--
-- Basic non-strict arrays.
--
+-- /Note:/ The "Data.Array.IArray" module provides more general interface
+-- to immutable arrays: it defines operations with the same names as
+-- those defined below, but with more general types, and also defines
+-- 'Array' instances of the relevant classes. To use that more general
+-- interface, import "Data.Array.IArray" but not "Data.Array".
-----------------------------------------------------------------------------
module Data.Array
(
+ -- * Immutable non-strict arrays
+ -- $intro
module Data.Ix -- export all of Ix
, Array -- Array type is abstract
+ -- * Array construction
, array -- :: (Ix a) => (a,a) -> [(a,b)] -> Array a b
, listArray -- :: (Ix a) => (a,a) -> [b] -> Array a b
+ , accumArray -- :: (Ix a) => (b -> c -> b) -> b -> (a,a) -> [(a,c)] -> Array a b
+ -- * Accessing arrays
, (!) -- :: (Ix a) => Array a b -> a -> b
, bounds -- :: (Ix a) => Array a b -> (a,a)
, indices -- :: (Ix a) => Array a b -> [a]
, elems -- :: (Ix a) => Array a b -> [b]
, assocs -- :: (Ix a) => Array a b -> [(a,b)]
- , accumArray -- :: (Ix a) => (b -> c -> b) -> b -> (a,a) -> [(a,c)] -> Array a b
+ -- * Incremental array updates
, (//) -- :: (Ix a) => Array a b -> [(a,b)] -> Array a b
, accum -- :: (Ix a) => (b -> c -> b) -> Array a b -> [(a,c)] -> Array a b
+ -- * Derived arrays
, ixmap -- :: (Ix a, Ix b) => (a,a) -> (a -> b) -> Array b c -> Array a b
-- Array instances:
#include "Typeable.h"
INSTANCE_TYPEABLE2(Array,arrayTc,"Array")
#endif
+
+{- $intro
+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', this module is exported
+from "Data.Array" so that modules need not import both "Data.Array" and
+"Data.Ix".
+-}
-----------------------------------------------------------------------------
-- Class of immutable arrays
--- | Class of array types with bounds
+-- | Class of array types with immutable bounds
+-- (even if the array elements are mutable).
class HasBounds a where
-- | Extracts the bounds of an array
bounds :: Ix i => a i e -> (i,i)
has bounds (1,10), and a one-origin 10 by 10 matrix has bounds
((1,1),(10,10)).
-An association is a pair of the form @(i,x)@, which defines the value
-of the array at index @i@ to be @x@. The array is undefined 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.
+An association is a pair of the form @(i,x)@, which defines the value of
+the array at index @i@ to be @x@. The array is undefined 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 implementation-dependent.
+(In GHC, the last value specified for that index is used.
+Other implementations will also do this for unboxed arrays, but Haskell
+98 requires that for 'Array' the value at such indices is bottom.)
Because the indices must be checked for these errors, 'array' is
strict in the bounds argument and in the indices of the association
{-|
Takes an array and a list of pairs and returns an array identical to
the left argument except that it has been updated by the associations
-in the right argument. (As with the array function, the indices in the
-association list must be unique for the updated elements to be
-defined.) 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.
+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.
+
+As with the 'array' function, if any two associations in the list have
+the same index, the value at that index is implementation-dependent.
+(In GHC, the last value specified for that index is used.
+Other implementations will also do this for unboxed arrays, but Haskell
+98 requires that for 'Array' the value at such indices is bottom.)
For most array types, this operation is O(/n/) where /n/ is the size
of the array. However, the 'Data.Array.Diff.DiffArray' type provides
-- Stability : experimental
-- Portability : non-portable
--
--- Immutable arrays, with an overloaded interface. For array types
--- which can be used with this interface, see "Data.Array",
--- "Data.Array.Unboxed", and "Data.Array.Diff".
+-- Immutable arrays, with an overloaded interface. For array types which
+-- can be used with this interface, see the 'Array' type exported by this
+-- module, and the "Data.Array.Unboxed" and "Data.Array.Diff" modules.
--
-----------------------------------------------------------------------------
module Data.Array.IArray (
- -- * Class of immutable array types
+ -- * Array classes
+ HasBounds, -- :: (* -> * -> *) -> class
IArray, -- :: (* -> * -> *) -> * -> class
- -- * Class of array types with immutable bounds
- HasBounds, -- :: (* -> * -> *) -> class
+ module Data.Ix,
- -- * Ordinary boxed\/lazy arrays
+ -- * Immutable non-strict (boxed) arrays
Array,
- -- * The @Ix@ class and operations
- module Data.Ix,
-
-- * Array construction
array, -- :: (IArray a e, Ix i) => (i,i) -> [(i, e)] -> a i e
listArray, -- :: (IArray a e, Ix i) => (i,i) -> [e] -> a i e
accumArray, -- :: (IArray a e, Ix i) => (e -> e' -> e) -> e -> (i,i) -> [(i, e')] -> a i e
- -- * Indexing arrays
+ -- * Accessing arrays
(!), -- :: (IArray a e, Ix i) => a i e -> i -> e
+ bounds, -- :: (HasBounds a, Ix i) => a i e -> (i,i)
+ indices, -- :: (HasBounds a, Ix i) => a i e -> [i]
+ elems, -- :: (IArray a e, Ix i) => a i e -> [e]
+ assocs, -- :: (IArray a e, Ix i) => a i e -> [(i, e)]
- -- * Incremental updates
+ -- * Incremental array updates
(//), -- :: (IArray a e, Ix i) => a i e -> [(i, e)] -> a i e
accum, -- :: (IArray a e, Ix i) => (e -> e' -> e) -> a i e -> [(i, e')] -> a i e
- -- * Derived Arrays
+ -- * Derived arrays
amap, -- :: (IArray a e', IArray a e, Ix i) => (e' -> e) -> a i e' -> a i e
ixmap, -- :: (IArray a e, Ix i, Ix j) => (i,i) -> (i -> j) -> a j e -> a i e
-
- -- * Deconstructing arrays
- bounds, -- :: (HasBounds a, Ix i) => a i e -> (i,i)
- indices, -- :: (HasBounds a, Ix i) => a i e -> [i]
- elems, -- :: (IArray a e, Ix i) => a i e -> [e]
- assocs, -- :: (IArray a e, Ix i) => a i e -> [(i, e)]
) where
import Prelude
\begin{code}
type IPr = (Int, Int)
+-- | The type of immutable non-strict (boxed) arrays
+-- with indices in @i@ and elements in @e@.
data Ix i => Array i e = Array !i !i (Array# e)
-- | Mutable, boxed, non-strict arrays in the 'ST' monad. The type
arrEleBottom :: a
arrEleBottom = error "(Array.!): undefined array element"
--- | Note: GHC's implementation of @array@ differs slightly from the Haskell 98
--- report. If the same index appears twice, GHC's implementation takes the
--- value from last (index,value) pair in the list, rather than yielding bottom
--- for that array slot.
+-- | 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. The Haskell 98 Report further specifies that if any
+-- two associations in the list have the same index, the value at that
+-- index is undefined (i.e. bottom). However in GHC's implementation,
+-- the value at such an index is the value part of the last association
+-- with that index in the list.
+--
+-- 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 nonstrict 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.
{-# INLINE array #-}
-array :: Ix i => (i,i) -> [(i, e)] -> Array i e
+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
array (l,u) ies = unsafeArray (l,u) [(index (l,u) i, e) | (i, e) <- ies]
{-# INLINE unsafeArray #-}
-- transformation on the list of elements; I guess it's impossible
-- using mechanisms currently available.
+-- | Construct an array from a pair of bounds and a list of values in
+-- index order.
{-# INLINE listArray #-}
listArray :: Ix i => (i,i) -> [e] -> Array i e
listArray (l,u) es = runST (ST $ \s1# ->
case fillFromList 0# es s2# of { s3# ->
done l u marr# s3# }}})
+-- | The value at the given index in an array.
{-# INLINE (!) #-}
(!) :: Ix i => Array i e -> i -> e
arr@(Array l u _) ! i = unsafeAt arr (index (l,u) i)
unsafeAt (Array _ _ arr#) (I# i#) =
case indexArray# arr# i# of (# e #) -> e
+-- | The bounds with which an array was constructed.
{-# INLINE bounds #-}
bounds :: Ix i => Array i e -> (i,i)
bounds (Array l u _) = (l,u)
+-- | The list of indices of an array in ascending order.
{-# INLINE indices #-}
indices :: Ix i => Array i e -> [i]
indices (Array l u _) = range (l,u)
+-- | The list of elements of an array in index order.
{-# INLINE elems #-}
elems :: Ix i => Array i e -> [e]
elems arr@(Array l u _) =
[unsafeAt arr i | i <- [0 .. rangeSize (l,u) - 1]]
+-- | The list of associations of an array in index order.
{-# INLINE assocs #-}
assocs :: Ix i => Array i e -> [(i, e)]
assocs arr@(Array l u _) =
[(i, unsafeAt arr (unsafeIndex (l,u) i)) | i <- range (l,u)]
+-- | The 'accumArray' 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.
{-# INLINE accumArray #-}
-accumArray :: Ix i => (e -> a -> e) -> e -> (i,i) -> [(i, a)] -> Array i e
+accumArray :: Ix i
+ => (e -> a -> e) -- ^ accumulating function
+ -> e -- ^ initial value
+ -> (i,i) -- ^ bounds of the array
+ -> [(i, a)] -- ^ association list
+ -> Array i e
accumArray f init (l,u) ies =
unsafeAccumArray f init (l,u) [(index (l,u) i, e) | (i, e) <- ies]
case writeArray# marr# i# (f old new) s2# of { s3# ->
next s3# }}
+-- | 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':
+-- Haskell 98 specifies that the resulting array is undefined (i.e. bottom),
+-- but GHC's implementation uses the last association for each index.
{-# INLINE (//) #-}
(//) :: Ix i => Array i e -> [(i, e)] -> Array i e
arr@(Array l u _) // ies =
STArray _ _ marr# <- thawSTArray arr
ST (foldr (fill marr#) (done l u marr#) ies))
+-- | @'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])
+--
{-# INLINE accum #-}
accum :: Ix i => (e -> a -> e) -> Array i e -> [(i, a)] -> Array i e
accum f arr@(Array l u _) ies =
amap f arr@(Array l u _) =
unsafeArray (l,u) [(i, f (unsafeAt arr i)) | i <- [0 .. rangeSize (l,u) - 1]]
+-- | '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.
{-# INLINE ixmap #-}
ixmap :: (Ix i, Ix j) => (i,i) -> (i -> j) -> Array j e -> Array i e
ixmap (l,u) f arr =
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