containers-0.5.0.0: Assorted concrete container types

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

Data.IntMap.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
- Min/Max
- Debugging

Description

An efficient implementation of maps from integer 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 `IntMap` type itself is shared between the lazy and strict modules, meaning that the same `IntMap` 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 Data.IntMap.Strict (IntMap)
      import qualified Data.IntMap.Strict as IntMap

The implementation is based on big-endian patricia trees. This data structure performs especially well on binary operations like `union` and `intersection`. However, my benchmarks show that it is also (much) faster on insertions and deletions when compared to a generic size-balanced map implementation (see Data.Map).

- Chris Okasaki and Andy Gill, "Fast Mergeable Integer Maps", Workshop on ML, September 1998, pages 77-86, http://citeseer.ist.psu.edu/okasaki98fast.html
- D.R. Morrison, "/PATRICIA -- Practical Algorithm To Retrieve Information Coded In Alphanumeric/", Journal of the ACM, 15(4), October 1968, pages 514-534.

Operation comments contain the operation time complexity in the Big-O notation http://en.wikipedia.org/wiki/Big_O_notation. Many operations have a worst-case complexity of O(min(n,W)). This means that the operation can become linear in the number of elements with a maximum of W -- the number of bits in an `Int` (32 or 64).

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 IntMap a
- type Key = Int
- (!) :: IntMap a -> Key -> a
- (\\) :: IntMap a -> IntMap b -> IntMap a
- null :: IntMap a -> Bool
- size :: IntMap a -> Int
- member :: Key -> IntMap a -> Bool
- notMember :: Key -> IntMap a -> Bool
- lookup :: Key -> IntMap a -> Maybe a
- findWithDefault :: a -> Key -> IntMap a -> a
- lookupLT :: Key -> IntMap a -> Maybe (Key, a)
- lookupGT :: Key -> IntMap a -> Maybe (Key, a)
- lookupLE :: Key -> IntMap a -> Maybe (Key, a)
- lookupGE :: Key -> IntMap a -> Maybe (Key, a)
- empty :: IntMap a
- singleton :: Key -> a -> IntMap a
- insert :: Key -> a -> IntMap a -> IntMap a
- insertWith :: (a -> a -> a) -> Key -> a -> IntMap a -> IntMap a
- insertWithKey :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> IntMap a
- insertLookupWithKey :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> (Maybe a, IntMap a)
- delete :: Key -> IntMap a -> IntMap a
- adjust :: (a -> a) -> Key -> IntMap a -> IntMap a
- adjustWithKey :: (Key -> a -> a) -> Key -> IntMap a -> IntMap a
- update :: (a -> Maybe a) -> Key -> IntMap a -> IntMap a
- updateWithKey :: (Key -> a -> Maybe a) -> Key -> IntMap a -> IntMap a
- updateLookupWithKey :: (Key -> a -> Maybe a) -> Key -> IntMap a -> (Maybe a, IntMap a)
- alter :: (Maybe a -> Maybe a) -> Key -> IntMap a -> IntMap a
- union :: IntMap a -> IntMap a -> IntMap a
- unionWith :: (a -> a -> a) -> IntMap a -> IntMap a -> IntMap a
- unionWithKey :: (Key -> a -> a -> a) -> IntMap a -> IntMap a -> IntMap a
- unions :: [IntMap a] -> IntMap a
- unionsWith :: (a -> a -> a) -> [IntMap a] -> IntMap a
- difference :: IntMap a -> IntMap b -> IntMap a
- differenceWith :: (a -> b -> Maybe a) -> IntMap a -> IntMap b -> IntMap a
- differenceWithKey :: (Key -> a -> b -> Maybe a) -> IntMap a -> IntMap b -> IntMap a
- intersection :: IntMap a -> IntMap b -> IntMap a
- intersectionWith :: (a -> b -> c) -> IntMap a -> IntMap b -> IntMap c
- intersectionWithKey :: (Key -> a -> b -> c) -> IntMap a -> IntMap b -> IntMap c
- mergeWithKey :: (Key -> a -> b -> Maybe c) -> (IntMap a -> IntMap c) -> (IntMap b -> IntMap c) -> IntMap a -> IntMap b -> IntMap c
- map :: (a -> b) -> IntMap a -> IntMap b
- mapWithKey :: (Key -> a -> b) -> IntMap a -> IntMap b
- traverseWithKey :: Applicative t => (Key -> a -> t b) -> IntMap a -> t (IntMap b)
- mapAccum :: (a -> b -> (a, c)) -> a -> IntMap b -> (a, IntMap c)
- mapAccumWithKey :: (a -> Key -> b -> (a, c)) -> a -> IntMap b -> (a, IntMap c)
- mapAccumRWithKey :: (a -> Key -> b -> (a, c)) -> a -> IntMap b -> (a, IntMap c)
- mapKeys :: (Key -> Key) -> IntMap a -> IntMap a
- mapKeysWith :: (a -> a -> a) -> (Key -> Key) -> IntMap a -> IntMap a
- mapKeysMonotonic :: (Key -> Key) -> IntMap a -> IntMap a
- foldr :: (a -> b -> b) -> b -> IntMap a -> b
- foldl :: (a -> b -> a) -> a -> IntMap b -> a
- foldrWithKey :: (Int -> a -> b -> b) -> b -> IntMap a -> b
- foldlWithKey :: (a -> Int -> b -> a) -> a -> IntMap b -> a
- foldr' :: (a -> b -> b) -> b -> IntMap a -> b
- foldl' :: (a -> b -> a) -> a -> IntMap b -> a
- foldrWithKey' :: (Int -> a -> b -> b) -> b -> IntMap a -> b
- foldlWithKey' :: (a -> Int -> b -> a) -> a -> IntMap b -> a
- elems :: IntMap a -> [a]
- keys :: IntMap a -> [Key]
- assocs :: IntMap a -> [(Key, a)]
- keysSet :: IntMap a -> IntSet
- fromSet :: (Key -> a) -> IntSet -> IntMap a
- toList :: IntMap a -> [(Key, a)]
- fromList :: [(Key, a)] -> IntMap a
- fromListWith :: (a -> a -> a) -> [(Key, a)] -> IntMap a
- fromListWithKey :: (Key -> a -> a -> a) -> [(Key, a)] -> IntMap a
- toAscList :: IntMap a -> [(Key, a)]
- toDescList :: IntMap a -> [(Key, a)]
- fromAscList :: [(Key, a)] -> IntMap a
- fromAscListWith :: (a -> a -> a) -> [(Key, a)] -> IntMap a
- fromAscListWithKey :: (Key -> a -> a -> a) -> [(Key, a)] -> IntMap a
- fromDistinctAscList :: [(Key, a)] -> IntMap a
- filter :: (a -> Bool) -> IntMap a -> IntMap a
- filterWithKey :: (Key -> a -> Bool) -> IntMap a -> IntMap a
- partition :: (a -> Bool) -> IntMap a -> (IntMap a, IntMap a)
- partitionWithKey :: (Key -> a -> Bool) -> IntMap a -> (IntMap a, IntMap a)
- mapMaybe :: (a -> Maybe b) -> IntMap a -> IntMap b
- mapMaybeWithKey :: (Key -> a -> Maybe b) -> IntMap a -> IntMap b
- mapEither :: (a -> Either b c) -> IntMap a -> (IntMap b, IntMap c)
- mapEitherWithKey :: (Key -> a -> Either b c) -> IntMap a -> (IntMap b, IntMap c)
- split :: Key -> IntMap a -> (IntMap a, IntMap a)
- splitLookup :: Key -> IntMap a -> (IntMap a, Maybe a, IntMap a)
- isSubmapOf :: Eq a => IntMap a -> IntMap a -> Bool
- isSubmapOfBy :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Bool
- isProperSubmapOf :: Eq a => IntMap a -> IntMap a -> Bool
- isProperSubmapOfBy :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Bool
- findMin :: IntMap a -> (Key, a)
- findMax :: IntMap a -> (Key, a)
- deleteMin :: IntMap a -> IntMap a
- deleteMax :: IntMap a -> IntMap a
- deleteFindMin :: IntMap a -> ((Key, a), IntMap a)
- deleteFindMax :: IntMap a -> ((Key, a), IntMap a)
- updateMin :: (a -> Maybe a) -> IntMap a -> IntMap a
- updateMax :: (a -> Maybe a) -> IntMap a -> IntMap a
- updateMinWithKey :: (Key -> a -> Maybe a) -> IntMap a -> IntMap a
- updateMaxWithKey :: (Key -> a -> Maybe a) -> IntMap a -> IntMap a
- minView :: IntMap a -> Maybe (a, IntMap a)
- maxView :: IntMap a -> Maybe (a, IntMap a)
- minViewWithKey :: IntMap a -> Maybe ((Key, a), IntMap a)
- maxViewWithKey :: IntMap a -> Maybe ((Key, a), IntMap a)
- showTree :: Show a => IntMap a -> String
- showTreeWith :: Show a => Bool -> Bool -> IntMap a -> String

# 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 IntMap a

A map of integers to values `a`.

Instances

Functor IntMap |
Typeable1 IntMap |
Foldable IntMap |
Traversable IntMap |
Eq a => Eq (IntMap a) |
Data a => Data (IntMap a) |
Ord a => Ord (IntMap a) |
Read e => Read (IntMap e) |
Show a => Show (IntMap a) |
Monoid (IntMap a) |
NFData a => NFData (IntMap a) |

type Key = Int

# Operators

(!) :: IntMap a -> Key -> a

O(min(n,W)). 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'

(\\) :: IntMap a -> IntMap b -> IntMap a

Same as `difference`.

# Query

null :: IntMap a -> Bool

O(1). Is the map empty?

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

size :: IntMap a -> Int

O(n). Number of elements in the map.

     size empty                                   == 0
     size (singleton 1 'a')                       == 1
     size (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3

member :: Key -> IntMap a -> Bool

O(min(n,W)). Is the key a member of the map?

     member 5 (fromList [(5,'a'), (3,'b')]) == True
     member 1 (fromList [(5,'a'), (3,'b')]) == False

notMember :: Key -> IntMap a -> Bool

O(min(n,W)). Is the key not a member of the map?

     notMember 5 (fromList [(5,'a'), (3,'b')]) == False
     notMember 1 (fromList [(5,'a'), (3,'b')]) == True

lookup :: Key -> IntMap a -> Maybe a

O(min(n,W)). Lookup the value at a key in the map. See also `lookup`.

findWithDefault :: a -> Key -> IntMap a -> a

O(min(n,W)). The expression `(`findWithDefault` def k map)` returns the value at key `k` or returns `def` when the key is not an element of the map.

     findWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x'
     findWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a'

lookupLT :: Key -> IntMap a -> Maybe (Key, a)

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 :: Key -> IntMap a -> Maybe (Key, a)

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 :: Key -> IntMap a -> Maybe (Key, a)

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 :: Key -> IntMap a -> Maybe (Key, a)

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 :: IntMap a

O(1). The empty map.

     empty      == fromList []
     size empty == 0

singleton :: Key -> a -> IntMap a

O(1). A map of one element.

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

## Insertion

insert :: Key -> a -> IntMap a -> IntMap a

O(min(n,W)). Insert a new key/value pair 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``.

     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 :: (a -> a -> a) -> Key -> a -> IntMap a -> IntMap a

O(min(n,W)). 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 `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 :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> IntMap a

O(min(n,W)). 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 `f key new_value old_value`.

     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"

If the key exists in the map, this function is lazy in `x` but strict in the result of `f`.

insertLookupWithKey :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> (Maybe a, IntMap a)

O(min(n,W)). 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 :: Key -> IntMap a -> IntMap a

O(min(n,W)). 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 :: (a -> a) -> Key -> IntMap a -> IntMap a

O(min(n,W)). Adjust a value at a specific key. 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 :: (Key -> a -> a) -> Key -> IntMap a -> IntMap a

O(min(n,W)). 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 :: (a -> Maybe a) -> Key -> IntMap a -> IntMap a

O(min(n,W)). 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 :: (Key -> a -> Maybe a) -> Key -> IntMap a -> IntMap a

O(min(n,W)). The expression (``update` 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 :: (Key -> a -> Maybe a) -> Key -> IntMap a -> (Maybe a, IntMap a)

O(min(n,W)). Lookup and update. The function returns original value, if it is updated. This is different behavior than `updateLookupWithKey`. 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 "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 :: (Maybe a -> Maybe a) -> Key -> IntMap a -> IntMap 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 an `IntMap`. In short : ``lookup` k (`alter` f k m) = f (`lookup` k m)`.

# Combine

## Union

union :: IntMap a -> IntMap a -> IntMap a

O(n+m). The (left-biased) union of two maps. It prefers the first map when duplicate keys are encountered, i.e. (``union` == `unionWith` `const``).

     union (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "a"), (7, "C")]

unionWith :: (a -> a -> a) -> IntMap a -> IntMap a -> IntMap a

O(n+m). The union with a combining function.

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

unionWithKey :: (Key -> a -> a -> a) -> IntMap a -> IntMap a -> IntMap a

O(n+m). The union with a combining function.

     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 :: [IntMap a] -> IntMap a

The union of a list of maps.

     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 :: (a -> a -> a) -> [IntMap a] -> IntMap a

The union of a list of maps, with a combining operation.

     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 :: IntMap a -> IntMap b -> IntMap a

O(n+m). Difference between two maps (based on keys).

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

differenceWith :: (a -> b -> Maybe a) -> IntMap a -> IntMap b -> IntMap a

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

     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 :: (Key -> a -> b -> Maybe a) -> IntMap a -> IntMap b -> IntMap 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`.

     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 :: IntMap a -> IntMap b -> IntMap a

O(n+m). The (left-biased) intersection of two maps (based on keys).

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

intersectionWith :: (a -> b -> c) -> IntMap a -> IntMap b -> IntMap c

O(n+m). The intersection with a combining function.

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

intersectionWithKey :: (Key -> a -> b -> c) -> IntMap a -> IntMap b -> IntMap c

O(n+m). The intersection with a combining function.

     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 :: (Key -> a -> b -> Maybe c) -> (IntMap a -> IntMap c) -> (IntMap b -> IntMap c) -> IntMap a -> IntMap b -> IntMap c

O(n+m). A high-performance universal combining function. Using `mergeWithKey`, all combining functions can be defined without any loss of efficiency (with exception of `union`, `difference` and `intersection`, where sharing of some nodes is lost with `mergeWithKey`).

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) -> IntMap a -> IntMap 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 :: (Key -> a -> b) -> IntMap a -> IntMap 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 => (Key -> a -> t b) -> IntMap a -> t (IntMap 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 -> IntMap b -> (a, IntMap 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 -> Key -> b -> (a, c)) -> a -> IntMap b -> (a, IntMap 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 -> Key -> b -> (a, c)) -> a -> IntMap b -> (a, IntMap c)

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

mapKeys :: (Key -> Key) -> IntMap a -> IntMap a

O(n*min(n,W)). ``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 :: (a -> a -> a) -> (Key -> Key) -> IntMap a -> IntMap 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 :: (Key -> Key) -> IntMap a -> IntMap a

O(n*min(n,W)). ``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 slightly better performance than `mapKeys`.

     mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")]

# Folds

foldr :: (a -> b -> b) -> b -> IntMap 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 -> IntMap 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 :: (Int -> a -> b -> b) -> b -> IntMap 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 -> Int -> b -> a) -> a -> IntMap 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 -> IntMap 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 -> IntMap 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' :: (Int -> a -> b -> b) -> b -> IntMap 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 -> Int -> b -> a) -> a -> IntMap 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 :: IntMap 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 :: IntMap a -> [Key]

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 :: IntMap a -> [(Key, a)]

O(n). An alias for `toAscList`. Returns 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 :: IntMap a -> IntSet

O(n*min(n,W)). The set of all keys of the map.

     keysSet (fromList [(5,"a"), (3,"b")]) == Data.IntSet.fromList [3,5]
     keysSet empty == Data.IntSet.empty

fromSet :: (Key -> a) -> IntSet -> IntMap 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.IntSet.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")]
     fromSet undefined Data.IntSet.empty == empty

## Lists

toList :: IntMap a -> [(Key, 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 :: [(Key, a)] -> IntMap a

O(n*min(n,W)). Create a map from a list of key/value pairs.

     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 :: (a -> a -> a) -> [(Key, a)] -> IntMap a

O(n*min(n,W)). Create 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 :: (Key -> a -> a -> a) -> [(Key, a)] -> IntMap a

O(n*min(n,W)). Build a map from a list of key/value pairs with a combining function. See also fromAscListWithKey'.

     fromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "ab"), (5, "aba")]
     fromListWith (++) [] == empty

## Ordered lists

toAscList :: IntMap a -> [(Key, 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 :: IntMap a -> [(Key, 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 :: [(Key, a)] -> IntMap a

O(n). Build a map from a list of key/value pairs where the keys are in ascending order.

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

fromAscListWith :: (a -> a -> a) -> [(Key, a)] -> IntMap a

O(n). Build a map from a list of key/value pairs where the keys are in ascending order, with a combining function on equal keys. The precondition (input list is ascending) is not checked.

     fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]

fromAscListWithKey :: (Key -> a -> a -> a) -> [(Key, a)] -> IntMap a

O(n). Build a map from a list of key/value pairs where the keys are in ascending order, with a combining function on equal keys. The precondition (input list is ascending) is not checked.

     fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]

fromDistinctAscList :: [(Key, a)] -> IntMap a

O(n). Build a map from a list of key/value pairs where the keys are in ascending order and all distinct. The precondition (input list is strictly ascending) is not checked.

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

# Filter

filter :: (a -> Bool) -> IntMap a -> IntMap a

O(n). Filter all values that satisfy some 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 :: (Key -> a -> Bool) -> IntMap a -> IntMap a

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

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

partition :: (a -> Bool) -> IntMap a -> (IntMap a, IntMap a)

O(n). Partition the map according to some 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 :: (Key -> a -> Bool) -> IntMap a -> (IntMap a, IntMap a)

O(n). Partition the map according to some 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) -> IntMap a -> IntMap 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 :: (Key -> a -> Maybe b) -> IntMap a -> IntMap 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) -> IntMap a -> (IntMap b, IntMap 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 :: (Key -> a -> Either b c) -> IntMap a -> (IntMap b, IntMap 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 :: Key -> IntMap a -> (IntMap a, IntMap a)

O(min(n,W)). The expression (``split` k map`) is a pair `(map1,map2)` where all keys in `map1` are lower than `k` and all 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 :: Key -> IntMap a -> (IntMap a, Maybe a, IntMap a)

O(min(n,W)). Performs a `split` but also returns whether the pivot key was found in the original 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 :: Eq a => IntMap a -> IntMap a -> Bool

O(n+m). Is this a submap? Defined as (``isSubmapOf` = `isSubmapOfBy` (==)`).

isSubmapOfBy :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Bool

O(n+m). The expression (``isSubmapOfBy` f m1 m2`) returns `True` if 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`:

     isSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
     isSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
     isSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])

But the following are all `False`:

     isSubmapOfBy (==) (fromList [(1,2)]) (fromList [(1,1),(2,2)])
     isSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
     isSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])

isProperSubmapOf :: Eq a => IntMap a -> IntMap a -> Bool

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

isProperSubmapOfBy :: (a -> b -> Bool) -> IntMap a -> IntMap 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)])

# Min/Max

findMin :: IntMap a -> (Key, a)

O(min(n,W)). The minimal key of the map.

findMax :: IntMap a -> (Key, a)

O(min(n,W)). The maximal key of the map.

deleteMin :: IntMap a -> IntMap a

O(min(n,W)). Delete the minimal key. An error is thrown if the IntMap is already empty. Note, this is not the same behavior Map.

deleteMax :: IntMap a -> IntMap a

O(min(n,W)). Delete the maximal key. An error is thrown if the IntMap is already empty. Note, this is not the same behavior Map.

deleteFindMin :: IntMap a -> ((Key, a), IntMap a)

O(min(n,W)). Delete and find the minimal element.

deleteFindMax :: IntMap a -> ((Key, a), IntMap a)

O(min(n,W)). Delete and find the maximal element.

updateMin :: (a -> Maybe a) -> IntMap a -> IntMap 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) -> IntMap a -> IntMap 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 :: (Key -> a -> Maybe a) -> IntMap a -> IntMap 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 :: (Key -> a -> Maybe a) -> IntMap a -> IntMap 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 :: IntMap a -> Maybe (a, IntMap a)

O(min(n,W)). Retrieves the minimal key of the map, and the map stripped of that element, or `Nothing` if passed an empty map.

maxView :: IntMap a -> Maybe (a, IntMap a)

O(min(n,W)). Retrieves the maximal key of the map, and the map stripped of that element, or `Nothing` if passed an empty map.

minViewWithKey :: IntMap a -> Maybe ((Key, a), IntMap a)

O(min(n,W)). 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 :: IntMap a -> Maybe ((Key, a), IntMap a)

O(min(n,W)). 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 a => IntMap a -> String

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

showTreeWith :: Show a => Bool -> Bool -> IntMap a -> String

O(n). The expression (``showTreeWith` hang wide map`) shows the tree that implements the map. 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.