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# Shape Polymorphic Arrays and Delayed Arrays

The library provides a layer on top of DPH unlifted arrays to support multi-dimensional arrays, and operations like maps, folds, permutations, shifts and so on. 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 explicitely convert a delayed array 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:

class (Show sh, U.Elt 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 index :: sh -> sh -> Int -- ^corresponding index into a linear, row-major -- representation of the array (first argument -- is the shape) indexInv:: 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 (index sh) (range sh) = [:0..(size sh)-1:] inRange :: sh -> sh -> Bool -- ^determines if a given index is in range zeroDim :: sh addDim :: sh -> sh -> sh -- ^adds two shapes of the same dimensionality modDim :: sh -> sh -> sh -- ^modulo operation lifted on shapes addModDim :: sh -> sh -> sh last :: (sh :*: Int) -> Int -- ^yields the innermost dimension of a shape inits :: (sh :*: Int) -> sh -- ^removes the innermost dimension from a shape

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.

For convenience, we provide type synonyms for dimensionality up to five:

type DIM0 = () type DIM1 = (DIM0 :*: Int) ....

## 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: (TODO there needs to be an darray constructor function accepting the shape and the function as arguments)

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

### Shape Invariant Computations on Arrays

The array operations described in this and the following subsection
are available on both strict and delayed arrays, and yield the same
result, with the exception that in case of delayed arrays, the result
is only calculated once its forced by calling `fromDArray`

. No
intermediate array structures are ever created.

The library provides a range of operation where the dimensionality of the result depends on the dimensionality of the argument in a non-trivial matter, 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 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

To express this relationship, the library provides the index GADT, which expresses a relationship between the inital and the projected dimensionality. It is defined as follows:

data Index a initialDim projectedDim where IndexNil :: Index a () () 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 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 DIM3 Double select arr (IndexFixed 3 (IndexAll (IndexAll IndexNil)))

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

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

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 somehow inconvenient to use, as the examples demonstrate. This is therefore another example where we 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.

## Array Operations

Backpermute and default backpermute are two general 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, Shape dim, Shape dim') => Array dim e -> dim' -> (dim' -> dim) -> Array dim' e backpermuteDft::(U.Elt e, Shape dim, Shape dim') => Array dim e -> e -> dim' -> (dim' -> Maybe dim) -> Array dim' e

map:: (U.Elt a, U.Elt b, Shape dim) => (a -> b) -> Array dim a -> Array dim b zip:: (U.Elt a, U.Elt b, Shape dim) => Array dim a -> Array dim b-> Array dim (a :*: b) zipWith:: (U.Elt a, U.Elt b, U.Elt c, Shape dim) => (a -> b -> c) -> Array dim a -> Array dim b-> Array dim c mapFold:: (U.Elt e, Shape dim) => (e -> e-> e) -> e -> Array (dim :*: Int) e -> Array dim e reshape:: (Shape dim', Shape dim, U.Elt e) => Array dim e -> dim' -> Array dim' e

The following operations could be (and in the sequential implementation indeed are) expressed in terms of backpermute and default backpermute. However, a programmer should aim at using more specialised functions when provided, as they carry more information about the pattern of reordering. In particular in the parallel case, this could be used to provide significantly more efficient implementation which make use of locality and communication patterns.

shift:: (Shape dim, U.Elt e) => Array dim e -> e -> dim -> Array dim e rotate:: (Shape dim, U.Elt e) => Array dim e -> e -> dim -> Array dim e tile:: (Shape dim, U.Elt e) => Array dim e -> dim -> dim -> Array dim e