----------------------------------------------------------------------------- -- | -- 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). -- -- This module is intended to be imported @qualified@, to avoid name -- clashes with Prelude functions. eg. -- -- > import 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. -----------------------------------------------------------------------------
module Data.Map ( -- * Map type Map -- instance Eq,Show -- * Operators , (!), (\\) -- * Query , null , size , member , lookup , findWithDefault -- * Construction , empty , singleton -- ** Insertion , insert , insertWith, insertWithKey, insertLookupWithKey -- ** Delete\/Update , delete , adjust , adjustWithKey , update , updateWithKey , updateLookupWithKey -- * 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 , 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 -- * 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.Typeable {- -- for quick check import qualified Prelude import qualified List import Debug.QuickCheck import List(nub,sort) -} {-------------------------------------------------------------------- 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 {-------------------------------------------------------------------- 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. 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 -- | /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)/. 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 [insert] is the inlined version of [insertWith (\k x y -> x)] --------------------------------------------------------------------} -- | /O(log n)/. Insert a new key and value in the map. 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 :: 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 :: 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 ky (f ky x y) 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 ky (f ky 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) {-------------------------------------------------------------------- Indexing --------------------------------------------------------------------} -- | /O(log n)/. Return the /index/ of a key. The index is a number from -- /0/ up to, but not including, the 'size' of the map. Calls 'error' when -- the key is not a 'member' of the map. findIndex :: Ord k => k -> Map k a -> Int findIndex k t = case lookupIndex k t of Nothing -> error "Map.findIndex: element is not in the map" Just idx -> idx -- | /O(log n)/. Lookup the /index/ of a key. The index is a number from -- /0/ up to, but not including, the 'size' of the map. lookupIndex :: (Monad m,Ord k) => k -> Map k a -> m Int lookupIndex k t = case lookup 0 t of Nothing -> fail "Data.Map.lookupIndex: Key not found." Just x -> return x where lookup idx Tip = Nothing lookup idx (Bin _ kx x l r) = case compare k kx of LT -> lookup idx l GT -> lookup (idx + size l + 1) r EQ -> Just (idx + size l) -- | /O(log n)/. Retrieve an element by /index/. Calls 'error' when an -- invalid index is used. elemAt :: Int -> Map k a -> (k,a) elemAt i Tip = error "Map.elemAt: index out of range" elemAt i (Bin _ kx x l r) = case compare i sizeL of LT -> elemAt i l GT -> elemAt (i-sizeL-1) r EQ -> (kx,x) where sizeL = size l -- | /O(log n)/. Update the element at /index/. Calls 'error' when an -- invalid index is used. updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a updateAt f i Tip = error "Map.updateAt: index out of range" updateAt f i (Bin sx kx x l r) = case compare i sizeL of LT -> updateAt f i l GT -> updateAt f (i-sizeL-1) r EQ -> case f kx x of Just x' -> Bin sx kx x' l r Nothing -> glue l r where sizeL = size l -- | /O(log n)/. Delete the element at /index/. -- Defined as (@'deleteAt' i map = 'updateAt' (\k x -> 'Nothing') i map@). deleteAt :: Int -> Map k a -> Map k a deleteAt i map = updateAt (\k x -> Nothing) i map {-------------------------------------------------------------------- Minimal, Maximal --------------------------------------------------------------------} -- | /O(log n)/. The minimal key of the map. findMin :: Map k a -> (k,a) findMin (Bin _ kx x Tip r) = (kx,x) findMin (Bin _ kx x l r) = findMin l findMin Tip = error "Map.findMin: empty tree 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 tree 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 {-------------------------------------------------------------------- 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 | size t1 >= size t2 = hedgeUnionL (const LT) (const GT) t1 t2 | otherwise = hedgeUnionR (const LT) (const GT) t2 t1 -- 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 | size t1 >= size t2 = hedgeUnionWithKey f (const LT) (const GT) t1 t2 | otherwise = hedgeUnionWithKey flipf (const LT) (const GT) t2 t1 where flipf k x y = f k y x 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 y -> case f kx y x of Nothing -> merge tl tr Just z -> join kx 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 | size t1 >= size t2 = intersectWithKey f t1 t2 | otherwise = intersectWithKey flipf t2 t1 where flipf k x y = f k y x 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 {-------------------------------------------------------------------- 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 {-------------------------------------------------------------------- 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 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 -> (lookup lo t, t) le -> trimLookupLo lo cmphi l GT -> trimLookupLo lo cmphi r EQ -> (Just 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) {-------------------------------------------------------------------- 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 (toList m1) (toList m2) {-------------------------------------------------------------------- Functor --------------------------------------------------------------------} instance Functor (Map k) where fmap f m = map f m {-------------------------------------------------------------------- Show --------------------------------------------------------------------} instance (Show k, Show a) => Show (Map k a) where showsPrec d m = showMap (toAscList 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) = showChar ',' . 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 --------------------------------------------------------------------} {- mapTc = mkTyCon "Map"; instance Typeable2 Map where { typeOf2 _ = mkTyConApp mapTc [] }; instance Typeable a => Typeable1 (Map a) where { typeOf1 = typeOf1Default }; instance (Typeable a, Typeable b) => Typeable (Map a b) where { typeOf = typeOfDefault } -} {-------------------------------------------------------------------- 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])]) -}
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