+++ /dev/null
-{-# OPTIONS_GHC -fno-bang-patterns #-}
-
------------------------------------------------------------------------------
--- |
--- Module : Data.Map
--- Copyright : (c) Daan Leijen 2002
--- License : BSD-style
--- Maintainer : libraries@haskell.org
--- 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 -> balance kx x (updateAt f i l) r
- GT -> balance kx x l (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])])
--}
projects / ghc-base.git / blobdiff