Data.IntMap.Strict

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 Data.IntMap.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 Data.IntMap.Lazy module, so if they are used on strict maps, the resulting maps will be lazy.

Strictness properties

This module satisfies the following strictness properties:

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

Here's an example illustrating the first property:

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

Methods

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

(<$) :: a -> IntMap b -> IntMap a

Foldable IntMap

Methods

fold :: Monoid m => IntMap m -> m

foldMap :: Monoid m => (a -> m) -> IntMap a -> m

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

foldr' :: (a -> b -> b) -> b -> IntMap a -> b

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

foldl' :: (b -> a -> b) -> b -> IntMap a -> b

foldr1 :: (a -> a -> a) -> IntMap a -> a

foldl1 :: (a -> a -> a) -> IntMap a -> a

toList :: IntMap a -> [a]

null :: IntMap a -> Bool

length :: IntMap a -> Int

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

maximum :: Ord a => IntMap a -> a

minimum :: Ord a => IntMap a -> a

sum :: Num a => IntMap a -> a

product :: Num a => IntMap a -> a

Traversable IntMap

Methods

traverse :: Applicative f => (a -> f b) -> IntMap a -> f (IntMap b)

sequenceA :: Applicative f => IntMap (f a) -> f (IntMap a)

mapM :: Monad m => (a -> m b) -> IntMap a -> m (IntMap b)

sequence :: Monad m => IntMap (m a) -> m (IntMap a)

IsList (IntMap a)

Associated Types

type Item (IntMap a) :: *

Methods

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

fromListN :: Int -> [Item (IntMap a)] -> IntMap a

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

Eq a => Eq (IntMap a)

Methods

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

(/=) :: IntMap a -> IntMap a -> Bool

Data a => Data (IntMap a)

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> IntMap a -> c (IntMap a)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (IntMap a)

toConstr :: IntMap a -> Constr

dataTypeOf :: IntMap a -> DataType

dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c (IntMap a))

dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (IntMap a))

gmapT :: (forall b. Data b => b -> b) -> IntMap a -> IntMap a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> IntMap a -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> IntMap a -> r

gmapQ :: (forall d. Data d => d -> u) -> IntMap a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> IntMap a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> IntMap a -> m (IntMap a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> IntMap a -> m (IntMap a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> IntMap a -> m (IntMap a)

Ord a => Ord (IntMap a)

Methods

compare :: IntMap a -> IntMap a -> Ordering

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

(<=) :: IntMap a -> IntMap a -> Bool

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

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

max :: IntMap a -> IntMap a -> IntMap a

min :: IntMap a -> IntMap a -> IntMap a

Read e => Read (IntMap e)

Methods

readsPrec :: Int -> ReadS (IntMap e)

readList :: ReadS [IntMap e]

readPrec :: ReadPrec (IntMap e)

readListPrec :: ReadPrec [IntMap e]

Show a => Show (IntMap a)

Methods

showsPrec :: Int -> IntMap a -> ShowS

show :: IntMap a -> String

showList :: [IntMap a] -> ShowS

Semigroup (IntMap a)

Methods

(<>) :: IntMap a -> IntMap a -> IntMap a

sconcat :: NonEmpty (IntMap a) -> IntMap a

stimes :: Integral b => b -> IntMap a -> IntMap a

Monoid (IntMap a)

Methods

mempty :: IntMap a

mappend :: IntMap a -> IntMap a -> IntMap a

mconcat :: [IntMap a] -> IntMap a

NFData a => NFData (IntMap a)

Methods

rnf :: IntMap a -> ()

type Item (IntMap a)

type Item (IntMap a) = (Key, 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 infixl 9

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(min(n,W)). 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).

alterF :: Functor f => (Maybe a -> f (Maybe a)) -> Key -> IntMap a -> f (IntMap a)

O(log n). The expression (alterF f k map) alters the value x at k, or absence thereof. alterF can be used to inspect, insert, delete, or update a value in an IntMap. In short : lookup k $ alterF f k m = f (lookup k m).

Example:

interactiveAlter :: Int -> IntMap String -> IO (IntMap String)
interactiveAlter k m = alterF f k m where
  f Nothing -> do
     putStrLn $ show k ++
         " was not found in the map. Would you like to add it?"
     getUserResponse1 :: IO (Maybe String)
  f (Just old) -> do
     putStrLn "The key is currently bound to " ++ show old ++
         ". Would you like to change or delete it?"
     getUserresponse2 :: IO (Maybe String)

alterF is the most general operation for working with an individual key that may or may not be in a given map.

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 IntMaps 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 :: (Key -> 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 -> Key -> 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)"

foldMapWithKey :: Monoid m => (Key -> a -> m) -> IntMap a -> m

O(n). Fold the keys and values in the map using the given monoid, such that

foldMapWithKey f = fold . mapWithKey f

This can be an asymptotically faster than foldrWithKey or foldlWithKey for some monoids.

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' :: (Key -> 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 -> Key -> 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"

restrictKeys :: IntMap a -> IntSet -> IntMap a

O(n+m). The restriction of a map to the keys in a set.

m restrictKeys s = filterWithKey (k _ -> k `member' s) m

Since: 0.5.8

withoutKeys :: IntMap a -> IntSet -> IntMap a

Remove all the keys in a given set from a map.

m withoutKeys s = filterWithKey (k _ -> k `notMember' s) m

Since: 0.5.8

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)

splitRoot :: IntMap a -> [IntMap a]

O(1). Decompose a map into pieces based on the structure of the underlying tree. This function is useful for consuming a map in parallel.

No guarantee is made as to the sizes of the pieces; an internal, but deterministic process determines this. However, it is guaranteed that the pieces returned will be in ascending order (all elements in the first submap less than all elements in the second, and so on).

Examples:

splitRoot (fromList (zip [1..6::Int] ['a'..])) ==
  [fromList [(1,'a'),(2,'b'),(3,'c')],fromList [(4,'d'),(5,'e'),(6,'f')]]

splitRoot empty == []

Note that the current implementation does not return more than two submaps, but you should not depend on this behaviour because it can change in the future without notice.

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. Returns an empty map if the map is empty.

Note that this is a change of behaviour for consistency with Map – versions prior to 0.5 threw an error if the IntMap was already empty.

deleteMax :: IntMap a -> IntMap a

O(min(n,W)). Delete the maximal key. Returns an empty map if the map is empty.

Note that this is a change of behaviour for consistency with Map – versions prior to 0.5 threw an error if the IntMap was already empty.

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.