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Data.Map

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

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`.

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 :: (Monad m, Ord k) => k -> Map k a -> m 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
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)
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
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)
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
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)]
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 :: (Monad m, Ord k) => k -> Map k a -> m 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 :: Monad m => Map k a -> m (Map k a, (k, a))
maxView :: Monad m => Map k a -> m (Map k a, (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
Foldable (Map k)
Functor (Map k)
Traversable (Map k)
(Data k, Data a, Ord k) => Data (Map k a)
(Eq k, Eq a) => Eq (Map k a)
Ord k => Monoid (Map k v)
(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)

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.
(\\) :: Ord k => Map k a -> Map k b -> Map k a
O(n+m). See `difference`.
Query
null :: Map k a -> Bool
O(1). Is the map empty?
size :: Map k a -> Int
O(1). The number of elements in the map.
member :: Ord k => k -> Map k a -> Bool
O(log n). Is the key a member of the map?
notMember :: Ord k => k -> Map k a -> Bool
O(log n). Is the key not a member of the map?
lookup :: (Monad m, Ord k) => k -> Map k a -> m a
O(log n). Lookup the value at a key in the map. The function will `return` the result in the monad or `fail` in it the key isn't in the map. Often, the monad to use is `Maybe`, so you get either `(`Just` result)` or ``Nothing``.
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 `def` when the key is not in the map.
Construction
empty :: Map k a
O(1). The empty map.
singleton :: k -> a -> Map k a
O(1). A map with a single element.
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, i.e. `insert` is equivalent to ``insertWith` `const``.
insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a
O(log n). Insert with a combining function. ``insertWith` f key value mp` will insert the pair (key, value) into `mp` if key does not exist in the map. If the key does exist, the function will insert the pair `(key, f new_value old_value)`.
insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
O(log n). Insert with a combining function. ``insertWithKey` f key value mp` will insert the pair (key, value) into `mp` if key does not exist in the map. If the key does exist, the function will insert the pair `(key,f key new_value old_value)`. Note that the key passed to f is the same key passed to `insertWithKey`.
insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)
O(log n). The expression (``insertLookupWithKey` f k x map`) is a pair where the first element is equal to (``lookup` k map`) and the second element equal to (``insertWithKey` f k x map`).
insertWith' :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a
Same as `insertWith`, but the combining function is applied strictly.
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.
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.
adjust :: Ord 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.
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.
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`.
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`.
updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a, Map k a)
O(log n). Lookup and update.
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)`
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)
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.
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).
unions :: Ord k => [Map k a] -> Map k a
The union of a list of maps: (``unions` == `foldl` `union` `empty``).
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``).
Difference
difference :: Ord k => Map k a -> Map k b -> Map k a
O(n+m). Difference of two maps. The implementation uses an efficient hedge algorithm comparable with hedge-union.
differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a
O(n+m). Difference with a combining function. The implementation uses an efficient hedge algorithm comparable with hedge-union.
differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a
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.
Intersection
intersection :: Ord k => Map k a -> Map k b -> Map k a
O(n+m). Intersection of two maps. The values in the first map are returned, i.e. (``intersection` m1 m2 == `intersectionWith` `const` m1 m2`).
intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k c
O(n+m). Intersection with a combining function.
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) intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c intersectionWithKey f Tip t = Tip intersectionWithKey f t Tip = Tip intersectionWithKey f t1 t2 = intersectWithKey f t1 t2

intersectWithKey f Tip t = Tip intersectWithKey f t Tip = Tip intersectWithKey f t (Bin _ kx x l r) = case found of Nothing -> merge tl tr Just y -> join kx (f kx y x) tl tr where (lt,found,gt) = splitLookup kx t tl = intersectWithKey f lt l tr = intersectWithKey f gt r

Traversal
Map
map :: (a -> b) -> Map k a -> Map k b
O(n). Map a function over all values in the map.
mapWithKey :: (k -> a -> b) -> Map k a -> Map k b
O(n). Map a function over all values in the map.
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.
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.
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.
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`.
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. 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

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

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

Conversion
elems :: Map k a -> [a]
O(n). Return all elements of the map in the ascending order of their keys.
keys :: Map k a -> [k]
O(n). Return all keys of the map in ascending order.
keysSet :: Map k a -> Set k
O(n). The set of all keys of the map.
assocs :: Map k a -> [(k, a)]
O(n). Return all key/value pairs in the map in ascending key order.
Lists
toList :: Map k a -> [(k, a)]
O(n). Convert to a list of key/value pairs.
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`.
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`.
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`.
Ordered lists
toAscList :: Map k a -> [(k, a)]
O(n). Convert to an ascending 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.
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.
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.
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.
Filter
filter :: Ord k => (a -> Bool) -> Map k a -> Map k a
O(n). Filter all values that satisfy the predicate.
filterWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> Map k a
O(n). Filter all keys/values that satisfy the predicate.
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`.
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`.
mapMaybe :: Ord k => (a -> Maybe b) -> Map k a -> Map k b
O(n). Map values and collect the `Just` results.
mapMaybeWithKey :: Ord k => (k -> a -> Maybe b) -> Map k a -> Map k b
O(n). Map keys/values and collect the `Just` results.
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.
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.
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`.
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`.
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 :: (Monad m, Ord k) => k -> Map k a -> m 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.
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.
elemAt :: Int -> Map k a -> (k, a)
O(log n). Retrieve an element by index. Calls `error` when an invalid index is used.
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.
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`).
Min/Max
findMin :: Map k a -> (k, a)
O(log n). The minimal key of the map.
findMax :: Map k a -> (k, a)
O(log n). The maximal key of the map.
deleteMin :: Map k a -> Map k a
O(log n). Delete the minimal key.
deleteMax :: Map k a -> Map k a
O(log n). Delete the maximal key.
deleteFindMin :: Map k a -> ((k, a), Map k a)
O(log n). Delete and find the minimal element.
deleteFindMax :: Map k a -> ((k, a), Map k a)
O(log n). Delete and find the maximal element.
updateMin :: (a -> Maybe a) -> Map k a -> Map k a
O(log n). Update the value at the minimal key.
updateMax :: (a -> Maybe a) -> Map k a -> Map k a
O(log n). Update the value at the maximal key.
updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
O(log n). Update the value at the minimal key.
updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
O(log n). Update the value at the maximal key.
minView :: Monad m => Map k a -> m (Map k a, (k, a))
O(log n). Retrieves the minimal key of the map, and the map stripped from that element `fail`s (in the monad) when passed an empty map.
maxView :: Monad m => Map k a -> m (Map k a, (k, a))
O(log n). Retrieves the maximal key of the map, and the map stripped from that element `fail`s (in the monad) when passed an empty map.
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.
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.
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