{-# OPTIONS_GHC -fno-bang-patterns #-}
-----------------------------------------------------------------------------
-- |
-- Module : Data.Map
-- Copyright : (c) Daan Leijen 2002
-- License : BSD-style
-- Maintainer : [email protected]
-- Stability : provisional
-- Portability : portable
--
-- An efficient implementation of maps from keys to values (dictionaries).
--
-- Since many function names (but not the type name) clash with
-- "Prelude" names, this module is usually imported @qualified@, e.g.
--
-- > import Data.Map (Map)
-- > import qualified Data.Map as Map
--
-- The implementation of 'Map' is based on /size balanced/ binary trees (or
-- trees of /bounded balance/) as described by:
--
-- * Stephen Adams, \"/Efficient sets: a balancing act/\",
-- Journal of Functional Programming 3(4):553-562, October 1993,
-- <http://www.swiss.ai.mit.edu/~adams/BB>.
--
-- * J. Nievergelt and E.M. Reingold,
-- \"/Binary search trees of bounded balance/\",
-- SIAM journal of computing 2(1), March 1973.
--
-- Note that the implementation is /left-biased/ -- the elements of a
-- first argument are always preferred to the second, for example in
-- 'union' or 'insert'.
-----------------------------------------------------------------------------
module Data.Map (
-- * Map type
Map -- instance Eq,Show,Read
-- * Operators
, (!), (\\)
-- * Query
, null
, size
, member
, notMember
, lookup
, findWithDefault
-- * Construction
, empty
, singleton
-- ** Insertion
, insert
, insertWith, insertWithKey, insertLookupWithKey
, insertWith', insertWithKey'
-- ** Delete\/Update
, delete
, adjust
, adjustWithKey
, update
, updateWithKey
, updateLookupWithKey
, alter
-- * Combine
-- ** Union
, union
, unionWith
, unionWithKey
, unions
, unionsWith
-- ** Difference
, difference
, differenceWith
, differenceWithKey
-- ** Intersection
, intersection
, intersectionWith
, intersectionWithKey
-- * Traversal
-- ** Map
, map
, mapWithKey
, mapAccum
, mapAccumWithKey
, mapKeys
, mapKeysWith
, mapKeysMonotonic
-- ** Fold
, fold
, foldWithKey
-- * Conversion
, elems
, keys
, keysSet
, assocs
-- ** Lists
, toList
, fromList
, fromListWith
, fromListWithKey
-- ** Ordered lists
, toAscList
, fromAscList
, fromAscListWith
, fromAscListWithKey
, fromDistinctAscList
-- * Filter
, filter
, filterWithKey
, partition
, partitionWithKey
, mapMaybe
, mapMaybeWithKey
, mapEither
, mapEitherWithKey
, split
, splitLookup
-- * Submap
, isSubmapOf, isSubmapOfBy
, isProperSubmapOf, isProperSubmapOfBy
-- * Indexed
, lookupIndex
, findIndex
, elemAt
, updateAt
, deleteAt
-- * Min\/Max
, findMin
, findMax
, deleteMin
, deleteMax
, deleteFindMin
, deleteFindMax
, updateMin
, updateMax
, updateMinWithKey
, updateMaxWithKey
, minView
, maxView
, minViewWithKey
, maxViewWithKey
-- * Debugging
, showTree
, showTreeWith
, valid
) where
import Prelude hiding (lookup,map,filter,foldr,foldl,null)
import qualified Data.Set as Set
import qualified Data.List as List
import Data.Monoid (Monoid(..))
import Data.Typeable
import Control.Applicative (Applicative(..), (<$>))
import Data.Traversable (Traversable(traverse))
import Data.Foldable (Foldable(foldMap))
{-
-- for quick check
import qualified Prelude
import qualified List
import Debug.QuickCheck
import List(nub,sort)
-}
#if __GLASGOW_HASKELL__
import Text.Read
import Data.Generics.Basics
import Data.Generics.Instances
#endif
{--------------------------------------------------------------------
Operators
--------------------------------------------------------------------}
infixl 9 !,\\ --
-- | /O(log n)/. Find the value at a key.
-- Calls 'error' when the element can not be found.
(!) :: Ord k => Map k a -> k -> a
m ! k = find k m
-- | /O(n+m)/. See 'difference'.
(\\) :: Ord k => Map k a -> Map k b -> Map k a
m1 \\ m2 = difference m1 m2
{--------------------------------------------------------------------
Size balanced trees.
--------------------------------------------------------------------}
-- | A Map from keys @k@ to values @a@.
data Map k a = Tip
| Bin {-# UNPACK #-} !Size !k a !(Map k a) !(Map k a)
type Size = Int
instance (Ord k) => Monoid (Map k v) where
mempty = empty
mappend = union
mconcat = unions
#if __GLASGOW_HASKELL__
{--------------------------------------------------------------------
A Data instance
--------------------------------------------------------------------}
-- This instance preserves data abstraction at the cost of inefficiency.
-- We omit reflection services for the sake of data abstraction.
instance (Data k, Data a, Ord k) => Data (Map k a) where
gfoldl f z map = z fromList `f` (toList map)
toConstr _ = error "toConstr"
gunfold _ _ = error "gunfold"
dataTypeOf _ = mkNorepType "Data.Map.Map"
dataCast2 f = gcast2 f
#endif
{--------------------------------------------------------------------
Query
--------------------------------------------------------------------}
-- | /O(1)/. Is the map empty?
null :: Map k a -> Bool
null t
= case t of
Tip -> True
Bin sz k x l r -> False
-- | /O(1)/. The number of elements in the map.
size :: Map k a -> Int
size t
= case t of
Tip -> 0
Bin sz k x l r -> sz
-- | /O(log n)/. Lookup the value at a key in the map.
--
-- The function will
-- @return@ the result in the monad or @fail@ in it the key isn't in the
-- map. Often, the monad to use is 'Maybe', so you get either
-- @('Just' result)@ or @'Nothing'@.
lookup :: (Monad m,Ord k) => k -> Map k a -> m a
lookup k t = case lookup' k t of
Just x -> return x
Nothing -> fail "Data.Map.lookup: Key not found"
lookup' :: Ord k => k -> Map k a -> Maybe a
lookup' k t
= case t of
Tip -> Nothing
Bin sz kx x l r
-> case compare k kx of
LT -> lookup' k l
GT -> lookup' k r
EQ -> Just x
lookupAssoc :: Ord k => k -> Map k a -> Maybe (k,a)
lookupAssoc k t
= case t of
Tip -> Nothing
Bin sz kx x l r
-> case compare k kx of
LT -> lookupAssoc k l
GT -> lookupAssoc k r
EQ -> Just (kx,x)
-- | /O(log n)/. Is the key a member of the map?
member :: Ord k => k -> Map k a -> Bool
member k m
= case lookup k m of
Nothing -> False
Just x -> True
-- | /O(log n)/. Is the key not a member of the map?
notMember :: Ord k => k -> Map k a -> Bool
notMember k m = not $ member k m
-- | /O(log n)/. Find the value at a key.
-- Calls 'error' when the element can not be found.
find :: Ord k => k -> Map k a -> a
find k m
= case lookup k m of
Nothing -> error "Map.find: element not in the map"
Just x -> x
-- | /O(log n)/. The expression @('findWithDefault' def k map)@ returns
-- the value at key @k@ or returns @def@ when the key is not in the map.
findWithDefault :: Ord k => a -> k -> Map k a -> a
findWithDefault def k m
= case lookup k m of
Nothing -> def
Just x -> x
{--------------------------------------------------------------------
Construction
--------------------------------------------------------------------}
-- | /O(1)/. The empty map.
empty :: Map k a
empty
= Tip
-- | /O(1)/. A map with a single element.
singleton :: k -> a -> Map k a
singleton k x
= Bin 1 k x Tip Tip
{--------------------------------------------------------------------
Insertion
--------------------------------------------------------------------}
-- | /O(log n)/. Insert a new key and value in the map.
-- If the key is already present in the map, the associated value is
-- replaced with the supplied value, i.e. 'insert' is equivalent to
-- @'insertWith' 'const'@.
insert :: Ord k => k -> a -> Map k a -> Map k a
insert kx x t
= case t of
Tip -> singleton kx x
Bin sz ky y l r
-> case compare kx ky of
LT -> balance ky y (insert kx x l) r
GT -> balance ky y l (insert kx x r)
EQ -> Bin sz kx x l r
-- | /O(log n)/. Insert with a combining function.
-- @'insertWith' f key value mp@
-- will insert the pair (key, value) into @mp@ if key does
-- not exist in the map. If the key does exist, the function will
-- insert the pair @(key, f new_value old_value)@.
insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a
insertWith f k x m
= insertWithKey (\k x y -> f x y) k x m
-- | Same as 'insertWith', but the combining function is applied strictly.
insertWith' :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a
insertWith' f k x m
= insertWithKey' (\k x y -> f x y) k x m
-- | /O(log n)/. Insert with a combining function.
-- @'insertWithKey' f key value mp@
-- will insert the pair (key, value) into @mp@ if key does
-- not exist in the map. If the key does exist, the function will
-- insert the pair @(key,f key new_value old_value)@.
-- Note that the key passed to f is the same key passed to 'insertWithKey'.
insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
insertWithKey f kx x t
= case t of
Tip -> singleton kx x
Bin sy ky y l r
-> case compare kx ky of
LT -> balance ky y (insertWithKey f kx x l) r
GT -> balance ky y l (insertWithKey f kx x r)
EQ -> Bin sy kx (f kx x y) l r
-- | Same as 'insertWithKey', but the combining function is applied strictly.
insertWithKey' :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
insertWithKey' f kx x t
= case t of
Tip -> singleton kx x
Bin sy ky y l r
-> case compare kx ky of
LT -> balance ky y (insertWithKey' f kx x l) r
GT -> balance ky y l (insertWithKey' f kx x r)
EQ -> let x' = f kx x y in seq x' (Bin sy kx x' l r)
-- | /O(log n)/. The expression (@'insertLookupWithKey' f k x map@)
-- is a pair where the first element is equal to (@'lookup' k map@)
-- and the second element equal to (@'insertWithKey' f k x map@).
insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a,Map k a)
insertLookupWithKey f kx x t
= case t of
Tip -> (Nothing, singleton kx x)
Bin sy ky y l r
-> case compare kx ky of
LT -> let (found,l') = insertLookupWithKey f kx x l in (found,balance ky y l' r)
GT -> let (found,r') = insertLookupWithKey f kx x r in (found,balance ky y l r')
EQ -> (Just y, Bin sy kx (f kx x y) l r)
{--------------------------------------------------------------------
Deletion
[delete] is the inlined version of [deleteWith (\k x -> Nothing)]
--------------------------------------------------------------------}
-- | /O(log n)/. Delete a key and its value from the map. When the key is not
-- a member of the map, the original map is returned.
delete :: Ord k => k -> Map k a -> Map k a
delete k t
= case t of
Tip -> Tip
Bin sx kx x l r
-> case compare k kx of
LT -> balance kx x (delete k l) r
GT -> balance kx x l (delete k r)
EQ -> glue l r
-- | /O(log n)/. Adjust a value at a specific key. When the key is not
-- a member of the map, the original map is returned.
adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k a
adjust f k m
= adjustWithKey (\k x -> f x) k m
-- | /O(log n)/. Adjust a value at a specific key. When the key is not
-- a member of the map, the original map is returned.
adjustWithKey :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k a
adjustWithKey f k m
= updateWithKey (\k x -> Just (f k x)) k m
-- | /O(log n)/. The expression (@'update' f k map@) updates the value @x@
-- at @k@ (if it is in the map). If (@f x@) is 'Nothing', the element is
-- deleted. If it is (@'Just' y@), the key @k@ is bound to the new value @y@.
update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k a
update f k m
= updateWithKey (\k x -> f x) k m
-- | /O(log n)/. The expression (@'updateWithKey' f k map@) updates the
-- value @x@ at @k@ (if it is in the map). If (@f k x@) is 'Nothing',
-- the element is deleted. If it is (@'Just' y@), the key @k@ is bound
-- to the new value @y@.
updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a
updateWithKey f k t
= case t of
Tip -> Tip
Bin sx kx x l r
-> case compare k kx of
LT -> balance kx x (updateWithKey f k l) r
GT -> balance kx x l (updateWithKey f k r)
EQ -> case f kx x of
Just x' -> Bin sx kx x' l r
Nothing -> glue l r
-- | /O(log n)/. Lookup and update.
updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a,Map k a)
updateLookupWithKey f k t
= case t of
Tip -> (Nothing,Tip)
Bin sx kx x l r
-> case compare k kx of
LT -> let (found,l') = updateLookupWithKey f k l in (found,balance kx x l' r)
GT -> let (found,r') = updateLookupWithKey f k r in (found,balance kx x l r')
EQ -> case f kx x of
Just x' -> (Just x',Bin sx kx x' l r)
Nothing -> (Just x,glue l r)
-- | /O(log n)/. The expression (@'alter' f k map@) alters the value @x@ at @k@, or absence thereof.
-- 'alter' can be used to insert, delete, or update a value in a 'Map'.
-- In short : @'lookup' k ('alter' f k m) = f ('lookup' k m)@
alter :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a
alter f k t
= case t of
Tip -> case f Nothing of
Nothing -> Tip
Just x -> singleton k x
Bin sx kx x l r
-> case compare k kx of
LT -> balance kx x (alter f k l) r
GT -> balance kx x l (alter f k r)
EQ -> case f (Just x) of
Just x' -> Bin sx kx x' l r
Nothing -> glue l r
{--------------------------------------------------------------------
Indexing
--------------------------------------------------------------------}
-- | /O(log n)/. Return the /index/ of a key. The index is a number from
-- /0/ up to, but not including, the 'size' of the map. Calls 'error' when
-- the key is not a 'member' of the map.
findIndex :: Ord k => k -> Map k a -> Int
findIndex k t
= case lookupIndex k t of
Nothing -> error "Map.findIndex: element is not in the map"
Just idx -> idx
-- | /O(log n)/. Lookup the /index/ of a key. The index is a number from
-- /0/ up to, but not including, the 'size' of the map.
lookupIndex :: (Monad m,Ord k) => k -> Map k a -> m Int
lookupIndex k t = case lookup 0 t of
Nothing -> fail "Data.Map.lookupIndex: Key not found."
Just x -> return x
where
lookup idx Tip = Nothing
lookup idx (Bin _ kx x l r)
= case compare k kx of
LT -> lookup idx l
GT -> lookup (idx + size l + 1) r
EQ -> Just (idx + size l)
-- | /O(log n)/. Retrieve an element by /index/. Calls 'error' when an
-- invalid index is used.
elemAt :: Int -> Map k a -> (k,a)
elemAt i Tip = error "Map.elemAt: index out of range"
elemAt i (Bin _ kx x l r)
= case compare i sizeL of
LT -> elemAt i l
GT -> elemAt (i-sizeL-1) r
EQ -> (kx,x)
where
sizeL = size l
-- | /O(log n)/. Update the element at /index/. Calls 'error' when an
-- invalid index is used.
updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a
updateAt f i Tip = error "Map.updateAt: index out of range"
updateAt f i (Bin sx kx x l r)
= case compare i sizeL of
LT -> updateAt f i l
GT -> updateAt f (i-sizeL-1) r
EQ -> case f kx x of
Just x' -> Bin sx kx x' l r
Nothing -> glue l r
where
sizeL = size l
-- | /O(log n)/. Delete the element at /index/.
-- Defined as (@'deleteAt' i map = 'updateAt' (\k x -> 'Nothing') i map@).
deleteAt :: Int -> Map k a -> Map k a
deleteAt i map
= updateAt (\k x -> Nothing) i map
{--------------------------------------------------------------------
Minimal, Maximal
--------------------------------------------------------------------}
-- | /O(log n)/. The minimal key of the map.
findMin :: Map k a -> (k,a)
findMin (Bin _ kx x Tip r) = (kx,x)
findMin (Bin _ kx x l r) = findMin l
findMin Tip = error "Map.findMin: empty map has no minimal element"
-- | /O(log n)/. The maximal key of the map.
findMax :: Map k a -> (k,a)
findMax (Bin _ kx x l Tip) = (kx,x)
findMax (Bin _ kx x l r) = findMax r
findMax Tip = error "Map.findMax: empty map has no maximal element"
-- | /O(log n)/. Delete the minimal key.
deleteMin :: Map k a -> Map k a
deleteMin (Bin _ kx x Tip r) = r
deleteMin (Bin _ kx x l r) = balance kx x (deleteMin l) r
deleteMin Tip = Tip
-- | /O(log n)/. Delete the maximal key.
deleteMax :: Map k a -> Map k a
deleteMax (Bin _ kx x l Tip) = l
deleteMax (Bin _ kx x l r) = balance kx x l (deleteMax r)
deleteMax Tip = Tip
-- | /O(log n)/. Update the value at the minimal key.
updateMin :: (a -> Maybe a) -> Map k a -> Map k a
updateMin f m
= updateMinWithKey (\k x -> f x) m
-- | /O(log n)/. Update the value at the maximal key.
updateMax :: (a -> Maybe a) -> Map k a -> Map k a
updateMax f m
= updateMaxWithKey (\k x -> f x) m
-- | /O(log n)/. Update the value at the minimal key.
updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
updateMinWithKey f t
= case t of
Bin sx kx x Tip r -> case f kx x of
Nothing -> r
Just x' -> Bin sx kx x' Tip r
Bin sx kx x l r -> balance kx x (updateMinWithKey f l) r
Tip -> Tip
-- | /O(log n)/. Update the value at the maximal key.
updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
updateMaxWithKey f t
= case t of
Bin sx kx x l Tip -> case f kx x of
Nothing -> l
Just x' -> Bin sx kx x' l Tip
Bin sx kx x l r -> balance kx x l (updateMaxWithKey f r)
Tip -> Tip
-- | /O(log n)/. Retrieves the minimal (key,value) pair of the map, and the map stripped from that element
-- @fail@s (in the monad) when passed an empty map.
minViewWithKey :: Monad m => Map k a -> m ((k,a), Map k a)
minViewWithKey Tip = fail "Map.minView: empty map"
minViewWithKey x = return (deleteFindMin x)
-- | /O(log n)/. Retrieves the maximal (key,value) pair of the map, and the map stripped from that element
-- @fail@s (in the monad) when passed an empty map.
maxViewWithKey :: Monad m => Map k a -> m ((k,a), Map k a)
maxViewWithKey Tip = fail "Map.maxView: empty map"
maxViewWithKey x = return (deleteFindMax x)
-- | /O(log n)/. Retrieves the minimal key\'s value of the map, and the map stripped from that element
-- @fail@s (in the monad) when passed an empty map.
minView :: Monad m => Map k a -> m (a, Map k a)
minView Tip = fail "Map.minView: empty map"
minView x = return (first snd $ deleteFindMin x)
-- | /O(log n)/. Retrieves the maximal key\'s value of the map, and the map stripped from that element
-- @fail@s (in the monad) when passed an empty map.
maxView :: Monad m => Map k a -> m (a, Map k a)
maxView Tip = fail "Map.maxView: empty map"
maxView x = return (first snd $ deleteFindMax x)
-- Update the 1st component of a tuple (special case of Control.Arrow.first)
first :: (a -> b) -> (a,c) -> (b,c)
first f (x,y) = (f x, y)
{--------------------------------------------------------------------
Union.
--------------------------------------------------------------------}
-- | The union of a list of maps:
-- (@'unions' == 'Prelude.foldl' 'union' 'empty'@).
unions :: Ord k => [Map k a] -> Map k a
unions ts
= foldlStrict union empty ts
-- | The union of a list of maps, with a combining operation:
-- (@'unionsWith' f == 'Prelude.foldl' ('unionWith' f) 'empty'@).
unionsWith :: Ord k => (a->a->a) -> [Map k a] -> Map k a
unionsWith f ts
= foldlStrict (unionWith f) empty ts
-- | /O(n+m)/.
-- The expression (@'union' t1 t2@) takes the left-biased union of @t1@ and @t2@.
-- It prefers @t1@ when duplicate keys are encountered,
-- i.e. (@'union' == 'unionWith' 'const'@).
-- The implementation uses the efficient /hedge-union/ algorithm.
-- Hedge-union is more efficient on (bigset `union` smallset)
union :: Ord k => Map k a -> Map k a -> Map k a
union Tip t2 = t2
union t1 Tip = t1
union t1 t2 = hedgeUnionL (const LT) (const GT) t1 t2
-- left-biased hedge union
hedgeUnionL cmplo cmphi t1 Tip
= t1
hedgeUnionL cmplo cmphi Tip (Bin _ kx x l r)
= join kx x (filterGt cmplo l) (filterLt cmphi r)
hedgeUnionL cmplo cmphi (Bin _ kx x l r) t2
= join kx x (hedgeUnionL cmplo cmpkx l (trim cmplo cmpkx t2))
(hedgeUnionL cmpkx cmphi r (trim cmpkx cmphi t2))
where
cmpkx k = compare kx k
-- right-biased hedge union
hedgeUnionR cmplo cmphi t1 Tip
= t1
hedgeUnionR cmplo cmphi Tip (Bin _ kx x l r)
= join kx x (filterGt cmplo l) (filterLt cmphi r)
hedgeUnionR cmplo cmphi (Bin _ kx x l r) t2
= join kx newx (hedgeUnionR cmplo cmpkx l lt)
(hedgeUnionR cmpkx cmphi r gt)
where
cmpkx k = compare kx k
lt = trim cmplo cmpkx t2
(found,gt) = trimLookupLo kx cmphi t2
newx = case found of
Nothing -> x
Just (_,y) -> y
{--------------------------------------------------------------------
Union with a combining function
--------------------------------------------------------------------}
-- | /O(n+m)/. Union with a combining function. The implementation uses the efficient /hedge-union/ algorithm.
unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a
unionWith f m1 m2
= unionWithKey (\k x y -> f x y) m1 m2
-- | /O(n+m)/.
-- Union with a combining function. The implementation uses the efficient /hedge-union/ algorithm.
-- Hedge-union is more efficient on (bigset `union` smallset).
unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a
unionWithKey f Tip t2 = t2
unionWithKey f t1 Tip = t1
unionWithKey f t1 t2 = hedgeUnionWithKey f (const LT) (const GT) t1 t2
hedgeUnionWithKey f cmplo cmphi t1 Tip
= t1
hedgeUnionWithKey f cmplo cmphi Tip (Bin _ kx x l r)
= join kx x (filterGt cmplo l) (filterLt cmphi r)
hedgeUnionWithKey f cmplo cmphi (Bin _ kx x l r) t2
= join kx newx (hedgeUnionWithKey f cmplo cmpkx l lt)
(hedgeUnionWithKey f cmpkx cmphi r gt)
where
cmpkx k = compare kx k
lt = trim cmplo cmpkx t2
(found,gt) = trimLookupLo kx cmphi t2
newx = case found of
Nothing -> x
Just (_,y) -> f kx x y
{--------------------------------------------------------------------
Difference
--------------------------------------------------------------------}
-- | /O(n+m)/. Difference of two maps.
-- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
difference :: Ord k => Map k a -> Map k b -> Map k a
difference Tip t2 = Tip
difference t1 Tip = t1
difference t1 t2 = hedgeDiff (const LT) (const GT) t1 t2
hedgeDiff cmplo cmphi Tip t
= Tip
hedgeDiff cmplo cmphi (Bin _ kx x l r) Tip
= join kx x (filterGt cmplo l) (filterLt cmphi r)
hedgeDiff cmplo cmphi t (Bin _ kx x l r)
= merge (hedgeDiff cmplo cmpkx (trim cmplo cmpkx t) l)
(hedgeDiff cmpkx cmphi (trim cmpkx cmphi t) r)
where
cmpkx k = compare kx k
-- | /O(n+m)/. Difference with a combining function.
-- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a
differenceWith f m1 m2
= differenceWithKey (\k x y -> f x y) m1 m2
-- | /O(n+m)/. Difference with a combining function. When two equal keys are
-- encountered, the combining function is applied to the key and both values.
-- If it returns 'Nothing', the element is discarded (proper set difference). If
-- it returns (@'Just' y@), the element is updated with a new value @y@.
-- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a
differenceWithKey f Tip t2 = Tip
differenceWithKey f t1 Tip = t1
differenceWithKey f t1 t2 = hedgeDiffWithKey f (const LT) (const GT) t1 t2
hedgeDiffWithKey f cmplo cmphi Tip t
= Tip
hedgeDiffWithKey f cmplo cmphi (Bin _ kx x l r) Tip
= join kx x (filterGt cmplo l) (filterLt cmphi r)
hedgeDiffWithKey f cmplo cmphi t (Bin _ kx x l r)
= case found of
Nothing -> merge tl tr
Just (ky,y) ->
case f ky y x of
Nothing -> merge tl tr
Just z -> join ky z tl tr
where
cmpkx k = compare kx k
lt = trim cmplo cmpkx t
(found,gt) = trimLookupLo kx cmphi t
tl = hedgeDiffWithKey f cmplo cmpkx lt l
tr = hedgeDiffWithKey f cmpkx cmphi gt r
{--------------------------------------------------------------------
Intersection
--------------------------------------------------------------------}
-- | /O(n+m)/. Intersection of two maps. The values in the first
-- map are returned, i.e. (@'intersection' m1 m2 == 'intersectionWith' 'const' m1 m2@).
intersection :: Ord k => Map k a -> Map k b -> Map k a
intersection m1 m2
= intersectionWithKey (\k x y -> x) m1 m2
-- | /O(n+m)/. Intersection with a combining function.
intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k c
intersectionWith f m1 m2
= intersectionWithKey (\k x y -> f x y) m1 m2
-- | /O(n+m)/. Intersection with a combining function.
-- Intersection is more efficient on (bigset `intersection` smallset)
--intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c
--intersectionWithKey f Tip t = Tip
--intersectionWithKey f t Tip = Tip
--intersectionWithKey f t1 t2 = intersectWithKey f t1 t2
--
--intersectWithKey f Tip t = Tip
--intersectWithKey f t Tip = Tip
--intersectWithKey f t (Bin _ kx x l r)
-- = case found of
-- Nothing -> merge tl tr
-- Just y -> join kx (f kx y x) tl tr
-- where
-- (lt,found,gt) = splitLookup kx t
-- tl = intersectWithKey f lt l
-- tr = intersectWithKey f gt r
intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c
intersectionWithKey f Tip t = Tip
intersectionWithKey f t Tip = Tip
intersectionWithKey f t1@(Bin s1 k1 x1 l1 r1) t2@(Bin s2 k2 x2 l2 r2) =
if s1 >= s2 then
let (lt,found,gt) = splitLookupWithKey k2 t1
tl = intersectionWithKey f lt l2
tr = intersectionWithKey f gt r2
in case found of
Just (k,x) -> join k (f k x x2) tl tr
Nothing -> merge tl tr
else let (lt,found,gt) = splitLookup k1 t2
tl = intersectionWithKey f l1 lt
tr = intersectionWithKey f r1 gt
in case found of
Just x -> join k1 (f k1 x1 x) tl tr
Nothing -> merge tl tr
{--------------------------------------------------------------------
Submap
--------------------------------------------------------------------}
-- | /O(n+m)/.
-- This function is defined as (@'isSubmapOf' = 'isSubmapOfBy' (==)@).
isSubmapOf :: (Ord k,Eq a) => Map k a -> Map k a -> Bool
isSubmapOf m1 m2
= isSubmapOfBy (==) m1 m2
{- | /O(n+m)/.
The expression (@'isSubmapOfBy' f t1 t2@) returns 'True' if
all keys in @t1@ are in tree @t2@, and when @f@ returns 'True' when
applied to their respective values. For example, the following
expressions are all 'True':
> isSubmapOfBy (==) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
> isSubmapOfBy (<=) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
> isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1),('b',2)])
But the following are all 'False':
> isSubmapOfBy (==) (fromList [('a',2)]) (fromList [('a',1),('b',2)])
> isSubmapOfBy (<) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
> isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1)])
-}
isSubmapOfBy :: Ord k => (a->b->Bool) -> Map k a -> Map k b -> Bool
isSubmapOfBy f t1 t2
= (size t1 <= size t2) && (submap' f t1 t2)
submap' f Tip t = True
submap' f t Tip = False
submap' f (Bin _ kx x l r) t
= case found of
Nothing -> False
Just y -> f x y && submap' f l lt && submap' f r gt
where
(lt,found,gt) = splitLookup kx t
-- | /O(n+m)/. Is this a proper submap? (ie. a submap but not equal).
-- Defined as (@'isProperSubmapOf' = 'isProperSubmapOfBy' (==)@).
isProperSubmapOf :: (Ord k,Eq a) => Map k a -> Map k a -> Bool
isProperSubmapOf m1 m2
= isProperSubmapOfBy (==) m1 m2
{- | /O(n+m)/. Is this a proper submap? (ie. a submap but not equal).
The expression (@'isProperSubmapOfBy' f m1 m2@) returns 'True' when
@m1@ and @m2@ are not equal,
all keys in @m1@ are in @m2@, and when @f@ returns 'True' when
applied to their respective values. For example, the following
expressions are all 'True':
> isProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
> isProperSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
But the following are all 'False':
> isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
> isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
> isProperSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
-}
isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool
isProperSubmapOfBy f t1 t2
= (size t1 < size t2) && (submap' f t1 t2)
{--------------------------------------------------------------------
Filter and partition
--------------------------------------------------------------------}
-- | /O(n)/. Filter all values that satisfy the predicate.
filter :: Ord k => (a -> Bool) -> Map k a -> Map k a
filter p m
= filterWithKey (\k x -> p x) m
-- | /O(n)/. Filter all keys\/values that satisfy the predicate.
filterWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> Map k a
filterWithKey p Tip = Tip
filterWithKey p (Bin _ kx x l r)
| p kx x = join kx x (filterWithKey p l) (filterWithKey p r)
| otherwise = merge (filterWithKey p l) (filterWithKey p r)
-- | /O(n)/. partition the map according to a predicate. The first
-- map contains all elements that satisfy the predicate, the second all
-- elements that fail the predicate. See also 'split'.
partition :: Ord k => (a -> Bool) -> Map k a -> (Map k a,Map k a)
partition p m
= partitionWithKey (\k x -> p x) m
-- | /O(n)/. partition the map according to a predicate. The first
-- map contains all elements that satisfy the predicate, the second all
-- elements that fail the predicate. See also 'split'.
partitionWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> (Map k a,Map k a)
partitionWithKey p Tip = (Tip,Tip)
partitionWithKey p (Bin _ kx x l r)
| p kx x = (join kx x l1 r1,merge l2 r2)
| otherwise = (merge l1 r1,join kx x l2 r2)
where
(l1,l2) = partitionWithKey p l
(r1,r2) = partitionWithKey p r
-- | /O(n)/. Map values and collect the 'Just' results.
mapMaybe :: Ord k => (a -> Maybe b) -> Map k a -> Map k b
mapMaybe f m
= mapMaybeWithKey (\k x -> f x) m
-- | /O(n)/. Map keys\/values and collect the 'Just' results.
mapMaybeWithKey :: Ord k => (k -> a -> Maybe b) -> Map k a -> Map k b
mapMaybeWithKey f Tip = Tip
mapMaybeWithKey f (Bin _ kx x l r) = case f kx x of
Just y -> join kx y (mapMaybeWithKey f l) (mapMaybeWithKey f r)
Nothing -> merge (mapMaybeWithKey f l) (mapMaybeWithKey f r)
-- | /O(n)/. Map values and separate the 'Left' and 'Right' results.
mapEither :: Ord k => (a -> Either b c) -> Map k a -> (Map k b, Map k c)
mapEither f m
= mapEitherWithKey (\k x -> f x) m
-- | /O(n)/. Map keys\/values and separate the 'Left' and 'Right' results.
mapEitherWithKey :: Ord k =>
(k -> a -> Either b c) -> Map k a -> (Map k b, Map k c)
mapEitherWithKey f Tip = (Tip, Tip)
mapEitherWithKey f (Bin _ kx x l r) = case f kx x of
Left y -> (join kx y l1 r1, merge l2 r2)
Right z -> (merge l1 r1, join kx z l2 r2)
where
(l1,l2) = mapEitherWithKey f l
(r1,r2) = mapEitherWithKey f r
{--------------------------------------------------------------------
Mapping
--------------------------------------------------------------------}
-- | /O(n)/. Map a function over all values in the map.
map :: (a -> b) -> Map k a -> Map k b
map f m
= mapWithKey (\k x -> f x) m
-- | /O(n)/. Map a function over all values in the map.
mapWithKey :: (k -> a -> b) -> Map k a -> Map k b
mapWithKey f Tip = Tip
mapWithKey f (Bin sx kx x l r)
= Bin sx kx (f kx x) (mapWithKey f l) (mapWithKey f r)
-- | /O(n)/. The function 'mapAccum' threads an accumulating
-- argument through the map in ascending order of keys.
mapAccum :: (a -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
mapAccum f a m
= mapAccumWithKey (\a k x -> f a x) a m
-- | /O(n)/. The function 'mapAccumWithKey' threads an accumulating
-- argument through the map in ascending order of keys.
mapAccumWithKey :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
mapAccumWithKey f a t
= mapAccumL f a t
-- | /O(n)/. The function 'mapAccumL' threads an accumulating
-- argument throught the map in ascending order of keys.
mapAccumL :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
mapAccumL f a t
= case t of
Tip -> (a,Tip)
Bin sx kx x l r
-> let (a1,l') = mapAccumL f a l
(a2,x') = f a1 kx x
(a3,r') = mapAccumL f a2 r
in (a3,Bin sx kx x' l' r')
-- | /O(n)/. The function 'mapAccumR' threads an accumulating
-- argument throught the map in descending order of keys.
mapAccumR :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
mapAccumR f a t
= case t of
Tip -> (a,Tip)
Bin sx kx x l r
-> let (a1,r') = mapAccumR f a r
(a2,x') = f a1 kx x
(a3,l') = mapAccumR f a2 l
in (a3,Bin sx kx x' l' r')
-- | /O(n*log n)/.
-- @'mapKeys' f s@ is the map obtained by applying @f@ to each key of @s@.
--
-- The size of the result may be smaller if @f@ maps two or more distinct
-- keys to the same new key. In this case the value at the smallest of
-- these keys is retained.
mapKeys :: Ord k2 => (k1->k2) -> Map k1 a -> Map k2 a
mapKeys = mapKeysWith (\x y->x)
-- | /O(n*log n)/.
-- @'mapKeysWith' c f s@ is the map obtained by applying @f@ to each key of @s@.
--
-- The size of the result may be smaller if @f@ maps two or more distinct
-- keys to the same new key. In this case the associated values will be
-- combined using @c@.
mapKeysWith :: Ord k2 => (a -> a -> a) -> (k1->k2) -> Map k1 a -> Map k2 a
mapKeysWith c f = fromListWith c . List.map fFirst . toList
where fFirst (x,y) = (f x, y)
-- | /O(n)/.
-- @'mapKeysMonotonic' f s == 'mapKeys' f s@, but works only when @f@
-- is strictly monotonic.
-- /The precondition is not checked./
-- Semi-formally, we have:
--
-- > and [x < y ==> f x < f y | x <- ls, y <- ls]
-- > ==> mapKeysMonotonic f s == mapKeys f s
-- > where ls = keys s
mapKeysMonotonic :: (k1->k2) -> Map k1 a -> Map k2 a
mapKeysMonotonic f Tip = Tip
mapKeysMonotonic f (Bin sz k x l r) =
Bin sz (f k) x (mapKeysMonotonic f l) (mapKeysMonotonic f r)
{--------------------------------------------------------------------
Folds
--------------------------------------------------------------------}
-- | /O(n)/. Fold the values in the map, such that
-- @'fold' f z == 'Prelude.foldr' f z . 'elems'@.
-- For example,
--
-- > elems map = fold (:) [] map
--
fold :: (a -> b -> b) -> b -> Map k a -> b
fold f z m
= foldWithKey (\k x z -> f x z) z m
-- | /O(n)/. Fold the keys and values in the map, such that
-- @'foldWithKey' f z == 'Prelude.foldr' ('uncurry' f) z . 'toAscList'@.
-- For example,
--
-- > keys map = foldWithKey (\k x ks -> k:ks) [] map
--
foldWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b
foldWithKey f z t
= foldr f z t
-- | /O(n)/. In-order fold.
foldi :: (k -> a -> b -> b -> b) -> b -> Map k a -> b
foldi f z Tip = z
foldi f z (Bin _ kx x l r) = f kx x (foldi f z l) (foldi f z r)
-- | /O(n)/. Post-order fold.
foldr :: (k -> a -> b -> b) -> b -> Map k a -> b
foldr f z Tip = z
foldr f z (Bin _ kx x l r) = foldr f (f kx x (foldr f z r)) l
-- | /O(n)/. Pre-order fold.
foldl :: (b -> k -> a -> b) -> b -> Map k a -> b
foldl f z Tip = z
foldl f z (Bin _ kx x l r) = foldl f (f (foldl f z l) kx x) r
{--------------------------------------------------------------------
List variations
--------------------------------------------------------------------}
-- | /O(n)/.
-- Return all elements of the map in the ascending order of their keys.
elems :: Map k a -> [a]
elems m
= [x | (k,x) <- assocs m]
-- | /O(n)/. Return all keys of the map in ascending order.
keys :: Map k a -> [k]
keys m
= [k | (k,x) <- assocs m]
-- | /O(n)/. The set of all keys of the map.
keysSet :: Map k a -> Set.Set k
keysSet m = Set.fromDistinctAscList (keys m)
-- | /O(n)/. Return all key\/value pairs in the map in ascending key order.
assocs :: Map k a -> [(k,a)]
assocs m
= toList m
{--------------------------------------------------------------------
Lists
use [foldlStrict] to reduce demand on the control-stack
--------------------------------------------------------------------}
-- | /O(n*log n)/. Build a map from a list of key\/value pairs. See also 'fromAscList'.
fromList :: Ord k => [(k,a)] -> Map k a
fromList xs
= foldlStrict ins empty xs
where
ins t (k,x) = insert k x t
-- | /O(n*log n)/. Build a map from a list of key\/value pairs with a combining function. See also 'fromAscListWith'.
fromListWith :: Ord k => (a -> a -> a) -> [(k,a)] -> Map k a
fromListWith f xs
= fromListWithKey (\k x y -> f x y) xs
-- | /O(n*log n)/. Build a map from a list of key\/value pairs with a combining function. See also 'fromAscListWithKey'.
fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k,a)] -> Map k a
fromListWithKey f xs
= foldlStrict ins empty xs
where
ins t (k,x) = insertWithKey f k x t
-- | /O(n)/. Convert to a list of key\/value pairs.
toList :: Map k a -> [(k,a)]
toList t = toAscList t
-- | /O(n)/. Convert to an ascending list.
toAscList :: Map k a -> [(k,a)]
toAscList t = foldr (\k x xs -> (k,x):xs) [] t
-- | /O(n)/.
toDescList :: Map k a -> [(k,a)]
toDescList t = foldl (\xs k x -> (k,x):xs) [] t
{--------------------------------------------------------------------
Building trees from ascending/descending lists can be done in linear time.
Note that if [xs] is ascending that:
fromAscList xs == fromList xs
fromAscListWith f xs == fromListWith f xs
--------------------------------------------------------------------}
-- | /O(n)/. Build a map from an ascending list in linear time.
-- /The precondition (input list is ascending) is not checked./
fromAscList :: Eq k => [(k,a)] -> Map k a
fromAscList xs
= fromAscListWithKey (\k x y -> x) xs
-- | /O(n)/. Build a map from an ascending list in linear time with a combining function for equal keys.
-- /The precondition (input list is ascending) is not checked./
fromAscListWith :: Eq k => (a -> a -> a) -> [(k,a)] -> Map k a
fromAscListWith f xs
= fromAscListWithKey (\k x y -> f x y) xs
-- | /O(n)/. Build a map from an ascending list in linear time with a
-- combining function for equal keys.
-- /The precondition (input list is ascending) is not checked./
fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(k,a)] -> Map k a
fromAscListWithKey f xs
= fromDistinctAscList (combineEq f xs)
where
-- [combineEq f xs] combines equal elements with function [f] in an ordered list [xs]
combineEq f xs
= case xs of
[] -> []
[x] -> [x]
(x:xx) -> combineEq' x xx
combineEq' z [] = [z]
combineEq' z@(kz,zz) (x@(kx,xx):xs)
| kx==kz = let yy = f kx xx zz in combineEq' (kx,yy) xs
| otherwise = z:combineEq' x xs
-- | /O(n)/. Build a map from an ascending list of distinct elements in linear time.
-- /The precondition is not checked./
fromDistinctAscList :: [(k,a)] -> Map k a
fromDistinctAscList xs
= build const (length xs) xs
where
-- 1) use continutations so that we use heap space instead of stack space.
-- 2) special case for n==5 to build bushier trees.
build c 0 xs = c Tip xs
build c 5 xs = case xs of
((k1,x1):(k2,x2):(k3,x3):(k4,x4):(k5,x5):xx)
-> c (bin k4 x4 (bin k2 x2 (singleton k1 x1) (singleton k3 x3)) (singleton k5 x5)) xx
build c n xs = seq nr $ build (buildR nr c) nl xs
where
nl = n `div` 2
nr = n - nl - 1
buildR n c l ((k,x):ys) = build (buildB l k x c) n ys
buildB l k x c r zs = c (bin k x l r) zs
{--------------------------------------------------------------------
Utility functions that return sub-ranges of the original
tree. Some functions take a comparison function as argument to
allow comparisons against infinite values. A function [cmplo k]
should be read as [compare lo k].
[trim cmplo cmphi t] A tree that is either empty or where [cmplo k == LT]
and [cmphi k == GT] for the key [k] of the root.
[filterGt cmp t] A tree where for all keys [k]. [cmp k == LT]
[filterLt cmp t] A tree where for all keys [k]. [cmp k == GT]
[split k t] Returns two trees [l] and [r] where all keys
in [l] are <[k] and all keys in [r] are >[k].
[splitLookup k t] Just like [split] but also returns whether [k]
was found in the tree.
--------------------------------------------------------------------}
{--------------------------------------------------------------------
[trim lo hi t] trims away all subtrees that surely contain no
values between the range [lo] to [hi]. The returned tree is either
empty or the key of the root is between @lo@ and @hi@.
--------------------------------------------------------------------}
trim :: (k -> Ordering) -> (k -> Ordering) -> Map k a -> Map k a
trim cmplo cmphi Tip = Tip
trim cmplo cmphi t@(Bin sx kx x l r)
= case cmplo kx of
LT -> case cmphi kx of
GT -> t
le -> trim cmplo cmphi l
ge -> trim cmplo cmphi r
trimLookupLo :: Ord k => k -> (k -> Ordering) -> Map k a -> (Maybe (k,a), Map k a)
trimLookupLo lo cmphi Tip = (Nothing,Tip)
trimLookupLo lo cmphi t@(Bin sx kx x l r)
= case compare lo kx of
LT -> case cmphi kx of
GT -> (lookupAssoc lo t, t)
le -> trimLookupLo lo cmphi l
GT -> trimLookupLo lo cmphi r
EQ -> (Just (kx,x),trim (compare lo) cmphi r)
{--------------------------------------------------------------------
[filterGt k t] filter all keys >[k] from tree [t]
[filterLt k t] filter all keys <[k] from tree [t]
--------------------------------------------------------------------}
filterGt :: Ord k => (k -> Ordering) -> Map k a -> Map k a
filterGt cmp Tip = Tip
filterGt cmp (Bin sx kx x l r)
= case cmp kx of
LT -> join kx x (filterGt cmp l) r
GT -> filterGt cmp r
EQ -> r
filterLt :: Ord k => (k -> Ordering) -> Map k a -> Map k a
filterLt cmp Tip = Tip
filterLt cmp (Bin sx kx x l r)
= case cmp kx of
LT -> filterLt cmp l
GT -> join kx x l (filterLt cmp r)
EQ -> l
{--------------------------------------------------------------------
Split
--------------------------------------------------------------------}
-- | /O(log n)/. The expression (@'split' k map@) is a pair @(map1,map2)@ where
-- the keys in @map1@ are smaller than @k@ and the keys in @map2@ larger than @k@. Any key equal to @k@ is found in neither @map1@ nor @map2@.
split :: Ord k => k -> Map k a -> (Map k a,Map k a)
split k Tip = (Tip,Tip)
split k (Bin sx kx x l r)
= case compare k kx of
LT -> let (lt,gt) = split k l in (lt,join kx x gt r)
GT -> let (lt,gt) = split k r in (join kx x l lt,gt)
EQ -> (l,r)
-- | /O(log n)/. The expression (@'splitLookup' k map@) splits a map just
-- like 'split' but also returns @'lookup' k map@.
splitLookup :: Ord k => k -> Map k a -> (Map k a,Maybe a,Map k a)
splitLookup k Tip = (Tip,Nothing,Tip)
splitLookup k (Bin sx kx x l r)
= case compare k kx of
LT -> let (lt,z,gt) = splitLookup k l in (lt,z,join kx x gt r)
GT -> let (lt,z,gt) = splitLookup k r in (join kx x l lt,z,gt)
EQ -> (l,Just x,r)
-- | /O(log n)/.
splitLookupWithKey :: Ord k => k -> Map k a -> (Map k a,Maybe (k,a),Map k a)
splitLookupWithKey k Tip = (Tip,Nothing,Tip)
splitLookupWithKey k (Bin sx kx x l r)
= case compare k kx of
LT -> let (lt,z,gt) = splitLookupWithKey k l in (lt,z,join kx x gt r)
GT -> let (lt,z,gt) = splitLookupWithKey k r in (join kx x l lt,z,gt)
EQ -> (l,Just (kx, x),r)
-- | /O(log n)/. Performs a 'split' but also returns whether the pivot
-- element was found in the original set.
splitMember :: Ord k => k -> Map k a -> (Map k a,Bool,Map k a)
splitMember x t = let (l,m,r) = splitLookup x t in
(l,maybe False (const True) m,r)
{--------------------------------------------------------------------
Utility functions that maintain the balance properties of the tree.
All constructors assume that all values in [l] < [k] and all values
in [r] > [k], and that [l] and [r] are valid trees.
In order of sophistication:
[Bin sz k x l r] The type constructor.
[bin k x l r] Maintains the correct size, assumes that both [l]
and [r] are balanced with respect to each other.
[balance k x l r] Restores the balance and size.
Assumes that the original tree was balanced and
that [l] or [r] has changed by at most one element.
[join k x l r] Restores balance and size.
Furthermore, we can construct a new tree from two trees. Both operations
assume that all values in [l] < all values in [r] and that [l] and [r]
are valid:
[glue l r] Glues [l] and [r] together. Assumes that [l] and
[r] are already balanced with respect to each other.
[merge l r] Merges two trees and restores balance.
Note: in contrast to Adam's paper, we use (<=) comparisons instead
of (<) comparisons in [join], [merge] and [balance].
Quickcheck (on [difference]) showed that this was necessary in order
to maintain the invariants. It is quite unsatisfactory that I haven't
been able to find out why this is actually the case! Fortunately, it
doesn't hurt to be a bit more conservative.
--------------------------------------------------------------------}
{--------------------------------------------------------------------
Join
--------------------------------------------------------------------}
join :: Ord k => k -> a -> Map k a -> Map k a -> Map k a
join kx x Tip r = insertMin kx x r
join kx x l Tip = insertMax kx x l
join kx x l@(Bin sizeL ky y ly ry) r@(Bin sizeR kz z lz rz)
| delta*sizeL <= sizeR = balance kz z (join kx x l lz) rz
| delta*sizeR <= sizeL = balance ky y ly (join kx x ry r)
| otherwise = bin kx x l r
-- insertMin and insertMax don't perform potentially expensive comparisons.
insertMax,insertMin :: k -> a -> Map k a -> Map k a
insertMax kx x t
= case t of
Tip -> singleton kx x
Bin sz ky y l r
-> balance ky y l (insertMax kx x r)
insertMin kx x t
= case t of
Tip -> singleton kx x
Bin sz ky y l r
-> balance ky y (insertMin kx x l) r
{--------------------------------------------------------------------
[merge l r]: merges two trees.
--------------------------------------------------------------------}
merge :: Map k a -> Map k a -> Map k a
merge Tip r = r
merge l Tip = l
merge l@(Bin sizeL kx x lx rx) r@(Bin sizeR ky y ly ry)
| delta*sizeL <= sizeR = balance ky y (merge l ly) ry
| delta*sizeR <= sizeL = balance kx x lx (merge rx r)
| otherwise = glue l r
{--------------------------------------------------------------------
[glue l r]: glues two trees together.
Assumes that [l] and [r] are already balanced with respect to each other.
--------------------------------------------------------------------}
glue :: Map k a -> Map k a -> Map k a
glue Tip r = r
glue l Tip = l
glue l r
| size l > size r = let ((km,m),l') = deleteFindMax l in balance km m l' r
| otherwise = let ((km,m),r') = deleteFindMin r in balance km m l r'
-- | /O(log n)/. Delete and find the minimal element.
deleteFindMin :: Map k a -> ((k,a),Map k a)
deleteFindMin t
= case t of
Bin _ k x Tip r -> ((k,x),r)
Bin _ k x l r -> let (km,l') = deleteFindMin l in (km,balance k x l' r)
Tip -> (error "Map.deleteFindMin: can not return the minimal element of an empty map", Tip)
-- | /O(log n)/. Delete and find the maximal element.
deleteFindMax :: Map k a -> ((k,a),Map k a)
deleteFindMax t
= case t of
Bin _ k x l Tip -> ((k,x),l)
Bin _ k x l r -> let (km,r') = deleteFindMax r in (km,balance k x l r')
Tip -> (error "Map.deleteFindMax: can not return the maximal element of an empty map", Tip)
{--------------------------------------------------------------------
[balance l x r] balances two trees with value x.
The sizes of the trees should balance after decreasing the
size of one of them. (a rotation).
[delta] is the maximal relative difference between the sizes of
two trees, it corresponds with the [w] in Adams' paper.
[ratio] is the ratio between an outer and inner sibling of the
heavier subtree in an unbalanced setting. It determines
whether a double or single rotation should be performed
to restore balance. It is correspondes with the inverse
of $\alpha$ in Adam's article.
Note that:
- [delta] should be larger than 4.646 with a [ratio] of 2.
- [delta] should be larger than 3.745 with a [ratio] of 1.534.
- A lower [delta] leads to a more 'perfectly' balanced tree.
- A higher [delta] performs less rebalancing.
- Balancing is automatic for random data and a balancing
scheme is only necessary to avoid pathological worst cases.
Almost any choice will do, and in practice, a rather large
[delta] may perform better than smaller one.
Note: in contrast to Adam's paper, we use a ratio of (at least) [2]
to decide whether a single or double rotation is needed. Allthough
he actually proves that this ratio is needed to maintain the
invariants, his implementation uses an invalid ratio of [1].
--------------------------------------------------------------------}
delta,ratio :: Int
delta = 5
ratio = 2
balance :: k -> a -> Map k a -> Map k a -> Map k a
balance k x l r
| sizeL + sizeR <= 1 = Bin sizeX k x l r
| sizeR >= delta*sizeL = rotateL k x l r
| sizeL >= delta*sizeR = rotateR k x l r
| otherwise = Bin sizeX k x l r
where
sizeL = size l
sizeR = size r
sizeX = sizeL + sizeR + 1
-- rotate
rotateL k x l r@(Bin _ _ _ ly ry)
| size ly < ratio*size ry = singleL k x l r
| otherwise = doubleL k x l r
rotateR k x l@(Bin _ _ _ ly ry) r
| size ry < ratio*size ly = singleR k x l r
| otherwise = doubleR k x l r
-- basic rotations
singleL k1 x1 t1 (Bin _ k2 x2 t2 t3) = bin k2 x2 (bin k1 x1 t1 t2) t3
singleR k1 x1 (Bin _ k2 x2 t1 t2) t3 = bin k2 x2 t1 (bin k1 x1 t2 t3)
doubleL k1 x1 t1 (Bin _ k2 x2 (Bin _ k3 x3 t2 t3) t4) = bin k3 x3 (bin k1 x1 t1 t2) (bin k2 x2 t3 t4)
doubleR k1 x1 (Bin _ k2 x2 t1 (Bin _ k3 x3 t2 t3)) t4 = bin k3 x3 (bin k2 x2 t1 t2) (bin k1 x1 t3 t4)
{--------------------------------------------------------------------
The bin constructor maintains the size of the tree
--------------------------------------------------------------------}
bin :: k -> a -> Map k a -> Map k a -> Map k a
bin k x l r
= Bin (size l + size r + 1) k x l r
{--------------------------------------------------------------------
Eq converts the tree to a list. In a lazy setting, this
actually seems one of the faster methods to compare two trees
and it is certainly the simplest :-)
--------------------------------------------------------------------}
instance (Eq k,Eq a) => Eq (Map k a) where
t1 == t2 = (size t1 == size t2) && (toAscList t1 == toAscList t2)
{--------------------------------------------------------------------
Ord
--------------------------------------------------------------------}
instance (Ord k, Ord v) => Ord (Map k v) where
compare m1 m2 = compare (toAscList m1) (toAscList m2)
{--------------------------------------------------------------------
Functor
--------------------------------------------------------------------}
instance Functor (Map k) where
fmap f m = map f m
instance Traversable (Map k) where
traverse f Tip = pure Tip
traverse f (Bin s k v l r)
= flip (Bin s k) <$> traverse f l <*> f v <*> traverse f r
instance Foldable (Map k) where
foldMap _f Tip = mempty
foldMap f (Bin _s _k v l r)
= foldMap f l `mappend` f v `mappend` foldMap f r
{--------------------------------------------------------------------
Read
--------------------------------------------------------------------}
instance (Ord k, Read k, Read e) => Read (Map k e) where
#ifdef __GLASGOW_HASKELL__
readPrec = parens $ prec 10 $ do
Ident "fromList" <- lexP
xs <- readPrec
return (fromList xs)
readListPrec = readListPrecDefault
#else
readsPrec p = readParen (p > 10) $ \ r -> do
("fromList",s) <- lex r
(xs,t) <- reads s
return (fromList xs,t)
#endif
-- parses a pair of things with the syntax a:=b
readPair :: (Read a, Read b) => ReadS (a,b)
readPair s = do (a, ct1) <- reads s
(":=", ct2) <- lex ct1
(b, ct3) <- reads ct2
return ((a,b), ct3)
{--------------------------------------------------------------------
Show
--------------------------------------------------------------------}
instance (Show k, Show a) => Show (Map k a) where
showsPrec d m = showParen (d > 10) $
showString "fromList " . shows (toList m)
showMap :: (Show k,Show a) => [(k,a)] -> ShowS
showMap []
= showString "{}"
showMap (x:xs)
= showChar '{' . showElem x . showTail xs
where
showTail [] = showChar '}'
showTail (x:xs) = showString ", " . showElem x . showTail xs
showElem (k,x) = shows k . showString " := " . shows x
-- | /O(n)/. Show the tree that implements the map. The tree is shown
-- in a compressed, hanging format.
showTree :: (Show k,Show a) => Map k a -> String
showTree m
= showTreeWith showElem True False m
where
showElem k x = show k ++ ":=" ++ show x
{- | /O(n)/. The expression (@'showTreeWith' showelem hang wide map@) shows
the tree that implements the map. Elements are shown using the @showElem@ function. If @hang@ is
'True', a /hanging/ tree is shown otherwise a rotated tree is shown. If
@wide@ is 'True', an extra wide version is shown.
> Map> let t = fromDistinctAscList [(x,()) | x <- [1..5]]
> Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True False t
> (4,())
> +--(2,())
> | +--(1,())
> | +--(3,())
> +--(5,())
>
> Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True True t
> (4,())
> |
> +--(2,())
> | |
> | +--(1,())
> | |
> | +--(3,())
> |
> +--(5,())
>
> Map> putStrLn $ showTreeWith (\k x -> show (k,x)) False True t
> +--(5,())
> |
> (4,())
> |
> | +--(3,())
> | |
> +--(2,())
> |
> +--(1,())
-}
showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> String
showTreeWith showelem hang wide t
| hang = (showsTreeHang showelem wide [] t) ""
| otherwise = (showsTree showelem wide [] [] t) ""
showsTree :: (k -> a -> String) -> Bool -> [String] -> [String] -> Map k a -> ShowS
showsTree showelem wide lbars rbars t
= case t of
Tip -> showsBars lbars . showString "|\n"
Bin sz kx x Tip Tip
-> showsBars lbars . showString (showelem kx x) . showString "\n"
Bin sz kx x l r
-> showsTree showelem wide (withBar rbars) (withEmpty rbars) r .
showWide wide rbars .
showsBars lbars . showString (showelem kx x) . showString "\n" .
showWide wide lbars .
showsTree showelem wide (withEmpty lbars) (withBar lbars) l
showsTreeHang :: (k -> a -> String) -> Bool -> [String] -> Map k a -> ShowS
showsTreeHang showelem wide bars t
= case t of
Tip -> showsBars bars . showString "|\n"
Bin sz kx x Tip Tip
-> showsBars bars . showString (showelem kx x) . showString "\n"
Bin sz kx x l r
-> showsBars bars . showString (showelem kx x) . showString "\n" .
showWide wide bars .
showsTreeHang showelem wide (withBar bars) l .
showWide wide bars .
showsTreeHang showelem wide (withEmpty bars) r
showWide wide bars
| wide = showString (concat (reverse bars)) . showString "|\n"
| otherwise = id
showsBars :: [String] -> ShowS
showsBars bars
= case bars of
[] -> id
_ -> showString (concat (reverse (tail bars))) . showString node
node = "+--"
withBar bars = "| ":bars
withEmpty bars = " ":bars
{--------------------------------------------------------------------
Typeable
--------------------------------------------------------------------}
#include "Typeable.h"
INSTANCE_TYPEABLE2(Map,mapTc,"Map")
{--------------------------------------------------------------------
Assertions
--------------------------------------------------------------------}
-- | /O(n)/. Test if the internal map structure is valid.
valid :: Ord k => Map k a -> Bool
valid t
= balanced t && ordered t && validsize t
ordered t
= bounded (const True) (const True) t
where
bounded lo hi t
= case t of
Tip -> True
Bin sz kx x l r -> (lo kx) && (hi kx) && bounded lo (<kx) l && bounded (>kx) hi r
-- | Exported only for "Debug.QuickCheck"
balanced :: Map k a -> Bool
balanced t
= case t of
Tip -> True
Bin sz kx x l r -> (size l + size r <= 1 || (size l <= delta*size r && size r <= delta*size l)) &&
balanced l && balanced r
validsize t
= (realsize t == Just (size t))
where
realsize t
= case t of
Tip -> Just 0
Bin sz kx x l r -> case (realsize l,realsize r) of
(Just n,Just m) | n+m+1 == sz -> Just sz
other -> Nothing
{--------------------------------------------------------------------
Utilities
--------------------------------------------------------------------}
foldlStrict f z xs
= case xs of
[] -> z
(x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)
{-
{--------------------------------------------------------------------
Testing
--------------------------------------------------------------------}
testTree xs = fromList [(x,"*") | x <- xs]
test1 = testTree [1..20]
test2 = testTree [30,29..10]
test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]
{--------------------------------------------------------------------
QuickCheck
--------------------------------------------------------------------}
qcheck prop
= check config prop
where
config = Config
{ configMaxTest = 500
, configMaxFail = 5000
, configSize = \n -> (div n 2 + 3)
, configEvery = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
}
{--------------------------------------------------------------------
Arbitrary, reasonably balanced trees
--------------------------------------------------------------------}
instance (Enum k,Arbitrary a) => Arbitrary (Map k a) where
arbitrary = sized (arbtree 0 maxkey)
where maxkey = 10000
arbtree :: (Enum k,Arbitrary a) => Int -> Int -> Int -> Gen (Map k a)
arbtree lo hi n
| n <= 0 = return Tip
| lo >= hi = return Tip
| otherwise = do{ x <- arbitrary
; i <- choose (lo,hi)
; m <- choose (1,30)
; let (ml,mr) | m==(1::Int)= (1,2)
| m==2 = (2,1)
| m==3 = (1,1)
| otherwise = (2,2)
; l <- arbtree lo (i-1) (n `div` ml)
; r <- arbtree (i+1) hi (n `div` mr)
; return (bin (toEnum i) x l r)
}
{--------------------------------------------------------------------
Valid tree's
--------------------------------------------------------------------}
forValid :: (Show k,Enum k,Show a,Arbitrary a,Testable b) => (Map k a -> b) -> Property
forValid f
= forAll arbitrary $ \t ->
-- classify (balanced t) "balanced" $
classify (size t == 0) "empty" $
classify (size t > 0 && size t <= 10) "small" $
classify (size t > 10 && size t <= 64) "medium" $
classify (size t > 64) "large" $
balanced t ==> f t
forValidIntTree :: Testable a => (Map Int Int -> a) -> Property
forValidIntTree f
= forValid f
forValidUnitTree :: Testable a => (Map Int () -> a) -> Property
forValidUnitTree f
= forValid f
prop_Valid
= forValidUnitTree $ \t -> valid t
{--------------------------------------------------------------------
Single, Insert, Delete
--------------------------------------------------------------------}
prop_Single :: Int -> Int -> Bool
prop_Single k x
= (insert k x empty == singleton k x)
prop_InsertValid :: Int -> Property
prop_InsertValid k
= forValidUnitTree $ \t -> valid (insert k () t)
prop_InsertDelete :: Int -> Map Int () -> Property
prop_InsertDelete k t
= (lookup k t == Nothing) ==> delete k (insert k () t) == t
prop_DeleteValid :: Int -> Property
prop_DeleteValid k
= forValidUnitTree $ \t ->
valid (delete k (insert k () t))
{--------------------------------------------------------------------
Balance
--------------------------------------------------------------------}
prop_Join :: Int -> Property
prop_Join k
= forValidUnitTree $ \t ->
let (l,r) = split k t
in valid (join k () l r)
prop_Merge :: Int -> Property
prop_Merge k
= forValidUnitTree $ \t ->
let (l,r) = split k t
in valid (merge l r)
{--------------------------------------------------------------------
Union
--------------------------------------------------------------------}
prop_UnionValid :: Property
prop_UnionValid
= forValidUnitTree $ \t1 ->
forValidUnitTree $ \t2 ->
valid (union t1 t2)
prop_UnionInsert :: Int -> Int -> Map Int Int -> Bool
prop_UnionInsert k x t
= union (singleton k x) t == insert k x t
prop_UnionAssoc :: Map Int Int -> Map Int Int -> Map Int Int -> Bool
prop_UnionAssoc t1 t2 t3
= union t1 (union t2 t3) == union (union t1 t2) t3
prop_UnionComm :: Map Int Int -> Map Int Int -> Bool
prop_UnionComm t1 t2
= (union t1 t2 == unionWith (\x y -> y) t2 t1)
prop_UnionWithValid
= forValidIntTree $ \t1 ->
forValidIntTree $ \t2 ->
valid (unionWithKey (\k x y -> x+y) t1 t2)
prop_UnionWith :: [(Int,Int)] -> [(Int,Int)] -> Bool
prop_UnionWith xs ys
= sum (elems (unionWith (+) (fromListWith (+) xs) (fromListWith (+) ys)))
== (sum (Prelude.map snd xs) + sum (Prelude.map snd ys))
prop_DiffValid
= forValidUnitTree $ \t1 ->
forValidUnitTree $ \t2 ->
valid (difference t1 t2)
prop_Diff :: [(Int,Int)] -> [(Int,Int)] -> Bool
prop_Diff xs ys
= List.sort (keys (difference (fromListWith (+) xs) (fromListWith (+) ys)))
== List.sort ((List.\\) (nub (Prelude.map fst xs)) (nub (Prelude.map fst ys)))
prop_IntValid
= forValidUnitTree $ \t1 ->
forValidUnitTree $ \t2 ->
valid (intersection t1 t2)
prop_Int :: [(Int,Int)] -> [(Int,Int)] -> Bool
prop_Int xs ys
= List.sort (keys (intersection (fromListWith (+) xs) (fromListWith (+) ys)))
== List.sort (nub ((List.intersect) (Prelude.map fst xs) (Prelude.map fst ys)))
{--------------------------------------------------------------------
Lists
--------------------------------------------------------------------}
prop_Ordered
= forAll (choose (5,100)) $ \n ->
let xs = [(x,()) | x <- [0..n::Int]]
in fromAscList xs == fromList xs
prop_List :: [Int] -> Bool
prop_List xs
= (sort (nub xs) == [x | (x,()) <- toList (fromList [(x,()) | x <- xs])])
-}
|
