A map of integers to values 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'

\(O(\min(n,W))\). Find the value at a key. Returns Nothing when the element can not be found.

fromList [(5,'a'), (3,'b')] !? 1 == Nothing
fromList [(5,'a'), (3,'b')] !? 5 == Just 'a'

Since: containers-0.5.11

Same as difference.

\(O(1)\). Is the map empty?

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

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

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

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

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

\(O(\min(n,W))\). 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

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

\(O(\min(n,W))\). 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'

\(O(\min(n,W))\). 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')

\(O(\min(n,W))\). 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

\(O(\min(n,W))\). 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')

\(O(\min(n,W))\). 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

\(O(n+m)\). Check whether the key sets of two maps are disjoint (i.e. their intersection is empty).

disjoint (fromList [(2,'a')]) (fromList [(1,()), (3,())])   == True
disjoint (fromList [(2,'a')]) (fromList [(1,'a'), (2,'b')]) == False
disjoint (fromList [])        (fromList [])                 == True

disjoint a b == null (intersection a b)

Since: containers-0.6.2.1

\(O(1)\). The empty map.

empty      == fromList []
size empty == 0

\(O(1)\). A map of one element.

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

\(O(\min(n,W))\). 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'

\(O(\min(n,W))\). 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"

Also see the performance note on fromListWith.

\(O(\min(n,W))\). 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"

Also see the performance note on fromListWith.

\(O(\min(n,W))\). 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")])

Also see the performance note on fromListWith.

\(O(\min(n,W))\). 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

\(O(\min(n,W))\). 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

\(O(\min(n,W))\). 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

\(O(\min(n,W))\). 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"

\(O(\min(n,W))\). 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"

\(O(\min(n,W))\). 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")

\(O(\min(n,W))\). 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).

\(O(\min(n,W))\). 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.

Note: alterF is a flipped version of the at combinator from Control.Lens.At.

Since: containers-0.5.8

\(O(n+m)\). 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")]

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

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

Also see the performance note on fromListWith.

\(O(n+m)\). 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")]

Also see the performance note on fromListWith.

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")]

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")]

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

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

\(O(n+m)\). 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"

\(O(n+m)\). 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"

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

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

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

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

\(O(n+m)\). 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"

Relate the keys of one map to the values of the other, by using the values of the former as keys for lookups in the latter.

Complexity: \( O(n * \min(m,W)) \), where \(m\) is the size of the first argument

compose (fromList [('a', "A"), ('b', "B")]) (fromList [(1,'a'),(2,'b'),(3,'z')]) = fromList [(1,"A"),(2,"B")]

(compose bc ab !?) = (bc !?) <=< (ab !?)

Note: Prior to v0.6.4, Data.IntMap.Strict exposed a version of compose that forced the values of the output IntMap. This version does not force these values.

Since: containers-0.6.3.1

A tactic for dealing with keys present in one map but not the other in merge.

A tactic of type SimpleWhenMissing x z is an abstract representation of a function of type Key -> x -> Maybe z.

Since: containers-0.5.9

A tactic for dealing with keys present in both maps in merge.

A tactic of type SimpleWhenMatched x y z is an abstract representation of a function of type Key -> x -> y -> Maybe z.

Since: containers-0.5.9

Along with zipWithMaybeAMatched, witnesses the isomorphism between WhenMatched f x y z and Key -> x -> y -> f (Maybe z).

Since: containers-0.5.9

Along with traverseMaybeMissing, witnesses the isomorphism between WhenMissing f x y and Key -> x -> f (Maybe y).

Since: containers-0.5.9

Merge two maps.

merge takes two WhenMissing tactics, a WhenMatched tactic and two maps. It uses the tactics to merge the maps. Its behavior is best understood via its fundamental tactics, mapMaybeMissing and zipWithMaybeMatched.

Consider

merge (mapMaybeMissing g1)
             (mapMaybeMissing g2)
             (zipWithMaybeMatched f)
             m1 m2

Take, for example,

m1 = [(0, 'a'), (1, 'b'), (3, 'c'), (4, 'd')]
m2 = [(1, "one"), (2, "two"), (4, "three")]

merge will first "align" these maps by key:

m1 = [(0, 'a'), (1, 'b'),               (3, 'c'), (4, 'd')]
m2 =           [(1, "one"), (2, "two"),           (4, "three")]

It will then pass the individual entries and pairs of entries to g1, g2, or f as appropriate:

maybes = [g1 0 'a', f 1 'b' "one", g2 2 "two", g1 3 'c', f 4 'd' "three"]

This produces a Maybe for each key:

keys =     0        1          2           3        4
results = [Nothing, Just True, Just False, Nothing, Just True]

Finally, the Just results are collected into a map:

return value = [(1, True), (2, False), (4, True)]

The other tactics below are optimizations or simplifications of mapMaybeMissing for special cases. Most importantly,

• dropMissing drops all the keys.
• preserveMissing leaves all the entries alone.

When merge is given three arguments, it is inlined at the call site. To prevent excessive inlining, you should typically use merge to define your custom combining functions.

Examples:

unionWithKey f = merge preserveMissing preserveMissing (zipWithMatched f)

intersectionWithKey f = merge dropMissing dropMissing (zipWithMatched f)

differenceWith f = merge diffPreserve diffDrop f

symmetricDifference = merge diffPreserve diffPreserve (\ _ _ _ -> Nothing)

mapEachPiece f g h = merge (diffMapWithKey f) (diffMapWithKey g)

Since: containers-0.5.9

When a key is found in both maps, apply a function to the key and values and maybe use the result in the merged map.

zipWithMaybeMatched
  :: (Key -> x -> y -> Maybe z)
  -> SimpleWhenMatched x y z

Since: containers-0.5.9

When a key is found in both maps, apply a function to the key and values and use the result in the merged map.

zipWithMatched
  :: (Key -> x -> y -> z)
  -> SimpleWhenMatched x y z

Since: containers-0.5.9

Map over the entries whose keys are missing from the other map, optionally removing some. This is the most powerful SimpleWhenMissing tactic, but others are usually more efficient.

mapMaybeMissing :: (Key -> x -> Maybe y) -> SimpleWhenMissing x y

mapMaybeMissing f = traverseMaybeMissing (\k x -> pure (f k x))

but mapMaybeMissing uses fewer unnecessary Applicative operations.

Since: containers-0.5.9

Drop all the entries whose keys are missing from the other map.

dropMissing :: SimpleWhenMissing x y

dropMissing = mapMaybeMissing (\_ _ -> Nothing)

but dropMissing is much faster.

Since: containers-0.5.9

Preserve, unchanged, the entries whose keys are missing from the other map.

preserveMissing :: SimpleWhenMissing x x

preserveMissing = Merge.Lazy.mapMaybeMissing (\_ x -> Just x)

but preserveMissing is much faster.

Since: containers-0.5.9

Map over the entries whose keys are missing from the other map.

mapMissing :: (k -> x -> y) -> SimpleWhenMissing x y

mapMissing f = mapMaybeMissing (\k x -> Just $ f k x)

but mapMissing is somewhat faster.

Since: containers-0.5.9

Filter the entries whose keys are missing from the other map.

filterMissing :: (k -> x -> Bool) -> SimpleWhenMissing x x

filterMissing f = Merge.Lazy.mapMaybeMissing $ \k x -> guard (f k x) *> Just x

but this should be a little faster.

Since: containers-0.5.9

A tactic for dealing with keys present in one map but not the other in merge or mergeA.

A tactic of type WhenMissing f k x z is an abstract representation of a function of type Key -> x -> f (Maybe z).

Since: containers-0.5.9

A tactic for dealing with keys present in both maps in merge or mergeA.

A tactic of type WhenMatched f x y z is an abstract representation of a function of type Key -> x -> y -> f (Maybe z).

Since: containers-0.5.9

An applicative version of merge.

mergeA takes two WhenMissing tactics, a WhenMatched tactic and two maps. It uses the tactics to merge the maps. Its behavior is best understood via its fundamental tactics, traverseMaybeMissing and zipWithMaybeAMatched.

Consider

mergeA (traverseMaybeMissing g1)
              (traverseMaybeMissing g2)
              (zipWithMaybeAMatched f)
              m1 m2

Take, for example,

m1 = [(0, 'a'), (1, 'b'), (3,'c'), (4, 'd')]
m2 = [(1, "one"), (2, "two"), (4, "three")]

mergeA will first "align" these maps by key:

m1 = [(0, 'a'), (1, 'b'),               (3, 'c'), (4, 'd')]
m2 =           [(1, "one"), (2, "two"),           (4, "three")]

It will then pass the individual entries and pairs of entries to g1, g2, or f as appropriate:

actions = [g1 0 'a', f 1 'b' "one", g2 2 "two", g1 3 'c', f 4 'd' "three"]

Next, it will perform the actions in the actions list in order from left to right.

keys =     0        1          2           3        4
results = [Nothing, Just True, Just False, Nothing, Just True]

Finally, the Just results are collected into a map:

return value = [(1, True), (2, False), (4, True)]

The other tactics below are optimizations or simplifications of traverseMaybeMissing for special cases. Most importantly,

• dropMissing drops all the keys.
• preserveMissing leaves all the entries alone.
• mapMaybeMissing does not use the Applicative context.

When mergeA is given three arguments, it is inlined at the call site. To prevent excessive inlining, you should generally only use mergeA to define custom combining functions.

Since: containers-0.5.9

When a key is found in both maps, apply a function to the key and values, perform the resulting action, and maybe use the result in the merged map.

This is the fundamental WhenMatched tactic.

Since: containers-0.5.9

When a key is found in both maps, apply a function to the key and values to produce an action and use its result in the merged map.

Since: containers-0.5.9

Traverse over the entries whose keys are missing from the other map, optionally producing values to put in the result. This is the most powerful WhenMissing tactic, but others are usually more efficient.

Since: containers-0.5.9

Traverse over the entries whose keys are missing from the other map.

Since: containers-0.5.9

Filter the entries whose keys are missing from the other map using some Applicative action.

filterAMissing f = Merge.Lazy.traverseMaybeMissing $
  \k x -> (\b -> guard b *> Just x) <$> f k x

but this should be a little faster.

Since: containers-0.5.9

\(O(n+m)\). 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.

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

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

\(O(n)\). 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")]

\(O(n)\). 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

\(O(n)\). Traverse keys/values and collect the Just results.

Since: containers-0.6.4

\(O(n)\). 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")])

\(O(n)\). 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")])

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

\(O(n \min(n,W))\). 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"

\(O(n \min(n,W))\). 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"

Also see the performance note on fromListWith.

\(O(n)\). mapKeysMonotonic f s == mapKeys f s, but works only when f is strictly monotonic. That is, for any values x and y, if x < y then f x < f y. The precondition is not checked. Semi-formally, we have:

and [x < y ==> f x < f y | x <- ls, y <- ls]
                    ==> mapKeysMonotonic f s == mapKeys f s
    where ls = keys s

This means that f maps distinct original keys to distinct resulting keys. This function has slightly better performance than mapKeys.

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

\(O(n)\). 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

\(O(n)\). 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

\(O(n)\). 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)"

\(O(n)\). 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)"

\(O(n)\). 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.

Since: containers-0.5.4

\(O(n)\). 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.

\(O(n)\). 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.

\(O(n)\). 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.

\(O(n)\). 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.

\(O(n)\). 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 == []

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

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

\(O(n)\). 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 == []

\(O(n)\). The set of all keys of the map.

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

\(O(n)\). 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

\(O(n)\). 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 == []

\(O(n \min(n,W))\). 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")]

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

fromListWith (++) [(5,"a"), (5,"b"), (3,"x"), (5,"c")] == fromList [(3, "x"), (5, "cba")]
fromListWith (++) [] == empty

Note the reverse ordering of "cba" in the example.

The symmetric combining function f is applied in a left-fold over the list, as f new old.

Performance

You should ensure that the given f is fast with this order of arguments.

Symmetric functions may be slow in one order, and fast in another. For the common case of collecting values of matching keys in a list, as above:

The complexity of (++) a b is \(O(a)\), so it is fast when given a short list as its first argument. Thus:

fromListWith       (++)  (replicate 1000000 (3, "x"))   -- O(n),  fast
fromListWith (flip (++)) (replicate 1000000 (3, "x"))   -- O(n²), extremely slow

because they evaluate as, respectively:

fromList [(3, "x" ++ ("x" ++ "xxxxx..xxxxx"))]   -- O(n)
fromList [(3, ("xxxxx..xxxxx" ++ "x") ++ "x")]   -- O(n²)

Thus, to get good performance with an operation like (++) while also preserving the same order as in the input list, reverse the input:

fromListWith (++) (reverse [(5,"a"), (5,"b"), (5,"c")]) == fromList [(5, "abc")]

and it is always fast to combine singleton-list values [v] with fromListWith (++), as in:

fromListWith (++) $ reverse $ map (\(k, v) -> (k, [v])) someListOfTuples

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

let f key new_value old_value = show key ++ ":" ++ new_value ++ "|" ++ old_value
fromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"c")] == fromList [(3, "3:a|b"), (5, "5:c|5:b|a")]
fromListWithKey f [] == empty

Also see the performance note on fromListWith.

\(O(n)\). 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")]

\(O(n)\). 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")]

\(O(n)\). 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")]

\(O(n)\). 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")]

Also see the performance note on fromListWith.

\(O(n)\). 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.

let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "5:b|a")]

Also see the performance note on fromListWith.

\(O(n)\). 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")]

\(O(n)\). 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

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

filterWithKey (\k _ -> k > 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "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: containers-0.5.8

\(O(n+m)\). Remove all the keys in a given set from a map.

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

Since: containers-0.5.8

\(O(n)\). 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")])

\(O(n)\). 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")])

\(O(\min(n,W))\). Take while a predicate on the keys holds. The user is responsible for ensuring that for all Ints, j < k ==> p j >= p k. See note at spanAntitone.

takeWhileAntitone p = fromDistinctAscList . takeWhile (p . fst) . toList
takeWhileAntitone p = filterWithKey (\k _ -> p k)

Since: containers-0.6.7

\(O(\min(n,W))\). Drop while a predicate on the keys holds. The user is responsible for ensuring that for all Ints, j < k ==> p j >= p k. See note at spanAntitone.

dropWhileAntitone p = fromDistinctAscList . dropWhile (p . fst) . toList
dropWhileAntitone p = filterWithKey (\k _ -> not (p k))

Since: containers-0.6.7

\(O(\min(n,W))\). Divide a map at the point where a predicate on the keys stops holding. The user is responsible for ensuring that for all Ints, j < k ==> p j >= p k.

spanAntitone p xs = (takeWhileAntitone p xs, dropWhileAntitone p xs)
spanAntitone p xs = partitionWithKey (\k _ -> p k) xs

Note: if p is not actually antitone, then spanAntitone will split the map at some unspecified point.

Since: containers-0.6.7

\(O(n)\). 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"

\(O(n)\). 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"

\(O(n)\). 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")])

\(O(n)\). 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")])

\(O(\min(n,W))\). 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)

\(O(\min(n,W))\). 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)

\(O(1)\). 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.

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

\(O(n+m)\). 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)])

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

\(O(n+m)\). Is this a proper submap? (ie. a submap but not equal). The expression (isProperSubmapOfBy f m1 m2) returns True when keys m1 and keys 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)])

\(O(\min(n,W))\). The minimal key of the map. Returns Nothing if the map is empty.

\(O(\min(n,W))\). The maximal key of the map. Returns Nothing if the map is empty.

\(O(\min(n,W))\). The minimal key of the map. Calls error if the map is empty. Use minViewWithKey if the map may be empty.

\(O(\min(n,W))\). The maximal key of the map. Calls error if the map is empty. Use maxViewWithKey if the map may be empty.

\(O(\min(n,W))\). 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.

\(O(\min(n,W))\). 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.

\(O(\min(n,W))\). Delete and find the minimal element. This function throws an error if the map is empty. Use minViewWithKey if the map may be empty.

\(O(\min(n,W))\). Delete and find the maximal element. This function throws an error if the map is empty. Use maxViewWithKey if the map may be empty.

\(O(\min(n,W))\). 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"

\(O(\min(n,W))\). 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"

\(O(\min(n,W))\). 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"

\(O(\min(n,W))\). 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"

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

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

\(O(\min(n,W))\). 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

\(O(\min(n,W))\). 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

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

\(O(n \min(n,W))\). 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.

Should this key follow the left subtree of a Bin with switching bit m? N.B., the answer is only valid when match i p m is true.

Does the key i differ from the prefix p before getting to the switching bit m?

Does the key i match the prefix p (up to but not including bit m)?

The prefix of key i up to (but not including) the switching bit m.

The prefix of key i up to (but not including) the switching bit m.

Does the left switching bit specify a shorter prefix?

The first switching bit where the two prefixes disagree.

Return a word where only the highest bit is set.

Map covariantly over a WhenMissing f x.

Since: containers-0.5.9

Map covariantly over a WhenMatched f x y.

Since: containers-0.5.9

Map contravariantly over a WhenMissing f _ x.

Since: containers-0.5.9

Map contravariantly over a WhenMatched f _ y z.

Since: containers-0.5.9

Map contravariantly over a WhenMatched f x _ z.

Since: containers-0.5.9

Map covariantly over a WhenMissing f x, using only a 'Functor f' constraint.

Map covariantly over a WhenMatched f k x, using only a 'Functor f' constraint.