wiki:DataParallel/Regular

Version 16 (modified by gckeller, 4 years ago) (diff)

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The library provides a layer on top of DPH unlifted arrays to support multi-dimensional arrays, and shape polymorphic operations for fast sequential and parallel execution. The interface for delayed arrays is similar, but in contrast to operations on the former, any operation on a delayed array is not evaluated. To force evaluation, the programmer has to explicitly convert them to a strict array.

The current implementation of the library exposes some implementation details the user of the library shouldn't have to worry about. Once the design of the library is finalised, most of these will be hidden by distinguishing between internal types and representation types.

Strict Arrays, Delayed Array and Shape Data Type

Both strict and delayed arrays are parametrised with their shape - that is, their dimensionality and size of each dimension. The shape type class has the following interface:

DArrays

DArrays are collections of values of `primitive' type, which are member of the class Data.Parallel.Array.Unlifted.Elt, which includes all basic types like Float, Double, Bool, Char, Integer, and pairs constructed with Data.Parallel.Array.Unlifted.(:*:), including ().

The Shape class

Values of class Shape serve two purposes: they can describe the dimensionality and size of an array (in which case we refer to them as 'shape'), and they can also refer to the position of a particular element in the array (in which case we refer to them as an 'index'). It provides the following methods:

class (Show sh, U.Elt sh, Eq sh, Rebox sh) => Shape sh where
  dim   :: sh -> Int           
  -- ^number of dimensions (>= 0)
  size  :: sh -> Int           
  -- ^for a *shape* yield the total number of  elements in that array
  toIndex :: sh -> sh -> Int     
  -- ^corresponding index into a linear representation of 
  -- the array (first argument is the shape)

  fromIndex:: sh -> Int -> sh   
  -- ^given index into linear row major representation, calculates 
  -- index into array                               

  range      :: sh -> U.Array sh
  -- ^all the valid indices in a shape. The following equality should hold: 
  -- map (toIndex sh) (range sh) = [:0..(size sh)-1:]

  
  inRange      :: sh -> sh -> Bool
  -- ^Checks if a given index is in the range of an array shape. I.e.,
  -- inRange sh ind == elem ind (range sh)

  zeroDim      :: sh
  -- ^shape of an array of size zero of the particular dimensionality

  intersectDim :: sh -> sh -> sh
  -- ^shape of an array of size zero of the particular dimensionality  

  next:: sh -> sh -> Maybe sh
  -- ^shape of an array of size zero of the particular dimensionality    

Note that a Shape has to be in the type class Elt imported from Data.Parallel.Array.Unboxed so that it can be an element of Data.Parallel.Array.Unboxed.Array.

The following instances are defined

instance Shape () 
instance Shape sh => Shape (sh :*: Int) 

so we have inductively defined n-tuples of Int values to represent shapes. This somewhat unusual representation is necessary to be able to inductively define operations on Shape. It should, however, be hidden from the library user in favour of the common tuple representation.

The multiparameter type class Subshape sh sh' contains all pairs of shapes sh and sh', for which the dimensionality of sh' is less than that of sh.

class (Shape sh, Shape sh') => Subshape sh sh' where
  addDim     :: sh -> sh' -> sh    
  modDim     :: sh -> sh' -> sh    
  inject     :: sh -> sh' -> sh

The method addDim adds the sizes of two shapes (or positions of two indices). If sh' is a strict subshape of sh, the fields of sh are copied when no corresponding fields of sh' exist, accordingly for modDim

Operations on Arrays and Delayed Arrays

Array Creation and Conversion

Strict arrays are simply defined as record containing a flat data array and shape information:

data Array dim e where
  Array:: { arrayShape    :: dim                -- ^extend of dimensions
          , arrayData     :: U.Array e          -- flat parallel array
           }  -> Array dim e
  deriving Show

toArray  :: (U.Elt e, Shape dim) => dim -> U.Array e -> Array dim e
fromArray:: (U.Elt e, Shape dim) => Array dim e -> U.Array e 

Delayed arrays, in contrast, in addition to the shape, only contain a function which, given an index, yields the corresponding element.

data DArray dim e where 
  DArray :: {dArrayShape::dim -> dArrayFn:: (dim -> e)} -> DArray dim e

Delayed arrays can be converted to and from strict arrays:

toDArray:: (U.Elt e, Array.Shape dim)   => Array.Array dim e -> DArray dim e
fromDArray:: (U.Elt e, Array.Shape dim) => DArray dim e      -> Array dim e

the result of toDArray is a DArray which contains an indexing function into an array. In general, the function dArrayFn can be much more complex. The function forceDArray (should this be called normalizeDArray?) forces the evaluation dArrayFn on every index of the range, and replaces dArrayFn by a simple indexing function into an array of the result values.

forceDArray:: (U.Elt e, A.Shape dim) => DArray dim e -> DArray dim e

Collection Oriented Operations

Elementwise Application of Functions

The map operation takes a function over element types and applies it to every data element of the DArray, which can have arbitrary dimensionality. Note that it is not possible to use this function to apply an operation for example to every row or column of a matrix. We will discuss how this can be done later on.

map:: (U.Elt a, U.Elt b, A.Shape dim) => (a -> b) -> DArray dim a -> DArray dim b

Similarily, zip and zipWith apply to every data element in the array as well. Both arguments have to have the same dimensionality (which is enforced by the type system). If they have a different shape, the result will have the intersection of both shapes. For example, zipping an array of shape (() :*: 4 :*: 6) and (() :*: 2 :*: 8) results in an array of shape (() :*: 2 :*: 6).

zipWith:: (U.Elt a, U.Elt b, U.Elt c, A.Shape dim) => 
  (a -> b -> c) -> DArray dim a -> DArray dim b-> DArray dim c
zip:: (U.Elt a, U.Elt b, A.Shape dim) => 
  DArray dim a -> DArray dim b-> DArray dim (a :*: b)

The function fold collapses the values of the innermost rows of an array of at least dimensionality 1.

fold :: (U.Elt e, A.Shape dim) => 
 (e -> e-> e) -> e -> DArray (dim :*: Int)  e  -> DArray dim e

Again, it's not possible to use fold directly to collapse an array along any other axis, but, as we will see shortly, this can be easily done using other functions in combination with fold.

TODO: MISSING: description of mapStencil

Shape Polymorphic Computations on Arrays

The library provides a range of operation where the dimensionality of the result depends on the dimensionality of the argument in a non-trivial manner, which we want to be reflected in the type system. Examples of such functions are generalised selection, which allows for extraction of subarrays of arbitrary dimension, and generalised replicate, which allows replication of an array in any dimension (or dimensions). For example, given a three dimensional matrix, we can use select to extract scalar element values, rows, columns, as well as two dimensional matrices along any of the three axes.

For selection, we can informally state the relationship between dimensionality of the argument, the selector, and the result as follows:

select:: Array dim e -> <select dim' of dim array> -> Array dim' e

Another example for such a generalised function would be a generalised map, which can apply a function to all elements, all rows, all columns, or submatrices of different orientation of a multidimensional array.

For the former example, we need a way to express the relationship between the shape of the argument and the shape and orientation of the result, as well as the numerical position of the structure (i.e., first, second, third element). In case of the generalised map, we don't need the numerical information, since the operation will be applied to all elements, rows, columns etc.

To express this dependency between input and output shape and orientation, as well as possibly a concrete position, the library provides the Index GADT, which expresses a relationship between the source and the projected dimension. It is defined as follows:

data Index a initialDim projectedDim where
  IndexNil   :: Index a initialDim initialDim
  IndexAll   :: (Shape init, Shape proj) =>      
                   Index a init proj -> Index a (init :*: Int) (proj :*: Int)
  IndexFixed :: (Shape init, Shape proj) => a -> 
                   Index a init proj -> Index a (init :*: Int)  proj

To refer to a specific element, the type parameter a is instantiated with the type Int, otherwise with the unit type:

type SelectIndex = Index Int
type MapIndex    = Index ()

Given this definition, the type of select now becomes

select:: (U.Elt e, Shape dim, Shape dim') => Array dim e -> SelectIndex dim dim'  -> Array dim' e

Example:

arr:: Array (() :*: Int :*: Int :*: Int) Double

arr' :: () :*: Int :*: Int
arr' = select arr (IndexFixed 3 (IndexAll (IndexAll IndexNil))) 

We could generalise this further, to extract from any array arr which is at least one dimensional the third element:

arr:: Shape dim => Array (dim :*: Int) Double

arr' :: Array dim Double
arr' = select arr (IndexFixed 3 IndexNil)

The index type is also used to express the type of generalised replicate

replicate:: Array dim' e -> SelectIndex dim dim'  -> Array dim e

which, given an array, can be used to expand it along any dimension. For example,

simpleReplicate:: (U.Elt e, Shape dim) => Array dim e -> Int -> Array (dim :*: Int) e
simpleReplicate arr n =
  replicate arr (IndexFixed n IndexNil)

replicates the argument array (which can of any dimensionality) n times and behaves thus similarly to list replicate, whereas

elementwiseReplicate:: (U.Elt e, Shape dim) => 
  Array (dim :*: Int) e -> Int -> Array (dim :*: Int :*: Int) e
elementwiseReplicate arr n =
  replicate arr (IndexAll (IndexFixed n IndexNil))

replicates each element of an array n times (similarly to map (replicate n) on lists).

Even though the index type serves well to express the relationship between the selector/multiplicator and the dimensionality of the argument and the result array, it is inconvenient to use, as the examples demonstrate. We therefore do need to add another layer to improve the usability of the library.

Note that the library provides no way to statically check the pre- and postconditions on the actual size of arguments and results. This has to be done at run time using assertions.

Reordering, Shifting, Tiling

Backpermute and default backpermute are two very versatile operations which allow the programmer to express all structural operations which reorder or extract elements based on their position in the argument array:

backpermute:: (U.Elt e, A.Shape dim, A.Shape dim') =>   
  DArray dim e -> dim' -> (dim' -> dim) -> DArray dim' e
backpermuteDft::(U.Elt e, A.Shape dim, A.Shape dim') => 
  DArray dim e -> e -> dim' -> (dim' -> Maybe dim) -> DArray dim' e

The function backpermute gets a source array, the shape of the new array, and a function which maps each index of the new array to an index of the source array (and thus indirectly provides a value for each index in the new array). Default backpermute is additionally provided with a default value which is inserted in the array in cases where the index function returns Nothing. (Remark: should probably be replaced by a default array instead of default value for more generality)

reshape arr newShape returns a new array with the same value as the argument array, but a new shape. The new shape has to have the same size as the original shape.

reshape:: (Shape dim', Shape dim, U.Elt e) => DArray dim e -> dim' -> DArray dim' e