## Chapter 14Data.Array

module Data.Array (
module Data.Ix,  Array,  array,  listArray,  accumArray,  (!),  bounds,
indices,  elems,  assocs,  (//),  accum,  ixmap
) where

### 14.1 Immutable non-strict arrays

Haskell provides indexable arrays, which may be thought of as functions whose domains are isomorphic to contiguous subsets of the integers. Functions restricted in this way can be implemented efficiently; in particular, a programmer may reasonably expect rapid access to the components. To ensure the possibility of such an implementation, arrays are treated as data, not as general functions.

Since most array functions involve the class Ix, the contents of the module Data.Ix are re-exported from Data.Array for convenience:

 module Data.Ix

 data Ix i => Array i e
The type of immutable non-strict (boxed) arrays with indices in i and elements in e.

 instance Ix i => Functor (Array i) instance (Ix i, Eq e) => Eq (Array i e) instance (Ix i, Ord e) => Ord (Array i e) instance (Ix a, Read a, Read b) => Read (Array a b) instance (Ix a, Show a, Show b) => Show (Array a b)

### 14.2 Array construction

 array
 :: Ix i => (i, i) a pair of bounds, each of the index type of the array. These bounds are the lowest and highest indices in the array, in that order. For example, a one-origin vector of length ’10’ has bounds ’(1,10)’, and a one-origin ’10’ by ’10’ matrix has bounds ’((1,1),(10,10))’. -> [(i, e)] a list of associations of the form (index, value). Typically, this list will be expressed as a comprehension. An association ’(i, x)’ defines the value of the array at index i to be x. -> Array i e

Construct an array with the specified bounds and containing values for given indices within these bounds.

The array is undefined (i.e. bottom) if any index in the list is out of bounds. If any two associations in the list have the same index, the value at that index is undefined (i.e. bottom).

Because the indices must be checked for these errors, array is strict in the bounds argument and in the indices of the association list, but non-strict in the values. Thus, recurrences such as the following are possible:

a = array (1,100) ((1,1) : [(i, i ⋆ a!(i-1)) | i <- [2..100]])

Not every index within the bounds of the array need appear in the association list, but the values associated with indices that do not appear will be undefined (i.e. bottom).

If, in any dimension, the lower bound is greater than the upper bound, then the array is legal, but empty. Indexing an empty array always gives an array-bounds error, but bounds still yields the bounds with which the array was constructed.

 listArray :: Ix i => (i, i) -> [e] -> Array i e
Construct an array from a pair of bounds and a list of values in index order.

 accumArray
 :: Ix i => (e -> a -> e) accumulating function -> e initial value -> (i, i) bounds of the array -> [(i, a)] association list -> Array i e

The accumArray function deals with repeated indices in the association list using an accumulating function which combines the values of associations with the same index. For example, given a list of values of some index type, hist produces a histogram of the number of occurrences of each index within a specified range:

hist :: (Ix a, Num b) => (a,a) -> [a] -> Array a b
hist bnds is = accumArray (+) 0 bnds [(i, 1) | i<-is, inRange bnds i]

If the accumulating function is strict, then accumArray is strict in the values, as well as the indices, in the association list. Thus, unlike ordinary arrays built with array, accumulated arrays should not in general be recursive.

### 14.3 Accessing arrays

 (!) :: Ix i => Array i e -> i -> e
The value at the given index in an array.

 bounds :: Ix i => Array i e -> (i, i)
The bounds with which an array was constructed.

 indices :: Ix i => Array i e -> [i]
The list of indices of an array in ascending order.

 elems :: Ix i => Array i e -> [e]
The list of elements of an array in index order.

 assocs :: Ix i => Array i e -> [(i, e)]
The list of associations of an array in index order.

 (//) :: Ix i => Array i e -> [(i, e)] -> Array i e
Constructs an array identical to the first argument except that it has been updated by the associations in the right argument. For example, if m is a 1-origin, n by n matrix, then

m//[((i,i), 0) | i <- [1..n]]

is the same matrix, except with the diagonal zeroed.

Repeated indices in the association list are handled as for array: the resulting array is undefined (i.e. bottom),

 accum :: Ix i => (e -> a -> e) -> Array i e -> [(i, a)] -> Array i e
accum f takes an array and an association list and accumulates pairs from the list into the array with the accumulating function f. Thus accumArray can be defined using accum:

accumArray f z b = accum f (array b [(i, z) | i <- range b])

### 14.5 Derived arrays

 ixmap :: (Ix i, Ix j) => (i, i) -> (i -> j) -> Array j e -> Array i e
ixmap allows for transformations on array indices. It may be thought of as providing function composition on the right with the mapping that the original array embodies.

A similar transformation of array values may be achieved using fmap from the Array instance of the Functor class.

### 14.6 Specification

module  Array (
module Data.Ix,  -- export all of Data.Ix
Array, array, listArray, (!), bounds, indices, elems, assocs,
accumArray, (//), accum, ixmap ) where

import Data.Ix
import Data.List( (\\) )

infixl 9  !, //

data (Ix a) => Array a b = MkArray (a,a) (a -> b) deriving ()

array       :: (Ix a) => (a,a) -> [(a,b)] -> Array a b
array b ivs
| any (not . inRange b. fst) ivs
= error "Data.Array.array: out-of-range array association"
| otherwise
= MkArray b arr
where
arr j = case [ v | (i,v) <- ivs, i == j ] of
[v]   -> v
[]    -> error "Data.Array.!: undefined array element"
_     -> error "Data.Array.!: multiply defined array element"

listArray             :: (Ix a) => (a,a) -> [b] -> Array a b
listArray b vs        =  array b (zipWith (\ a b -> (a,b)) (range b) vs)

(!)                   :: (Ix a) => Array a b -> a -> b
(!) (MkArray _ f)     =  f

bounds                :: (Ix a) => Array a b -> (a,a)
bounds (MkArray b _)  =  b

indices               :: (Ix a) => Array a b -> [a]
indices               =  range . bounds

elems                 :: (Ix a) => Array a b -> [b]
elems a               =  [a!i | i <- indices a]

assocs                :: (Ix a) => Array a b -> [(a,b)]
assocs a              =  [(i, a!i) | i <- indices a]

(//)                  :: (Ix a) => Array a b -> [(a,b)] -> Array a b
a // new_ivs          = array (bounds a) (old_ivs ++ new_ivs)
where
old_ivs = [(i,a!i) | i <- indices a,
i ‘notElem‘ new_is]
new_is  = [i | (i,_) <- new_ivs]

accum                 :: (Ix a) => (b -> c -> b) -> Array a b -> [(a,c)]
-> Array a b
accum f               =  foldl (\a (i,v) -> a // [(i,f (a!i) v)])

accumArray            :: (Ix a) => (b -> c -> b) -> b -> (a,a) -> [(a,c)]
-> Array a b
accumArray f z b      =  accum f (array b [(i,z) | i <- range b])

ixmap                 :: (Ix a, Ix b) => (a,a) -> (a -> b) -> Array b c
-> Array a c
ixmap b f a           = array b [(i, a ! f i) | i <- range b]

instance  (Ix a)          => Functor (Array a) where
fmap fn (MkArray b f) =  MkArray b (fn . f)

instance  (Ix a, Eq b)  => Eq (Array a b)  where
a == a' =  assocs a == assocs a'

instance  (Ix a, Ord b) => Ord (Array a b)  where
a <= a' =  assocs a <= assocs a'

instance  (Ix a, Show a, Show b) => Show (Array a b)  where
showsPrec p a = showParen (p > arrPrec) (
showString "array " .
showsPrec (arrPrec+1) (bounds a) . showChar ' ' .
showsPrec (arrPrec+1) (assocs a)                  )