An efficient implementation of maps from integer keys to values.

Since many function names (but not the type name) clash with Prelude names, this module is usually imported qualified, e.g.

  import Data.IntMap (IntMap)
  import qualified Data.IntMap 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).

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(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(log n). 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)). 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(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"

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"

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")])

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 Data.Map.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(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")]

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")]

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"

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). 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). Fold the values in the map, such that fold f z == Prelude.foldr f z . elems. For example,

 elems map = fold (:) [] map

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

O(n). Fold the keys and values in the map, such that foldWithKey f z == Prelude.foldr (uncurry f) z . toAscList. For example,

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

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

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

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

O(n). Return all keys of the map in ascending order.

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

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

O(n). Return all key/value pairs in the map in ascending key order.

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

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

 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)). 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

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

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

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

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")]

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")]

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). 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(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(log n). 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(log n). 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(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). 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)])

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"

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"

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"

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"

O(log n). Retrieves the minimal (key,value) pair of the map, and the map stripped of that element, or Nothing if passed an empty map.

 minViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((3,"b"), singleton 5 "a")
 minViewWithKey empty == Nothing

O(log n). Retrieves the maximal (key,value) pair of the map, and the map stripped of that element, or Nothing if passed an empty map.

 maxViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((5,"a"), singleton 3 "b")
 maxViewWithKey empty == Nothing