containers-0.5.0.0: Assorted concrete container types

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

Data.Map.Strict

Contents

- Strictness properties
- Map type
- Operators
- Query
- Construction
  - Insertion
  - Delete/Update
- Combine
  - Union
  - Difference
  - Intersection
  - Universal combining function
- Traversal
  - Map
- Folds
  - Strict folds
- Conversion
  - Lists
  - Ordered lists
- Filter
- Submap
- Indexed
- Min/Max
- Debugging

Description

An efficient implementation of ordered maps from keys to values (dictionaries).

API of this module is strict in both the keys and the values. If you need value-lazy maps, use `Lazy` instead. The `Map` type is shared between the lazy and strict modules, meaning that the same `Map` value can be passed to functions in both modules (although that is rarely needed).

These modules are intended to be imported qualified, to avoid name clashes with Prelude functions, e.g.

      import qualified Data.Map.Strict 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).

Be aware that the `Functor`, `Traversable` and `Data` instances are the same as for the `Lazy` module, so if they are used on strict maps, the resulting maps will be lazy.

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
- lookupLT :: Ord k => k -> Map k v -> Maybe (k, v)
- lookupGT :: Ord k => k -> Map k v -> Maybe (k, v)
- lookupLE :: Ord k => k -> Map k v -> Maybe (k, v)
- lookupGE :: Ord k => k -> Map k v -> Maybe (k, v)
- 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
- 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)
- 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
- mergeWithKey :: Ord k => (k -> a -> b -> Maybe c) -> (Map k a -> Map k c) -> (Map k b -> Map k 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
- traverseWithKey :: Applicative t => (k -> a -> t b) -> Map k a -> t (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
- foldr :: (a -> b -> b) -> b -> Map k a -> b
- foldl :: (a -> b -> a) -> a -> Map k b -> a
- foldrWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b
- foldlWithKey :: (a -> k -> b -> a) -> a -> Map k b -> a
- foldr' :: (a -> b -> b) -> b -> Map k a -> b
- foldl' :: (a -> b -> a) -> a -> Map k b -> a
- foldrWithKey' :: (k -> a -> b -> b) -> b -> Map k a -> b
- foldlWithKey' :: (a -> k -> b -> a) -> a -> Map k b -> a
- elems :: Map k a -> [a]
- keys :: Map k a -> [k]
- assocs :: Map k a -> [(k, a)]
- keysSet :: Map k a -> Set k
- fromSet :: (k -> a) -> Set k -> Map 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 :: (a -> Bool) -> Map k a -> Map k a
- filterWithKey :: (k -> a -> Bool) -> Map k a -> Map k a
- partition :: (a -> Bool) -> Map k a -> (Map k a, Map k a)
- partitionWithKey :: (k -> a -> Bool) -> Map k a -> (Map k a, Map k a)
- mapMaybe :: (a -> Maybe b) -> Map k a -> Map k b
- mapMaybeWithKey :: (k -> a -> Maybe b) -> Map k a -> Map k b
- mapEither :: (a -> Either b c) -> Map k a -> (Map k b, Map k c)
- mapEitherWithKey :: (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

# Strictness properties

This module satisfies the following strictness properties:

1. Key and value arguments are evaluated to WHNF;
2. Keys and values are evaluated to WHNF before they are stored in the map.

Here are some examples that illustrate the first property:

     insertWith (\ new old -> old) k undefined m  ==  undefined
     delete undefined m  ==  undefined

Here are some examples that illustrate the second property:

     map (\ v -> undefined) m  ==  undefined      -- m is not empty
     mapKeys (\ k -> undefined) m  ==  undefined  -- m is not empty

# 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) |
(NFData k, NFData a) => NFData (Map k a) |

# 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'

lookupLT :: Ord k => k -> Map k v -> Maybe (k, v)

O(log n). Find largest key smaller than the given one and return the corresponding (key, value) pair.

     lookupLT 3 (fromList [(3,'a'), (5,'b')]) == Nothing
     lookupLT 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')

lookupGT :: Ord k => k -> Map k v -> Maybe (k, v)

O(log n). Find smallest key greater than the given one and return the corresponding (key, value) pair.

     lookupGT 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
     lookupGT 5 (fromList [(3,'a'), (5,'b')]) == Nothing

lookupLE :: Ord k => k -> Map k v -> Maybe (k, v)

O(log n). Find largest key smaller or equal to the given one and return the corresponding (key, value) pair.

     lookupLE 2 (fromList [(3,'a'), (5,'b')]) == Nothing
     lookupLE 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
     lookupLE 5 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')

lookupGE :: Ord k => k -> Map k v -> Maybe (k, v)

O(log n). Find smallest key greater or equal to the given one and return the corresponding (key, value) pair.

     lookupGE 3 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
     lookupGE 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
     lookupGE 6 (fromList [(3,'a'), (5,'b')]) == Nothing

# 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"

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"

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")])

## 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"

## Universal combining function

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

O(n+m). A high-performance universal combining function. This function is used to define `unionWith`, `unionWithKey`, `differenceWith`, `differenceWithKey`, `intersectionWith`, `intersectionWithKey` and can be used to define other custom combine functions.

Please make sure you know what is going on when using `mergeWithKey`, otherwise you can be surprised by unexpected code growth or even corruption of the data structure.

When `mergeWithKey` is given three arguments, it is inlined to the call site. You should therefore use `mergeWithKey` only to define your custom combining functions. For example, you could define `unionWithKey`, `differenceWithKey` and `intersectionWithKey` as

     myUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) id id m1 m2
     myDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2
     myIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) (const empty) (const empty) m1 m2

When calling ``mergeWithKey` combine only1 only2`, a function combining two `IntMap`s is created, such that

- if a key is present in both maps, it is passed with both corresponding values to the `combine` function. Depending on the result, the key is either present in the result with specified value, or is left out;
- a nonempty subtree present only in the first map is passed to `only1` and the output is added to the result;
- a nonempty subtree present only in the second map is passed to `only2` and the output is added to the result.

The `only1` and `only2` methods must return a map with a subset (possibly empty) of the keys of the given map. The values can be modified arbitrarily. Most common variants of `only1` and `only2` are `id` and ``const` `empty``, but for example ``map` f` or ``filterWithKey` f` could be used for any `f`.

# 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")]

traverseWithKey :: Applicative t => (k -> a -> t b) -> Map k a -> t (Map k b)

O(n). ``traverseWithKey` f s == `fromList` $ `traverse` ((k, v) -> (,) k $ f k v) (`toList` m)` That is, behaves exactly like a regular `traverse` except that the traversing function also has access to the key associated with a value.

     traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])
     traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')])           == Nothing

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 greatest of the original 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

# Folds

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

O(n). Fold the values in the map using the given right-associative binary operator, such that ``foldr` f z == `foldr` f z . `elems``.

For example,

     elems map = foldr (:) [] map

     let f a len = len + (length a)
     foldr f 0 (fromList [(5,"a"), (3,"bbb")]) == 4

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

O(n). Fold the values in the map using the given left-associative binary operator, such that ``foldl` f z == `foldl` f z . `elems``.

For example,

     elems = reverse . foldl (flip (:)) []

     let f len a = len + (length a)
     foldl f 0 (fromList [(5,"a"), (3,"bbb")]) == 4

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

O(n). Fold the keys and values in the map using the given right-associative binary operator, such that ``foldrWithKey` f z == `foldr` (`uncurry` f) z . `toAscList``.

For example,

     keys map = foldrWithKey (\k x ks -> k:ks) [] map

     let f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
     foldrWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)"

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

O(n). Fold the keys and values in the map using the given left-associative binary operator, such that ``foldlWithKey` f z == `foldl` (\z' (kx, x) -> f z' kx x) z . `toAscList``.

For example,

     keys = reverse . foldlWithKey (\ks k x -> k:ks) []

     let f result k a = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
     foldlWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (3:b)(5:a)"

## Strict folds

foldr' :: (a -> b -> b) -> b -> Map k a -> b

O(n). A strict version of `foldr`. Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value.

foldl' :: (a -> b -> a) -> a -> Map k b -> a

O(n). A strict version of `foldl`. Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value.

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

O(n). A strict version of `foldrWithKey`. Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value.

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

O(n). A strict version of `foldlWithKey`. Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value.

# Conversion

elems :: Map k a -> [a]

O(n). Return all elements of the map in the ascending order of their keys. Subject to list fusion.

     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. Subject to list fusion.

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

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

O(n). An alias for `toAscList`. Return all key/value pairs in the map in ascending key order. Subject to list fusion.

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

fromSet :: (k -> a) -> Set k -> Map k a

O(n). Build a map from a set of keys and a function which for each key computes its value.

     fromSet (\k -> replicate k 'a') (Data.Set.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")]
     fromSet undefined Data.Set.empty == empty

## Lists

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

O(n). Convert the map to a list of key/value pairs. Subject to list fusion.

     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 the map to a list of key/value pairs where the keys are in ascending order. Subject to list fusion.

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

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

O(n). Convert the map to a list of key/value pairs where the keys are in descending order. Subject to list fusion.

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

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 :: (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 :: (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 :: (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 :: (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 :: (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 :: (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 :: (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 :: (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` if 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` if 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