Data.Map

containers-0.4.0.0: Assorted concrete container types

Portability | portable
Stability | provisional
Maintainer | libraries@haskell.org

Data.Map

Contents

- Map type
- Operators
- Query
- Construction
  - Insertion
  - Delete/Update
- Combine
  - Union
  - Difference
  - Intersection
- Traversal
  - Map
  - Fold
- Conversion
  - Lists
  - Ordered lists
- Filter
- Submap
- Indexed
- Min/Max
- Debugging

Description

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.

Synopsis

- data Map k a
- (!) :: Ord k => Map k a -> k -> a
- (\\) :: Ord k => Map k a -> Map k b -> Map k a
- null :: Map k a -> Bool
- size :: Map k a -> Int
- member :: Ord k => k -> Map k a -> Bool
- notMember :: Ord k => k -> Map k a -> Bool
- lookup :: Ord k => k -> Map k a -> Maybe a
- findWithDefault :: Ord k => a -> k -> Map k a -> a
- empty :: Map k a
- singleton :: k -> a -> Map k a
- insert :: Ord k => k -> a -> Map k a -> Map k a
- insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a
- insertWith' :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a
- insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
- insertWithKey' :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
- insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)
- insertLookupWithKey' :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)
- delete :: Ord k => k -> Map k a -> Map k a
- adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k a
- adjustWithKey :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k a
- update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k a
- updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a
- updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a, Map k a)
- alter :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a
- union :: Ord k => Map k a -> Map k a -> Map k a
- unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a
- unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a
- unions :: Ord k => [Map k a] -> Map k a
- unionsWith :: Ord k => (a -> a -> a) -> [Map k a] -> Map k a
- difference :: Ord k => Map k a -> Map k b -> Map k a
- differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a
- differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a
- intersection :: Ord k => Map k a -> Map k b -> Map k a
- intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k c
- intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c
- map :: (a -> b) -> Map k a -> Map k b
- mapWithKey :: (k -> a -> b) -> Map k a -> Map k b
- mapAccum :: (a -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)
- mapAccumWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)
- mapAccumRWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)
- mapKeys :: Ord k2 => (k1 -> k2) -> Map k1 a -> Map k2 a
- mapKeysWith :: Ord k2 => (a -> a -> a) -> (k1 -> k2) -> Map k1 a -> Map k2 a
- mapKeysMonotonic :: (k1 -> k2) -> Map k1 a -> Map k2 a
- fold :: (a -> b -> b) -> b -> Map k a -> b
- foldWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b
- foldrWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b
- foldlWithKey :: (b -> k -> a -> b) -> b -> Map k a -> b
- elems :: Map k a -> [a]
- keys :: Map k a -> [k]
- keysSet :: Map k a -> Set k
- assocs :: Map k a -> [(k, a)]
- toList :: Map k a -> [(k, a)]
- fromList :: Ord k => [(k, a)] -> Map k a
- fromListWith :: Ord k => (a -> a -> a) -> [(k, a)] -> Map k a
- fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k, a)] -> Map k a
- toAscList :: Map k a -> [(k, a)]
- toDescList :: Map k a -> [(k, a)]
- fromAscList :: Eq k => [(k, a)] -> Map k a
- fromAscListWith :: Eq k => (a -> a -> a) -> [(k, a)] -> Map k a
- fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(k, a)] -> Map k a
- fromDistinctAscList :: [(k, a)] -> Map k a
- filter :: Ord k => (a -> Bool) -> Map k a -> Map k a
- filterWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> Map k a
- partition :: Ord k => (a -> Bool) -> Map k a -> (Map k a, Map k a)
- partitionWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> (Map k a, Map k a)
- mapMaybe :: Ord k => (a -> Maybe b) -> Map k a -> Map k b
- mapMaybeWithKey :: Ord k => (k -> a -> Maybe b) -> Map k a -> Map k b
- mapEither :: Ord k => (a -> Either b c) -> Map k a -> (Map k b, Map k c)
- mapEitherWithKey :: Ord k => (k -> a -> Either b c) -> Map k a -> (Map k b, Map k c)
- split :: Ord k => k -> Map k a -> (Map k a, Map k a)
- splitLookup :: Ord k => k -> Map k a -> (Map k a, Maybe a, Map k a)
- isSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool
- isSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool
- isProperSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool
- isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool
- lookupIndex :: Ord k => k -> Map k a -> Maybe Int
- findIndex :: Ord k => k -> Map k a -> Int
- elemAt :: Int -> Map k a -> (k, a)
- updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a
- deleteAt :: Int -> Map k a -> Map k a
- findMin :: Map k a -> (k, a)
- findMax :: Map k a -> (k, a)
- deleteMin :: Map k a -> Map k a
- deleteMax :: Map k a -> Map k a
- deleteFindMin :: Map k a -> ((k, a), Map k a)
- deleteFindMax :: Map k a -> ((k, a), Map k a)
- updateMin :: (a -> Maybe a) -> Map k a -> Map k a
- updateMax :: (a -> Maybe a) -> Map k a -> Map k a
- updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
- updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
- minView :: Map k a -> Maybe (a, Map k a)
- maxView :: Map k a -> Maybe (a, Map k a)
- minViewWithKey :: Map k a -> Maybe ((k, a), Map k a)
- maxViewWithKey :: Map k a -> Maybe ((k, a), Map k a)
- showTree :: (Show k, Show a) => Map k a -> String
- showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> String
- valid :: Ord k => Map k a -> Bool

# Map type

data Map k a

A Map from keys `k` to values `a`.

Instances

Typeable2 Map |
Functor (Map k) |
Foldable (Map k) |
Traversable (Map k) |
(Eq k, Eq a) => Eq (Map k a) |
(Data k, Data a, Ord k) => Data (Map k a) |
(Ord k, Ord v) => Ord (Map k v) |
(Ord k, Read k, Read e) => Read (Map k e) |
(Show k, Show a) => Show (Map k a) |
Ord k => Monoid (Map k v) |

# Operators

(!) :: Ord k => Map k a -> k -> a

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 -> Map k b -> Map k a

Same as `difference`.

# Query

null :: Map k a -> Bool

O(1). Is the map empty?

     Data.Map.null (empty)           == True
     Data.Map.null (singleton 1 'a') == False

size :: Map k a -> Int

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

member :: Ord k => k -> Map k a -> Bool

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

notMember :: Ord k => k -> Map k a -> Bool

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

lookup :: Ord k => k -> Map k a -> Maybe a

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

findWithDefault :: Ord k => a -> k -> Map k a -> a

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'

# Construction

empty :: Map k a

O(1). The empty map.

     empty      == fromList []
     size empty == 0

singleton :: k -> a -> Map k a

O(1). A map with a single element.

     singleton 1 'a'        == fromList [(1, 'a')]
     size (singleton 1 'a') == 1

## Insertion

insert :: Ord k => k -> a -> Map k a -> Map k a

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'

insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a

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

Same as `insertWith`, but the combining function is applied strictly. This is often the most desirable behavior.

For example, to update a counter:

     insertWith' (+) k 1 m

insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a

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

Same as `insertWithKey`, but the combining function is applied strictly.

insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)

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)

O(log n). A strict version of `insertLookupWithKey`.

## Delete/Update

delete :: Ord k => k -> Map k a -> Map k a

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

adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k a

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

adjustWithKey :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k a

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

update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k a

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"

updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a

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"

updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a, Map k a)

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")

alter :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a

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")]

# Combine

## Union

union :: Ord k => Map k a -> Map k a -> Map k a

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")]

unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a

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")]

unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a

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")]

unions :: Ord k => [Map k a] -> Map k a

The union of a list of maps: (``unions` == `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")]

unionsWith :: Ord k => (a -> a -> a) -> [Map k a] -> Map k a

The union of a list of maps, with a combining operation: (``unionsWith` f == `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")]

## Difference

difference :: Ord k => Map k a -> Map k b -> Map k a

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"

differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a

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"

differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a

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"

## Intersection

intersection :: Ord k => Map k a -> Map k b -> Map k a

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"

intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k c

O(n+m). Intersection with a combining function.

     intersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"

intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c

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"

# Traversal

## Map

map :: (a -> b) -> Map k a -> Map k b

O(n). Map a function over all values in the map.

     map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]

mapWithKey :: (k -> a -> b) -> Map k a -> Map k b

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")]

mapAccum :: (a -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)

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")])

mapAccumWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)

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")])

mapAccumRWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)

O(n). The function `mapAccumR` threads an accumulating argument through the map in descending order of keys.

mapKeys :: Ord k2 => (k1 -> k2) -> Map k1 a -> Map k2 a

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"

mapKeysWith :: Ord k2 => (a -> a -> a) -> (k1 -> k2) -> Map k1 a -> Map k2 a

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"

mapKeysMonotonic :: (k1 -> k2) -> Map k1 a -> Map k2 a

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

## Fold

fold :: (a -> b -> b) -> b -> Map k a -> b

O(n). Fold the values in the map, such that ``fold` f z == `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

foldWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b

O(n). Fold the keys and values in the map, such that ``foldWithKey` f z == `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.

foldrWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b

O(n). Post-order fold. The function will be applied from the lowest value to the highest.

foldlWithKey :: (b -> k -> a -> b) -> b -> Map k a -> b

O(n). Pre-order fold. The function will be applied from the highest value to the lowest.

# Conversion

elems :: Map k a -> [a]

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 == []

keys :: Map k a -> [k]

O(n). Return all keys of the map in ascending order.

     keys (fromList [(5,"a"), (3,"b")]) == [3,5]
     keys empty == []

keysSet :: Map k a -> Set k

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

assocs :: Map k a -> [(k, a)]

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 == []

## Lists

toList :: Map k a -> [(k, a)]

O(n). Convert to a list of key/value pairs.

     toList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
     toList empty == []

fromList :: Ord k => [(k, a)] -> Map k a

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")]

fromListWith :: Ord k => (a -> a -> a) -> [(k, a)] -> Map k a

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

fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k, a)] -> Map k a

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

## Ordered lists

toAscList :: Map k a -> [(k, a)]

O(n). Convert to an ascending list.

     toAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]

toDescList :: Map k a -> [(k, a)]

O(n). Convert to a descending list.

fromAscList :: Eq k => [(k, a)] -> Map k a

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

fromAscListWith :: Eq k => (a -> a -> a) -> [(k, a)] -> Map k a

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

fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(k, a)] -> Map k a

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

fromDistinctAscList :: [(k, a)] -> Map k a

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

# Filter

filter :: Ord k => (a -> Bool) -> Map k a -> Map k a

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

filterWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> Map k a

O(n). Filter all keys/values that satisfy the predicate.

     filterWithKey (\k _ -> k > 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"

partition :: Ord k => (a -> Bool) -> Map k a -> (Map k a, Map k a)

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")])

partitionWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> (Map k a, Map k a)

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")])

mapMaybe :: Ord k => (a -> Maybe b) -> Map k a -> Map k b

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"

mapMaybeWithKey :: Ord k => (k -> a -> Maybe b) -> Map k a -> Map k b

O(n). Map keys/values and collect the `Just` results.

     let f k _ = if k < 5 then Just ("key : " ++ (show k)) else Nothing
     mapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key : 3"

mapEither :: Ord k => (a -> Either b c) -> Map k a -> (Map k b, Map k c)

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")])

mapEitherWithKey :: Ord k => (k -> a -> Either b c) -> Map k a -> (Map k b, Map k c)

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")])

split :: Ord k => k -> Map k a -> (Map k a, Map k a)

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)

splitLookup :: Ord k => k -> Map k a -> (Map k a, Maybe a, Map k a)

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)

# Submap

isSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool

O(n+m). This function is defined as (``isSubmapOf` = `isSubmapOfBy` (==)`).

isSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool

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)])

isProperSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool

O(n+m). Is this a proper submap? (ie. a submap but not equal). Defined as (``isProperSubmapOf` = `isProperSubmapOfBy` (==)`).

isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool

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)])

# Indexed

lookupIndex :: Ord k => k -> Map k a -> Maybe Int

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

findIndex :: Ord k => k -> Map k a -> Int

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

elemAt :: Int -> Map k a -> (k, a)

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

updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a

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

deleteAt :: Int -> Map k a -> Map k a

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

# Min/Max

findMin :: Map k a -> (k, a)

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

findMax :: Map k a -> (k, a)

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

deleteMin :: Map k a -> Map k a

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

deleteMax :: Map k a -> Map k a

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

deleteFindMin :: Map k a -> ((k, a), Map k a)

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

deleteFindMax :: Map k a -> ((k, a), Map k a)

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

updateMin :: (a -> Maybe a) -> Map k a -> Map k a

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"

updateMax :: (a -> Maybe a) -> Map k a -> Map k a

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"

updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a

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"

updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a

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"

minView :: Map k a -> Maybe (a, Map k a)

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

maxView :: Map k a -> Maybe (a, Map k a)

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

minViewWithKey :: Map k a -> Maybe ((k, a), Map k a)

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

maxViewWithKey :: Map k a -> Maybe ((k, a), Map k a)

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

# Debugging

showTree :: (Show k, Show a) => Map k a -> String

O(n). Show the tree that implements the map. The tree is shown in a compressed, hanging format. See `showTreeWith`.

showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> String

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,())

valid :: Ord k => Map k a -> Bool

O(n). Test if the internal map structure is valid.

     valid (fromAscList [(3,"b"), (5,"a")]) == True
     valid (fromAscList [(5,"a"), (3,"b")]) == False