, but there is no need to fully understand how it works in order to start using it. GHC might ask you for the DataKinds language extension if IxN n is used in a type signature, but there are type and pattern synonyms for the first five dimensions: Ix1, Ix2, Ix3, Ix4 and Ix5.

There are three distinguishable constructors for the index:

λ> makeArrayR D Seq (Sz (3 :. 5)) (\ (i :. j) -> i * j)
Array D Seq (Sz (3 :. 5))
  [ [ 0, 0, 0, 0, 0 ]
  , [ 0, 1, 2, 3, 4 ]
  , [ 0, 2, 4, 6, 8 ]
  ]
λ> arr3 = makeArrayR P Seq (Sz (3 :> 2 :. 5)) (\ (i :> j :. k) -> i * j + k)
λ> :t arr3
arr3 :: Array P (IxN 3) Int
λ> arr3
Array P Seq (Sz (3 :> 2 :. 5))
  [ [ [ 0, 1, 2, 3, 4 ]
    , [ 0, 1, 2, 3, 4 ]
    ]
  , [ [ 0, 1, 2, 3, 4 ]
    , [ 1, 2, 3, 4, 5 ]
    ]
  , [ [ 0, 1, 2, 3, 4 ]
    , [ 2, 3, 4, 5, 6 ]
    ]
  ]
λ> arr3 ! (2 :> 1 :. 4)
6
λ> ix10 = 10 :> 9 :> 8 :> 7 :> 6 :> 5 :> 4 :> 3 :> 2 :. 1
λ> :t ix10
ix10 :: IxN 10
λ> ix10 -- 10-dimensional index
10 :> 9 :> 8 :> 7 :> 6 :> 5 :> 4 :> 3 :> 2 :. 1

Here is how we can construct a 4-dimensional array and sum its elements in constant memory:

λ> arr = makeArrayR D Seq (Sz (10 :> 20 :> 30 :. 40)) $ \ (i :> j :> k :. l) -> (i * j + k) * k + l
λ> :t arr -- a 4-dimensional array
arr :: Array D (IxN 4) Int
λ> A.sum arr
221890000

There are quite a few helper functions that can operate on indices, but these are only needed when writing functions that work for arrays of arbitrary dimension, as such they are scarcely used:

λ> pullOutDim' ix10 5
(5,10 :> 9 :> 8 :> 7 :> 6 :> 4 :> 3 :> 2 :. 1)
λ> unconsDim ix10
(10,9 :> 8 :> 7 :> 6 :> 5 :> 4 :> 3 :> 2 :. 1)
λ> unsnocDim ix10
(10 :> 9 :> 8 :> 7 :> 6 :> 5 :> 4 :> 3 :. 2,1)

All of the Ix n indices are instances of Num so basic numeric operations are made easier:

λ> (1 :> 2 :. 3) + (4 :> 5 :. 6)
5 :> 7 :. 9
λ> 5 :: Ix4
5 :> 5 :> 5 :. 5

It is important to note that the size type is distinct from the index by the newtype wrapper Sz ix. There is a constructor Sz, which will make sure that none of the dimensions are negative:

λ> Sz (2 :> 3 :. 4)
Sz (2 :> 3 :. 4)
λ> Sz (10 :> 2 :> -30 :. 4)
Sz (10 :> 2 :> 0 :. 4)

Same as with indices, there are helper pattern synonyms: Sz1, Sz2, Sz3, Sz4 and Sz5.

λ> Sz3 2 3 4
Sz (2 :> 3 :. 4)
λ> Sz4 10 2 (-30) 4
Sz (10 :> 2 :> 0 :. 4)

As well as the Num instance:

λ> 4 :: Sz5
Sz (4 :> 4 :> 4 :> 4 :. 4)
λ> (Sz2 1 2) + 3
Sz (4 :. 5)
λ> (Sz2 1 2) - 3
Sz (0 :. 0)

Alternatively tuples of Ints can be used for working with arrays, up to and including 5-tuples (type synonyms: Ix2T .. Ix5T), but since tuples are polymorphic it is necessary to restrict the resulting array type. Not all operations in the library support tuples, so it is advised to avoid them for indexing.

λ> makeArray Seq (4, 20) (uncurry (*)) :: Array P Ix2T Int
(Array P Seq ((4,20))
  [ [ 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 ]
  , [ 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19 ]
  , [ 0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38 ]
  , [ 0,3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57 ]
  ])
λ> :i Ix2T
type Ix2T = (Int, Int)

There are helper functions that can go back and forth between tuples and Ix n indices.

λ> fromIx4 (3 :> 4 :> 5 :. 6)
(3,4,5,6)
λ> toIx5 (3, 4, 5, 6, 7)
3 :> 4 :> 5 :> 6 :. 7

Slicing

In order to get a subsection of an array there is no need to recompute it, unless we want to free up the no longer memory, of course. So, there are a few slicing, resizing and extraction operators that can do it all in constant time, modulo the index manipulation:

λ> arr = makeArrayR U Seq (Sz (4 :> 2 :. 6)) fromIx3
λ> arr !> 3 !> 1
Array M Seq (Sz1 6)
  [ (3,1,0), (3,1,1), (3,1,2), (3,1,3), (3,1,4), (3,1,5) ]

As you might suspect all of the slicing, indexing, extracting, resizing operations are partial, and those are frowned upon in Haskell. So there are matching functions that can do the same operations safely by using MonadThrow and thus returning Nothing, Left SomeException or throwing an exception in case of IO on failure, for example:

λ> arr !?> 3 ??> 1
Array M Seq (Sz1 6)
  [ (3,1,0), (3,1,1), (3,1,2), (3,1,3), (3,1,4), (3,1,5) ]
λ> arr !?> 3 ??> 1 ?? 0 :: Maybe (Int, Int, Int)
Just (3,1,0)

In above examples we first take a slice at the 4th page (index 3, since we start at 0), then another one at the 2nd row (index 1). While in the last example we also take 1st element at position 0. Pretty neat, huh? Naturally, by doing a slice we always reduce dimension by one. We can do slicing from the outside as well as from the inside:

λ> Ix1 1 ... 9
Array D Seq (Sz1 10)
  [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ]
λ> a <- resizeM (Sz (3 :> 2 :. 4)) $ Ix1 11 ... 34
λ> a
Array D Seq (Sz (3 :> 2 :. 4))
  [ [ [ 11, 12, 13, 14 ]
    , [ 15, 16, 17, 18 ]
    ]
  , [ [ 19, 20, 21, 22 ]
    , [ 23, 24, 25, 26 ]
    ]
  , [ [ 27, 28, 29, 30 ]
    , [ 31, 32, 33, 34 ]
    ]
  ]
λ> a !> 0
Array D Seq (Sz (2 :. 4))
  [ [ 11, 12, 13, 14 ]
  , [ 15, 16, 17, 18 ]
  ]
λ> a <! 0
Array D Seq (Sz (3 :. 2))
  [ [ 11, 15 ]
  , [ 19, 23 ]
  , [ 27, 31 ]
  ]

Or we can slice along any other available dimension:

λ> a <!> (Dim 2, 0)
Array D Seq (Sz (3 :. 4))
  [ [ 11, 12, 13, 14 ]
  , [ 19, 20, 21, 22 ]
  , [ 27, 28, 29, 30 ]
  ]

In order to extract sub-array while preserving dimensionality we can use extractM or extractFromToM.

λ> extractM (0 :> 1 :. 1) (Sz (3 :> 1 :. 2)) a
Array D Seq (Sz (3 :> 1 :. 2))
  [ [ [ 16, 17 ]
    ]
  , [ [ 24, 25 ]
    ]
  , [ [ 32, 33 ]
    ]
  ]
λ> extractFromToM (1 :> 0 :. 1) (3 :> 2 :. 4) a
Array D Seq (Sz (2 :> 2 :. 3))
  [ [ [ 20, 21, 22 ]
    , [ 24, 25, 26 ]
    ]
  , [ [ 28, 29, 30 ]
    , [ 32, 33, 34 ]
    ]
  ]

Computation and parallelism

There is a data type Comp that controls how elements will be computed when calling the compute function. It has a few constructors, although most of the time either Seq or Par will be sufficient:

Just to make sure a simple novice mistake is prevented, which I have seen in the past, make sure your source code is compiled with ghc -O2 -threaded -with-rtsopts=-N, otherwise no parallelization and poor performance are waiting for you. Also a bit later you might notice the `{-# INLINE funcName

-}` pragma being used, oftentimes it is a good idea to do that, but not always required. It is

worthwhile to benchmark and experiment.

Stencil

Instead of manually iterating over a multi-dimensional array and applying a function to each element, while reading its neighboring elements (as you would do in an imperative language) in a functional language it is much more efficient to apply a stencil function and let the library take care of all of bounds checking and iterating in a cache friendly manner.

What's a stencil? It is a declarative way of specifying a pattern for how elements of an array in a neighborhood will be used in order to update each element of the newly created array. In massiv a Stencil is a function that can read the neighboring elements of the stencil's center (the zero index), and only those, and then outputs a new value for the center element.

stencil

Let's create a simple, but somewhat meaningful array and create an averaging stencil. There is nothing special about the array itself, but the averaging filter is a stencil that sums the elements in a Moore neighborhood and divides the result by 9, i.e. finds the average of a 3 by 3 square.

arrLightIx2 :: Comp -> Sz Ix2 -> Array D Ix2 Double
arrLightIx2 comp arrSz = makeArray comp arrSz $ \ (i :. j) -> sin (fromIntegral (i * i + j * j))
{-# INLINE arrLightIx2 #-}

average3x3Filter :: Fractional a => Stencil Ix2 a a
average3x3Filter = makeStencil (Sz (3 :. 3)) (1 :. 1) $ \ get ->
  (  get (-1 :. -1) + get (-1 :. 0) + get (-1 :. 1) +
     get ( 0 :. -1) + get ( 0 :. 0) + get ( 0 :. 1) +
     get ( 1 :. -1) + get ( 1 :. 0) + get ( 1 :. 1)   ) / 9
{-# INLINE average3x3Filter #-}

Here is what it would look like in GHCi. We create a delayed array with some funky periodic function, and make sure it is computed prior to mapping an average stencil over it:

λ> arr = computeAs U $ arrLightIx2 Par (Sz (600 :. 800))
λ> :t arr
arr :: Array U Ix2 Double
λ> :t mapStencil Edge average3x3Filter arr
mapStencil Edge average3x3Filter arr :: Array DW Ix2 Double

As you can see, that operation produced an array of the earlier mentioned representation Delayed Windowed DW. In its essence DW is an array type that does no bounds checking in order to gain performance, except when it's near the border, where it uses a border resolution technique supplied by the user (Edge in the example above). Currently it is used only in stencils and not much else can be done to an array of this type besides further computing it into a manifest representation.

This example will be continued in the next section, but before that I would like to mention that some might notice that it looks very much like convolution, and in fact convolution can be implemented with a stencil. There is a helper function makeConvolutionStencil that lets you do just that. For the sake of example we'll do a sum of all neighbors by hand instead:

sum3x3Filter :: Fractional a => Stencil Ix2 a a
sum3x3Filter = makeConvolutionStencil (Sz (3 :. 3)) (1 :. 1) $ \ get ->
  get (-1 :. -1) 1 . get (-1 :. 0) 1 . get (-1 :. 1) 1 .
  get ( 0 :. -1) 1 . get ( 0 :. 0) 1 . get ( 0 :. 1) 1 .
  get ( 1 :. -1) 1 . get ( 1 :. 0) 1 . get ( 1 :. 1) 1
{-# INLINE sum3x3Filter #-}

There is not a single plus or multiplication sign, that is because convolutions is actually summation of elements multiplied by a kernel element, so instead we have composition of functions applied to an offset index and a multiplier. After we map that stencil, we can further divide each element of the array by 9 in order to get the average. Yeah, I lied a bit, Array DW ix is an instance of Functor class, so we can map functions over it, which will be fused as with a regular Delayed array:

computeAs U $ fmap (/9) $ mapStencil Edge sum3x3Filter arr

If you are still confused of what a stencil is, but you are familiar with Conway's Game of Life this should hopefully clarify it a bit more. The function life below is a single iteration of Game of Life:

lifeRules :: Word8 -> Word8 -> Word8
lifeRules 0 3 = 1
lifeRules 1 2 = 1
lifeRules 1 3 = 1
lifeRules _ _ = 0

lifeStencil :: Stencil Ix2 Word8 Word8
lifeStencil = makeStencil (Sz (3 :. 3)) (1 :. 1) $ \ get ->
  lifeRules (get (0 :. 0)) $ get (-1 :. -1) + get (-1 :. 0) + get (-1 :. 1) +
                             get ( 0 :. -1)         +         get ( 0 :. 1) +
                             get ( 1 :. -1) + get ( 1 :. 0) + get ( 1 :. 1)

life :: Array S Ix2 Word8 -> Array S Ix2 Word8
life = compute . mapStencil Wrap lifeStencil

The full working example that uses GLUT and OpenGL is located in GameOfLife. You can run it if you have the GLUT dependencies installed:

$ cd massiv-examples && stack run GameOfLife

massiv-io

In order to do anything useful with arrays we often need to be able to read some data from a file. Considering that most common array-like files are images, massiv-io provides an interface to read, write and display images in common formats using Haskell native JuicyPixels and Netpbm packages.

Color package provides a variety of color spaces and conversions between them, which are used by massiv-io package as pixels during reading and writing images.

An earlier example wasn't particularly interesting, since we couldn't visualize what is actually going on, so let's expand on it:

import Data.Massiv.Array
import Data.Massiv.Array.IO

main :: IO ()
main = do
  let arr = computeAs S $ arrLightIx2 Par (600 :. 800)
      toImage ::
           (Functor (Array r Ix2), Load r Ix2 (Pixel (Y' SRGB) Word8))
        => Array r Ix2 Double
        -> Image S (Y' SRGB) Word8
      toImage = computeAs S . fmap (PixelY' . toWord8)
      lightPath = "files/light.png"
      lightImage = toImage $ delay arr
      lightAvgPath = "files/light_avg.png"
      lightAvgImage = toImage $ mapStencil Edge (avgStencil 3) arr
      lightSumPath = "files/light_sum.png"
      lightSumImage = toImage $ mapStencil Edge (sumStencil 3) arr
  writeImage lightPath lightImage
  putStrLn $ "written: " ++ lightPath
  writeImage lightAvgPath lightAvgImage
  putStrLn $ "written: " ++ lightAvgPath
  writeImage lightSumPath lightSumImage
  putStrLn $ "written: " ++ lightSumPath
  displayImageUsing defaultViewer True . computeAs S
    =<< concatM 1 [lightAvgImage, lightImage, lightSumImage]

massiv-examples/vision/files/light.png:

Light

massiv-examples/vision/files/light_avg.png:

Light Average

The full example is in the example vision package and if you have stack installed you can run it as:

$ cd massiv-examples && stack run avg-sum

Other libraries

A natural question might come to mind: Why even bother with a new array library when we already have a few really good ones in the Haskell world? The main reasons for me are performance and usability. I personally felt like there was much room for improvement before I even started working on this package, and it seems like it turned out to be true. For example, the most common goto library for dealing with multidimensional arrays and parallel computation used to be Repa, which I personally was a big fan of for quite some time, to the point that I even wrote a Haskell Image Processing library based on top of it.

Here is a quick summary of how massiv is better than Repa:

As far as usability of the library goes, it is very subjective, thus I'll let you be a judge of that. When talking about performance it is the facts that do matter. Thus, let's not continue this discussion in pure abstract words, below is a glimpse into benchmarks against Repa library running with GHC 8.8.4 on Intel® Core™ i7-3740QM CPU @ 2.70GHz × 8

Matrix multiplication:

benchmarking Repa/MxM U Double - (500x800 X 800x500)/Par
time                 120.5 ms   (115.0 ms .. 127.2 ms)
                     0.998 R²   (0.996 R² .. 1.000 R²)
mean                 124.1 ms   (121.2 ms .. 127.3 ms)
std dev              5.212 ms   (2.422 ms .. 6.620 ms)
variance introduced by outliers: 11% (moderately inflated)

benchmarking Massiv/MxM U Double - (500x800 X 800x500)/Par
time                 41.46 ms   (40.67 ms .. 42.45 ms)
                     0.998 R²   (0.994 R² .. 0.999 R²)
mean                 38.45 ms   (37.22 ms .. 39.68 ms)
std dev              2.342 ms   (1.769 ms .. 3.010 ms)
variance introduced by outliers: 19% (moderately inflated)

Sobel operator:

benchmarking Sobel/Par/Operator - Repa
time                 17.82 ms   (17.30 ms .. 18.32 ms)
                     0.997 R²   (0.994 R² .. 0.998 R²)
mean                 17.42 ms   (17.21 ms .. 17.69 ms)
std dev              593.0 μs   (478.1 μs .. 767.5 μs)
variance introduced by outliers: 12% (moderately inflated)

benchmarking Sobel/Par/Operator - Massiv
time                 7.421 ms   (7.230 ms .. 7.619 ms)
                     0.994 R²   (0.991 R² .. 0.997 R²)
mean                 7.537 ms   (7.422 ms .. 7.635 ms)
std dev              334.3 μs   (281.3 μs .. 389.9 μs)
variance introduced by outliers: 20% (moderately inflated)

Sum all elements of a 2D array:

benchmarking Sum/Seq/Repa
time                 539.7 ms   (523.2 ms .. 547.9 ms)
                     1.000 R²   (1.000 R² .. 1.000 R²)
mean                 540.1 ms   (535.7 ms .. 543.2 ms)
std dev              4.727 ms   (2.208 ms .. 6.609 ms)
variance introduced by outliers: 19% (moderately inflated)

benchmarking Sum/Seq/Vector
time                 16.95 ms   (16.78 ms .. 17.07 ms)
                     0.999 R²   (0.998 R² .. 1.000 R²)
mean                 17.23 ms   (17.13 ms .. 17.43 ms)
std dev              331.4 μs   (174.1 μs .. 490.0 μs)

benchmarking Sum/Seq/Massiv
time                 16.78 ms   (16.71 ms .. 16.85 ms)
                     1.000 R²   (1.000 R² .. 1.000 R²)
mean                 16.80 ms   (16.76 ms .. 16.88 ms)
std dev              127.8 μs   (89.95 μs .. 186.2 μs)

benchmarking Sum/Par/Repa
time                 81.76 ms   (78.52 ms .. 84.37 ms)
                     0.997 R²   (0.990 R² .. 1.000 R²)
mean                 79.20 ms   (78.03 ms .. 80.91 ms)
std dev              2.613 ms   (1.565 ms .. 3.736 ms)

benchmarking Sum/Par/Massiv
time                 8.102 ms   (7.971 ms .. 8.216 ms)
                     0.999 R²   (0.998 R² .. 1.000 R²)
mean                 7.967 ms   (7.852 ms .. 8.028 ms)
std dev              236.4 μs   (168.4 μs .. 343.2 μs)
variance introduced by outliers: 11% (moderately inflated)

Here is also a blog post that compares Performance of Haskell Array libraries through Canny edge detection

Further resources on learning massiv: