Dear libraries, for Data.Map, I needed a "unionWithMaybe" function for my sparse system of linear equations (unionWithMaybe :: Ord k => (a -> a -> Maybe a) -> Map k a -> Map k a -> Map k a). My usage: (.+.) = Map.unionWithMaybe (\a b->case a+b of {0->Nothing;s->Just s}) (I do not think Map.unionWithMaybe can be expressed in terms of other functions without loosing performance, so I recon it would be a nice addition to the library.) I built this function myself by modifying the Data.Map implementation, but it would be nice to see it in the official version of Data.Map. Here is the file including the modifications I made to it. Feel free to use it, I will agree to whatever license you need to make it public. (PS: this is the first time I return modified source code to the maintainer, feel free to instruct me on how to do this in the future.) Sincerely, Sebastiaan J.C. Joosten {-# OPTIONS_GHC -XNoBangPatterns -cpp -XStandaloneDeriving -XDeriveDataTypeable #-} ----------------------------------------------------------------------------- -- | -- Module : Data.Map -- Copyright : (c) Daan Leijen 2002 -- (c) Andriy Palamarchuk 2008 -- 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'. -- -- Operation comments contain the operation time complexity in -- the Big-O notation http://en.wikipedia.org/wiki/Big_O_notation. ----------------------------------------------------------------------------- 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 , unionWithMaybe , unionWithMaybeKey , unions , unionsWith -- ** Difference , difference , differenceWith , differenceWithKey -- ** Intersection , intersection , intersectionWith , intersectionWithKey -- * Traversal -- ** Map , map , mapWithKey , mapAccum , mapAccumWithKey , mapAccumRWithKey , mapKeys , mapKeysWith , mapKeysMonotonic -- ** Fold , fold , foldWithKey , foldrWithKey , foldlWithKey -- * Conversion , elems , keys , keysSet , assocs -- ** Lists , toList , fromList , fromListWith , fromListWithKey -- ** Ordered lists , toAscList , toDescList , fromAscList , fromAscListWith , fromAscListWithKey , fromDistinctAscList -- * Filter , filter , filterWithKey , partition , partitionWithKey , mapMaybe , mapWithMaybeKey , 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,null) import qualified Data.Set as Set import qualified Data.List as List import Data.Monoid (Monoid(..)) import Control.Applicative (Applicative(..), (<$>)) import Data.Traversable (Traversable(traverse)) import Data.Foldable (Foldable(foldMap)) #ifndef __GLASGOW_HASKELL__ import Data.Typeable ( Typeable, typeOf, typeOfDefault , Typeable1, typeOf1, typeOf1Default) #endif import Data.Typeable (Typeable2(..), TyCon, mkTyCon, mkTyConApp) {- -- for quick check import qualified Prelude import qualified List import Debug.QuickCheck import List(nub,sort) -} #if __GLASGOW_HASKELL__ import Text.Read import Data.Data (Data(..), mkNoRepType, gcast2) #endif {-------------------------------------------------------------------- Operators --------------------------------------------------------------------} infixl 9 !,\\ -- -- | /O(log n)/. Find the value at a key. -- Calls 'error' when the element can not be found. -- -- > fromList [(5,'a'), (3,'b')] ! 1 Error: element not in the map -- > fromList [(5,'a'), (3,'b')] ! 5 == 'a' (!) :: Ord k => Map k a -> k -> a m ! k = find k m -- | Same as '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 m = z fromList `f` toList m toConstr _ = error "toConstr" gunfold _ _ = error "gunfold" dataTypeOf _ = mkNoRepType "Data.Map.Map" dataCast2 f = gcast2 f #endif {-------------------------------------------------------------------- Query --------------------------------------------------------------------} -- | /O(1)/. Is the map empty? -- -- > Data.Map.null (empty) == True -- > Data.Map.null (singleton 1 'a') == False null :: Map k a -> Bool null t = case t of Tip -> True Bin {} -> False -- | /O(1)/. The number of elements in the map. -- -- > size empty == 0 -- > size (singleton 1 'a') == 1 -- > size (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3 size :: Map k a -> Int size t = case t of Tip -> 0 Bin sz _ _ _ _ -> sz -- | /O(log n)/. Lookup the value at a key in the map. -- -- The function will return the corresponding value as @('Just' value)@, -- or 'Nothing' if the key isn't in the map. -- -- An example of using @lookup@: -- -- > import Prelude hiding (lookup) -- > import Data.Map -- > -- > employeeDept = fromList([("John","Sales"), ("Bob","IT")]) -- > deptCountry = fromList([("IT","USA"), ("Sales","France")]) -- > countryCurrency = fromList([("USA", "Dollar"), ("France", "Euro")]) -- > -- > employeeCurrency :: String -> Maybe String -- > employeeCurrency name = do -- > dept <- lookup name employeeDept -- > country <- lookup dept deptCountry -- > lookup country countryCurrency -- > -- > main = do -- > putStrLn $ "John's currency: " ++ (show (employeeCurrency "John")) -- > putStrLn $ "Pete's currency: " ++ (show (employeeCurrency "Pete")) -- -- The output of this program: -- -- > John's currency: Just "Euro" -- > Pete's currency: Nothing lookup :: Ord k => k -> Map k a -> Maybe a lookup k t = case t of Tip -> Nothing Bin _ 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 _ 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? See also 'notMember'. -- -- > member 5 (fromList [(5,'a'), (3,'b')]) == True -- > member 1 (fromList [(5,'a'), (3,'b')]) == False member :: Ord k => k -> Map k a -> Bool member k m = case lookup k m of Nothing -> False Just _ -> True -- | /O(log n)/. Is the key not a member of the map? See also 'member'. -- -- > notMember 5 (fromList [(5,'a'), (3,'b')]) == False -- > notMember 1 (fromList [(5,'a'), (3,'b')]) == True 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 default value @def@ -- when the key is not in the map. -- -- > findWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x' -- > findWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a' 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 == fromList [] -- > size empty == 0 empty :: Map k a empty = Tip -- | /O(1)/. A map with a single element. -- -- > singleton 1 'a' == fromList [(1, 'a')] -- > size (singleton 1 'a') == 1 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. 'insert' is equivalent to -- @'insertWith' 'const'@. -- -- > insert 5 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'x')] -- > insert 7 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'a'), (7, 'x')] -- > insert 5 'x' empty == singleton 5 'x' 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 function, combining new value and old value. -- @'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 (++) 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "xxxa")] -- > insertWith (++) 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")] -- > insertWith (++) 5 "xxx" empty == singleton 5 "xxx" insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a insertWith f k x m = insertWithKey (\_ 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' (\_ x' y' -> f x' y') k x m -- | /O(log n)/. Insert with a function, combining key, new value and old value. -- @'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'. -- -- > let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value -- > insertWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:xxx|a")] -- > insertWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")] -- > insertWithKey f 5 "xxx" empty == singleton 5 "xxx" 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)/. Combines insert operation with old value retrieval. -- 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@). -- -- > let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value -- > insertLookupWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:xxx|a")]) -- > insertLookupWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a"), (7, "xxx")]) -- > insertLookupWithKey f 5 "xxx" empty == (Nothing, singleton 5 "xxx") -- -- This is how to define @insertLookup@ using @insertLookupWithKey@: -- -- > let insertLookup kx x t = insertLookupWithKey (\_ a _ -> a) kx x t -- > insertLookup 5 "x" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "x")]) -- > insertLookup 7 "x" (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a"), (7, "x")]) 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 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" -- > delete 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] -- > delete 5 empty == empty delete :: Ord k => k -> Map k a -> Map k a delete k t = case t of Tip -> Tip Bin _ 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)/. Update a value at a specific key with the result of the provided function. -- When the key is not -- a member of the map, the original map is returned. -- -- > adjust ("new " ++) 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")] -- > adjust ("new " ++) 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] -- > adjust ("new " ++) 7 empty == empty adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k a adjust f k m = adjustWithKey (\_ 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. -- -- > let f key x = (show key) ++ ":new " ++ x -- > adjustWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")] -- > adjustWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] -- > adjustWithKey f 7 empty == empty 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@. -- -- > let f x = if x == "a" then Just "new a" else Nothing -- > update f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")] -- > update f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] -- > update f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k a update f k m = updateWithKey (\_ 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@. -- -- > let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing -- > updateWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")] -- > updateWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] -- > updateWithKey f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" 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. See also 'updateWithKey'. -- The function returns changed value, if it is updated. -- Returns the original key value if the map entry is deleted. -- -- > let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing -- > updateLookupWithKey f 5 (fromList [(5,"a"), (3,"b")]) == (Just "5:new a", fromList [(3, "b"), (5, "5:new a")]) -- > updateLookupWithKey f 7 (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a")]) -- > updateLookupWithKey f 3 (fromList [(5,"a"), (3,"b")]) == (Just "b", singleton 5 "a") 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)@. -- -- > let f _ = Nothing -- > alter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] -- > alter f 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" -- > -- > let f _ = Just "c" -- > alter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "c")] -- > alter f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "c")] 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 2 (fromList [(5,"a"), (3,"b")]) Error: element is not in the map -- > findIndex 3 (fromList [(5,"a"), (3,"b")]) == 0 -- > findIndex 5 (fromList [(5,"a"), (3,"b")]) == 1 -- > findIndex 6 (fromList [(5,"a"), (3,"b")]) Error: element is not in 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. -- -- > isJust (lookupIndex 2 (fromList [(5,"a"), (3,"b")])) == False -- > fromJust (lookupIndex 3 (fromList [(5,"a"), (3,"b")])) == 0 -- > fromJust (lookupIndex 5 (fromList [(5,"a"), (3,"b")])) == 1 -- > isJust (lookupIndex 6 (fromList [(5,"a"), (3,"b")])) == False lookupIndex :: Ord k => k -> Map k a -> Maybe Int lookupIndex k t = f 0 t where f _ Tip = Nothing f idx (Bin _ kx _ l r) = case compare k kx of LT -> f idx l GT -> f (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 0 (fromList [(5,"a"), (3,"b")]) == (3,"b") -- > elemAt 1 (fromList [(5,"a"), (3,"b")]) == (5, "a") -- > elemAt 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range elemAt :: Int -> Map k a -> (k,a) elemAt _ 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 (\ _ _ -> Just "x") 0 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "x"), (5, "a")] -- > updateAt (\ _ _ -> Just "x") 1 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "x")] -- > updateAt (\ _ _ -> Just "x") 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range -- > updateAt (\ _ _ -> Just "x") (-1) (fromList [(5,"a"), (3,"b")]) Error: index out of range -- > updateAt (\_ _ -> Nothing) 0 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" -- > updateAt (\_ _ -> Nothing) 1 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" -- > updateAt (\_ _ -> Nothing) 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range -- > updateAt (\_ _ -> Nothing) (-1) (fromList [(5,"a"), (3,"b")]) Error: index out of range updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a updateAt _ _ 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 0 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" -- > deleteAt 1 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" -- > deleteAt 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range -- > deleteAt (-1) (fromList [(5,"a"), (3,"b")]) Error: index out of range deleteAt :: Int -> Map k a -> Map k a deleteAt i m = updateAt (\_ _ -> Nothing) i m {-------------------------------------------------------------------- Minimal, Maximal --------------------------------------------------------------------} -- | /O(log n)/. The minimal key of the map. Calls 'error' is the map is empty. -- -- > findMin (fromList [(5,"a"), (3,"b")]) == (3,"b") -- > findMin empty Error: empty map has no minimal element findMin :: Map k a -> (k,a) findMin (Bin _ kx x Tip _) = (kx,x) findMin (Bin _ _ _ l _) = findMin l findMin Tip = error "Map.findMin: empty map has no minimal element" -- | /O(log n)/. The maximal key of the map. Calls 'error' is the map is empty. -- -- > findMax (fromList [(5,"a"), (3,"b")]) == (5,"a") -- > findMax empty Error: empty map has no maximal element findMax :: Map k a -> (k,a) findMax (Bin _ kx x _ Tip) = (kx,x) findMax (Bin _ _ _ _ r) = findMax r findMax Tip = error "Map.findMax: empty map has no maximal element" -- | /O(log n)/. Delete the minimal key. Returns an empty map if the map is empty. -- -- > deleteMin (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(5,"a"), (7,"c")] -- > deleteMin empty == empty deleteMin :: Map k a -> Map k a deleteMin (Bin _ _ _ 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. Returns an empty map if the map is empty. -- -- > deleteMax (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(3,"b"), (5,"a")] -- > deleteMax empty == empty deleteMax :: Map k a -> Map k a deleteMax (Bin _ _ _ 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 -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "Xb"), (5, "a")] -- > updateMin (\ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" updateMin :: (a -> Maybe a) -> Map k a -> Map k a updateMin f m = updateMinWithKey (\_ x -> f x) m -- | /O(log n)/. Update the value at the maximal key. -- -- > updateMax (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "Xa")] -- > updateMax (\ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" updateMax :: (a -> Maybe a) -> Map k a -> Map k a updateMax f m = updateMaxWithKey (\_ x -> f x) m -- | /O(log n)/. Update the value at the minimal key. -- -- > updateMinWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"3:b"), (5,"a")] -- > updateMinWithKey (\ _ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" 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 _ 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 -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"b"), (5,"5:a")] -- > updateMaxWithKey (\ _ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" 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 _ 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 of that element, or 'Nothing' if passed an empty map. -- -- > minViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((3,"b"), singleton 5 "a") -- > minViewWithKey empty == Nothing minViewWithKey :: Map k a -> Maybe ((k,a), Map k a) minViewWithKey Tip = Nothing minViewWithKey x = Just (deleteFindMin x) -- | /O(log n)/. Retrieves the maximal (key,value) pair of the map, and -- the map stripped of that element, or 'Nothing' if passed an empty map. -- -- > maxViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((5,"a"), singleton 3 "b") -- > maxViewWithKey empty == Nothing maxViewWithKey :: Map k a -> Maybe ((k,a), Map k a) maxViewWithKey Tip = Nothing maxViewWithKey x = Just (deleteFindMax x) -- | /O(log n)/. Retrieves the value associated with minimal key of the -- map, and the map stripped of that element, or 'Nothing' if passed an -- empty map. -- -- > minView (fromList [(5,"a"), (3,"b")]) == Just ("b", singleton 5 "a") -- > minView empty == Nothing minView :: Map k a -> Maybe (a, Map k a) minView Tip = Nothing minView x = Just (first snd $ deleteFindMin x) -- | /O(log n)/. Retrieves the value associated with maximal key of the -- map, and the map stripped of that element, or 'Nothing' if passed an -- -- > maxView (fromList [(5,"a"), (3,"b")]) == Just ("a", singleton 3 "b") -- > maxView empty == Nothing maxView :: Map k a -> Maybe (a, Map k a) maxView Tip = Nothing maxView x = Just (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 [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])] -- > == fromList [(3, "b"), (5, "a"), (7, "C")] -- > unions [(fromList [(5, "A3"), (3, "B3")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "a"), (3, "b")])] -- > == fromList [(3, "B3"), (5, "A3"), (7, "C")] 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 (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])] -- > == fromList [(3, "bB3"), (5, "aAA3"), (7, "C")] 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 (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "a"), (7, "C")] 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 :: Ord a => (a -> Ordering) -> (a -> Ordering) -> Map a b -> Map a b -> Map a b hedgeUnionL _ _ 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 {- XXX unused code -- right-biased hedge union hedgeUnionR :: Ord a => (a -> Ordering) -> (a -> Ordering) -> Map a b -> Map a b -> Map a b hedgeUnionR _ _ 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 (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")] unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a unionWith f m1 m2 = unionWithKey (\_ x y -> f x y) m1 m2 unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a unionWithKey _ Tip t2 = t2 unionWithKey _ t1 Tip = t1 unionWithKey f t1 t2 = hedgeUnionWithKey f (const LT) (const GT) t1 t2 hedgeUnionWithKey :: Ord a => (a -> b -> b -> b) -> (a -> Ordering) -> (a -> Ordering) -> Map a b -> Map a b -> Map a b hedgeUnionWithKey _ _ _ t1 Tip = t1 hedgeUnionWithKey _ 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. -- Return elements of the first map not existing in the second map. -- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/. -- -- > difference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b" difference :: Ord k => Map k a -> Map k b -> Map k a difference Tip _ = Tip difference t1 Tip = t1 difference t1 t2 = hedgeDiff (const LT) (const GT) t1 t2 hedgeDiff :: Ord a => (a -> Ordering) -> (a -> Ordering) -> Map a b -> Map a c -> Map a b hedgeDiff _ _ Tip _ = 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 _ 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. -- When two equal keys are -- encountered, the combining function is applied to the values of these keys. -- 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/. -- -- > let f al ar = if al == "b" then Just (al ++ ":" ++ ar) else Nothing -- > differenceWith f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (7, "C")]) -- > == singleton 3 "b:B" differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a differenceWith f m1 m2 = differenceWithKey (\_ 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/. -- -- > let f k al ar = if al == "b" then Just ((show k) ++ ":" ++ al ++ "|" ++ ar) else Nothing -- > differenceWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (10, "C")]) -- > == singleton 3 "3:b|B" differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a differenceWithKey _ Tip _ = Tip differenceWithKey _ t1 Tip = t1 differenceWithKey f t1 t2 = hedgeDiffWithKey f (const LT) (const GT) t1 t2 hedgeDiffWithKey :: Ord a => (a -> b -> c -> Maybe b) -> (a -> Ordering) -> (a -> Ordering) -> Map a b -> Map a c -> Map a b hedgeDiffWithKey _ _ _ Tip _ = Tip hedgeDiffWithKey _ 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. -- Return data in the first map for the keys existing in both maps. -- (@'intersection' m1 m2 == 'intersectionWith' 'const' m1 m2@). -- -- > intersection (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "a" intersection :: Ord k => Map k a -> Map k b -> Map k a intersection m1 m2 = intersectionWithKey (\_ x _ -> x) m1 m2 -- | /O(n+m)/. Intersection with a combining function. -- -- > intersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA" intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k c intersectionWith f m1 m2 = intersectionWithKey (\_ x y -> f x y) m1 m2 -- | /O(n+m)/. Intersection with a combining function. -- Intersection is more efficient on (bigset \``intersection`\` smallset). -- -- > let f k al ar = (show k) ++ ":" ++ al ++ "|" ++ ar -- > intersectionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "5:a|A" --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 _ Tip _ = Tip intersectionWithKey _ _ 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' :: Ord a => (b -> c -> Bool) -> Map a b -> Map a c -> Bool submap' _ Tip _ = True submap' _ _ 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 (> "a") (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" -- > filter (> "x") (fromList [(5,"a"), (3,"b")]) == empty -- > filter (< "a") (fromList [(5,"a"), (3,"b")]) == empty filter :: Ord k => (a -> Bool) -> Map k a -> Map k a filter p m = filterWithKey (\_ x -> p x) m -- | /O(n)/. Filter all keys\/values that satisfy the predicate. -- -- > filterWithKey (\k _ -> k > 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" filterWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> Map k a filterWithKey _ 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 (> "a") (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a") -- > partition (< "x") (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty) -- > partition (> "x") (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")]) partition :: Ord k => (a -> Bool) -> Map k a -> (Map k a,Map k a) partition p m = partitionWithKey (\_ 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 (\ k _ -> k > 3) (fromList [(5,"a"), (3,"b")]) == (singleton 5 "a", singleton 3 "b") -- > partitionWithKey (\ k _ -> k < 7) (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty) -- > partitionWithKey (\ k _ -> k > 7) (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")]) partitionWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> (Map k a,Map k a) partitionWithKey _ 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. -- -- > let f x = if x == "a" then Just "new a" else Nothing -- > mapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a" mapMaybe :: Ord k => (a -> Maybe b) -> Map k a -> Map k b mapMaybe f m = mapWithMaybeKey (\_ x -> f x) m -- | /O(n)/. Map keys\/values and collect the 'Just' results. -- -- > let f k _ = if k < 5 then Just ("key : " ++ (show k)) else Nothing -- > mapWithMaybeKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key : 3" mapWithMaybeKey :: Ord k => (k -> a -> Maybe b) -> Map k a -> Map k b mapWithMaybeKey _ Tip = Tip mapWithMaybeKey f (Bin _ kx x l r) = case f kx x of Just y -> join kx y (mapWithMaybeKey f l) (mapWithMaybeKey f r) Nothing -> merge (mapWithMaybeKey f l) (mapWithMaybeKey f r) -- | /O(n)/. Map values and separate the 'Left' and 'Right' results. -- -- > let f a = if a < "c" then Left a else Right a -- > mapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) -- > == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")]) -- > -- > mapEither (\ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) -- > == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) mapEither :: Ord k => (a -> Either b c) -> Map k a -> (Map k b, Map k c) mapEither f m = mapEitherWithKey (\_ x -> f x) m -- | /O(n)/. Map keys\/values and separate the 'Left' and 'Right' results. -- -- > let f k a = if k < 5 then Left (k * 2) else Right (a ++ a) -- > mapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) -- > == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")]) -- > -- > mapEitherWithKey (\_ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) -- > == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")]) mapEitherWithKey :: Ord k => (k -> a -> Either b c) -> Map k a -> (Map k b, Map k c) mapEitherWithKey _ 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 (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")] map :: (a -> b) -> Map k a -> Map k b map f m = mapWithKey (\_ x -> f x) m -- | /O(n)/. Map a function over all values in the map. -- -- > let f key x = (show key) ++ ":" ++ x -- > mapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")] mapWithKey :: (k -> a -> b) -> Map k a -> Map k b mapWithKey _ 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. -- -- > let f a b = (a ++ b, b ++ "X") -- > mapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")]) mapAccum :: (a -> b -> (a,c)) -> a -> Map k b -> (a,Map k c) mapAccum f a m = mapAccumWithKey (\a' _ x' -> f a' x') a m -- | /O(n)/. The function 'mapAccumWithKey' threads an accumulating -- argument through the map in ascending order of keys. -- -- > let f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X") -- > mapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")]) 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 through the map in descending order of keys. mapAccumRWithKey :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c) mapAccumRWithKey f a t = case t of Tip -> (a,Tip) Bin sx kx x l r -> let (a1,r') = mapAccumRWithKey f a r (a2,x') = f a1 kx x (a3,l') = mapAccumRWithKey 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 (+ 1) (fromList [(5,"a"), (3,"b")]) == fromList [(4, "b"), (6, "a")] -- > mapKeys (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c" -- > mapKeys (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c" mapKeys :: Ord k2 => (k1->k2) -> Map k1 a -> Map k2 a mapKeys = mapKeysWith (\x _ -> 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 (++) (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab" -- > mapKeysWith (++) (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab" 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. -- That is, for any values @x@ and @y@, if @x@ < @y@ then @f x@ < @f y@. -- /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 -- -- This means that @f@ maps distinct original keys to distinct resulting keys. -- This function has better performance than 'mapKeys'. -- -- > mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")] -- > valid (mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")])) == True -- > valid (mapKeysMonotonic (\ _ -> 1) (fromList [(5,"a"), (3,"b")])) == False mapKeysMonotonic :: (k1->k2) -> Map k1 a -> Map k2 a mapKeysMonotonic _ 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 -- -- > let f a len = len + (length a) -- > fold f 0 (fromList [(5,"a"), (3,"bbb")]) == 4 fold :: (a -> b -> b) -> b -> Map k a -> b fold f z m = foldWithKey (\_ 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 -- -- > let f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")" -- > foldWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)" -- -- This is identical to 'foldrWithKey', and you should use that one instead of -- this one. This name is kept for backward compatibility. foldWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b foldWithKey f z t = foldrWithKey f z t {- XXX unused code -- | /O(n)/. In-order fold. foldi :: (k -> a -> b -> b -> b) -> b -> Map k a -> b foldi _ 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. The function will be applied from the lowest -- value to the highest. foldrWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b foldrWithKey _ z Tip = z foldrWithKey f z (Bin _ kx x l r) = foldrWithKey f (f kx x (foldrWithKey f z r)) l -- | /O(n)/. Pre-order fold. The function will be applied from the highest -- value to the lowest. foldlWithKey :: (b -> k -> a -> b) -> b -> Map k a -> b foldlWithKey _ z Tip = z foldlWithKey f z (Bin _ kx x l r) = foldlWithKey f (f (foldlWithKey f z l) kx x) r {-------------------------------------------------------------------- List variations --------------------------------------------------------------------} -- | /O(n)/. -- Return all elements of the map in the ascending order of their keys. -- -- > elems (fromList [(5,"a"), (3,"b")]) == ["b","a"] -- > elems empty == [] elems :: Map k a -> [a] elems m = [x | (_,x) <- assocs m] -- | /O(n)/. Return all keys of the map in ascending order. -- -- > keys (fromList [(5,"a"), (3,"b")]) == [3,5] -- > keys empty == [] keys :: Map k a -> [k] keys m = [k | (k,_) <- assocs m] -- | /O(n)/. The set of all keys of the map. -- -- > keysSet (fromList [(5,"a"), (3,"b")]) == Data.Set.fromList [3,5] -- > keysSet empty == Data.Set.empty 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 (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")] -- > assocs empty == [] 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'. -- If the list contains more than one value for the same key, the last value -- for the key is retained. -- -- > fromList [] == empty -- > fromList [(5,"a"), (3,"b"), (5, "c")] == fromList [(5,"c"), (3,"b")] -- > fromList [(5,"c"), (3,"b"), (5, "a")] == fromList [(5,"a"), (3,"b")] 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 (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "ab"), (5, "aba")] -- > fromListWith (++) [] == empty fromListWith :: Ord k => (a -> a -> a) -> [(k,a)] -> Map k a fromListWith f xs = fromListWithKey (\_ 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'. -- -- > let f k a1 a2 = (show k) ++ a1 ++ a2 -- > fromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "3ab"), (5, "5a5ba")] -- > fromListWithKey f [] == empty 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 (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")] -- > toList empty == [] toList :: Map k a -> [(k,a)] toList t = toAscList t -- | /O(n)/. Convert to an ascending list. -- -- > toAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")] toAscList :: Map k a -> [(k,a)] toAscList t = foldrWithKey (\k x xs -> (k,x):xs) [] t -- | /O(n)/. Convert to a descending list. toDescList :: Map k a -> [(k,a)] toDescList t = foldlWithKey (\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 [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")] -- > fromAscList [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "b")] -- > valid (fromAscList [(3,"b"), (5,"a"), (5,"b")]) == True -- > valid (fromAscList [(5,"a"), (3,"b"), (5,"b")]) == False fromAscList :: Eq k => [(k,a)] -> Map k a fromAscList xs = fromAscListWithKey (\_ x _ -> 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 (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")] -- > valid (fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")]) == True -- > valid (fromAscListWith (++) [(5,"a"), (3,"b"), (5,"b")]) == False fromAscListWith :: Eq k => (a -> a -> a) -> [(k,a)] -> Map k a fromAscListWith f xs = fromAscListWithKey (\_ 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./ -- -- > let f k a1 a2 = (show k) ++ ":" ++ a1 ++ a2 -- > fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")] == fromList [(3, "b"), (5, "5:b5:ba")] -- > valid (fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")]) == True -- > valid (fromAscListWithKey f [(5,"a"), (3,"b"), (5,"b"), (5,"b")]) == False 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 _ 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 [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")] -- > valid (fromDistinctAscList [(3,"b"), (5,"a")]) == True -- > valid (fromDistinctAscList [(3,"b"), (5,"a"), (5,"b")]) == False 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 _ -> error "fromDistinctAscList build" 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 buildR _ _ _ [] = error "fromDistinctAscList buildR []" 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 _ _ Tip = Tip trim cmplo cmphi t@(Bin _ kx _ l r) = case cmplo kx of LT -> case cmphi kx of GT -> t _ -> trim cmplo cmphi l _ -> trim cmplo cmphi r trimLookupLo :: Ord k => k -> (k -> Ordering) -> Map k a -> (Maybe (k,a), Map k a) trimLookupLo _ _ Tip = (Nothing,Tip) trimLookupLo lo cmphi t@(Bin _ kx x l r) = case compare lo kx of LT -> case cmphi kx of GT -> (lookupAssoc lo t, t) _ -> 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 _ Tip = Tip filterGt cmp (Bin _ 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 _ Tip = Tip filterLt cmp (Bin _ 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 2 (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3,"b"), (5,"a")]) -- > split 3 (fromList [(5,"a"), (3,"b")]) == (empty, singleton 5 "a") -- > split 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a") -- > split 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", empty) -- > split 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], empty) split :: Ord k => k -> Map k a -> (Map k a,Map k a) split _ Tip = (Tip,Tip) split k (Bin _ 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 2 (fromList [(5,"a"), (3,"b")]) == (empty, Nothing, fromList [(3,"b"), (5,"a")]) -- > splitLookup 3 (fromList [(5,"a"), (3,"b")]) == (empty, Just "b", singleton 5 "a") -- > splitLookup 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Nothing, singleton 5 "a") -- > splitLookup 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Just "a", empty) -- > splitLookup 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], Nothing, empty) splitLookup :: Ord k => k -> Map k a -> (Map k a,Maybe a,Map k a) splitLookup _ Tip = (Tip,Nothing,Tip) splitLookup k (Bin _ 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 _ Tip = (Tip,Nothing,Tip) splitLookupWithKey k (Bin _ 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) {- XXX unused code -- | /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 _ ky y l r -> balance ky y l (insertMax kx x r) insertMin kx x t = case t of Tip -> singleton kx x Bin _ 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 (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((3,"b"), fromList[(5,"a"), (10,"c")]) -- > deleteFindMin Error: can not return the minimal element of an empty map 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 (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((10,"c"), fromList [(3,"b"), (5,"a")]) -- > deleteFindMax empty Error: can not return the maximal element of an empty map 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 :: a -> b -> Map a b -> Map a b -> Map a b rotateL k x l r@(Bin _ _ _ ly ry) | size ly < ratio*size ry = singleL k x l r | otherwise = doubleL k x l r rotateL _ _ _ Tip = error "rotateL Tip" rotateR :: a -> b -> Map a b -> Map a b -> Map a b rotateR k x l@(Bin _ _ _ ly ry) r | size ry < ratio*size ly = singleR k x l r | otherwise = doubleR k x l r rotateR _ _ Tip _ = error "rotateR Tip" -- basic rotations singleL, singleR :: a -> b -> Map a b -> Map a b -> Map a b singleL k1 x1 t1 (Bin _ k2 x2 t2 t3) = bin k2 x2 (bin k1 x1 t1 t2) t3 singleL _ _ _ Tip = error "singleL Tip" singleR k1 x1 (Bin _ k2 x2 t1 t2) t3 = bin k2 x2 t1 (bin k1 x1 t2 t3) singleR _ _ Tip _ = error "singleR Tip" doubleL, doubleR :: a -> b -> Map a b -> Map a b -> Map a b 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) doubleL _ _ _ _ = error "doubleL" 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) doubleR _ _ _ _ = error "doubleR" {-------------------------------------------------------------------- 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 _ 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 {- XXX unused code -- 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) {- XXX unused code 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. See 'showTreeWith'. 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 _ kx x Tip Tip -> showsBars lbars . showString (showelem kx x) . showString "\n" Bin _ 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 _ kx x Tip Tip -> showsBars bars . showString (showelem kx x) . showString "\n" Bin _ 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 :: Bool -> [String] -> String -> String 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 :: String node = "+--" withBar, withEmpty :: [String] -> [String] 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 (fromAscList [(3,"b"), (5,"a")]) == True -- > valid (fromAscList [(5,"a"), (3,"b")]) == False valid :: Ord k => Map k a -> Bool valid t = balanced t && ordered t && validsize t ordered :: Ord a => Map a b -> Bool ordered t = bounded (const True) (const True) t where bounded lo hi t' = case t' of Tip -> True Bin _ kx _ l r -> (lo kx) && (hi kx) && bounded lo (kx) hi r -- | Exported only for "Debug.QuickCheck" balanced :: Map k a -> Bool balanced t = case t of Tip -> True Bin _ _ _ l r -> (size l + size r <= 1 || (size l <= delta*size r && size r <= delta*size l)) && balanced l && balanced r validsize :: Map a b -> Bool validsize t = (realsize t == Just (size t)) where realsize t' = case t' of Tip -> Just 0 Bin sz _ _ l r -> case (realsize l,realsize r) of (Just n,Just m) | n+m+1 == sz -> Just sz _ -> Nothing {-------------------------------------------------------------------- Utilities --------------------------------------------------------------------} foldlStrict :: (a -> b -> a) -> a -> [b] -> a foldlStrict f z xs = case xs of [] -> z (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx) unionWithMaybe :: Ord k => (a -> a -> Maybe a) -> Map k a -> Map k a -> Map k a unionWithMaybe f m1 m2 = unionWithMaybeKey (\_ 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). -- -- > let f key left_value right_value = (show key) ++ ":" ++ left_value ++ "|" ++ right_value -- > unionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")] unionWithMaybeKey :: Ord k => (k -> a -> a -> Maybe a) -> Map k a -> Map k a -> Map k a unionWithMaybeKey _ Tip t2 = t2 unionWithMaybeKey _ t1 Tip = t1 unionWithMaybeKey f t1 t2 = hedgeUnionWithMaybeKey f (const LT) (const GT) t1 t2 hedgeUnionWithMaybeKey :: Ord a => (a -> b -> b -> Maybe b) -> (a -> Ordering) -> (a -> Ordering) -> Map a b -> Map a b -> Map a b hedgeUnionWithMaybeKey _ _ _ t1 Tip = t1 hedgeUnionWithMaybeKey _ cmplo cmphi Tip (Bin _ kx x l r) = join kx x (filterGt cmplo l) (filterLt cmphi r) hedgeUnionWithMaybeKey f cmplo cmphi (Bin _ kx x l r) t2 = case newx of Nothing -> merge (hedgeUnionWithMaybeKey f cmplo cmpkx l lt) (hedgeUnionWithMaybeKey f cmpkx cmphi r gt) (Just nx) -> join kx nx (hedgeUnionWithMaybeKey f cmplo cmpkx l lt) (hedgeUnionWithMaybeKey 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 -> Just x Just (_,y) -> f kx x y