An efficient implementation of maps from keys to values (dictionaries).

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

  import Data.Map (Map)

  import qualified Data.Map as Map

The implementation of Map is based on size balanced binary trees (or trees of bounded balance) as described by:

• Stephen Adams, "Efficient sets: a balancing act", Journal of Functional Programming 3(4):553-562, October 1993, http://www.swiss.ai.mit.edu/~adams/BB/.
• J. Nievergelt and E.M. Reingold, "Binary search trees of bounded balance", SIAM journal of computing 2(1), March 1973.

Note that the implementation is left-biased -- the elements of a first argument are always preferred to the second, for example in union or insert.

Operation comments contain the operation time complexity in the Big-O notation http://en.wikipedia.org/wiki/Big_O_notation.

O(log n). 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.Map.null (empty)           == True

 Data.Map.null (singleton 1 'a') == False

O(1). The number of elements in the map.

 size empty                                   == 0

 size (singleton 1 'a')                       == 1

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

O(log n). Is the key a member of the map? See also notMember.

 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? See also member.

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

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

O(log n). Lookup the value at a key in the map.

The function will return the corresponding value as (Just value), or Nothing if the key isn't in the map.

An example of using lookup:

 import Prelude hiding (lookup)

 import Data.Map



 employeeDept = fromList([("John","Sales"), ("Bob","IT")])

 deptCountry = fromList([("IT","USA"), ("Sales","France")])

 countryCurrency = fromList([("USA", "Dollar"), ("France", "Euro")])



 employeeCurrency :: String -> Maybe String

 employeeCurrency name = do

     dept <- lookup name employeeDept

     country <- lookup dept deptCountry

     lookup country countryCurrency



 main = do

     putStrLn $ "John's currency: " ++ (show (employeeCurrency "John"))

     putStrLn $ "Pete's currency: " ++ (show (employeeCurrency "Pete"))

The output of this program:

   John's currency: Just "Euro"

   Pete's currency: Nothing

O(log n). The expression (findWithDefault def k map) returns the value at key k or returns default value def when the key is not in 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 with a single element.

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

 size (singleton 1 'a') == 1

O(log n). Insert a new key and value in the map. If the key is already present in the map, the associated value is replaced with the supplied value. 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(log n). Insert with a function, combining new value and old value. insertWith f key value mp will insert the pair (key, value) into mp if key does not exist in the map. If the key does exist, the function will insert the pair (key, f new_value old_value).

 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(log n). Insert with a function, combining key, new value and old value. insertWithKey f key value mp will insert the pair (key, value) into mp if key does not exist in the map. If the key does exist, the function will insert the pair (key,f key new_value old_value). Note that the key passed to f is the same key passed to insertWithKey.

 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(log n). Combines insert operation with old value retrieval. 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(log n). Delete a key and its value from the map. When the key is not a member of the map, the original map is returned.

 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(log n). Update a value at a specific key with the result of the provided function. 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(log n). 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(log n). The expression (update f k map) updates the value x at k (if it is in the map). If (f x) is Nothing, the element is deleted. If it is (Just y), the key k is bound to the new value y.

 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(log n). The expression (updateWithKey f k map) updates the value x at k (if it is in the map). If (f k x) is Nothing, the element is deleted. If it is (Just y), the key k is bound to the new value y.

 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(log n). Lookup and update. See also updateWithKey. The function returns changed value, if it is updated. 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 "5:new 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(log n). The expression (alter f k map) alters the value x at k, or absence thereof. alter can be used to insert, delete, or update a value in a Map. In short : lookup k (alter f k m) = f (lookup k m).

 let f _ = Nothing

 alter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]

 alter f 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"



 let f _ = Just "c"

 alter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "c")]

 alter f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "c")]

O(n+m). The expression (union t1 t2) takes the left-biased union of t1 and t2. It prefers t1 when duplicate keys are encountered, i.e. (union == unionWith const). The implementation uses the efficient hedge-union algorithm. Hedge-union is more efficient on (bigset `union` smallset).

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

O(n+m). Union with a combining function. The implementation uses the efficient hedge-union algorithm.

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

O(n+m). Union with a combining function. The implementation uses the efficient hedge-union algorithm. Hedge-union is more efficient on (bigset `union` smallset).

 let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_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 == Prelude.foldl union empty).

 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 f == Prelude.foldl (unionWith f) empty).

 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 of two maps. Return elements of the first map not existing in the second map. The implementation uses an efficient hedge algorithm comparable with hedge-union.

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

O(n+m). Difference with a combining function. When two equal keys are encountered, the combining function is applied to the values of these keys. If it returns Nothing, the element is discarded (proper set difference). If it returns (Just y), the element is updated with a new value y. The implementation uses an efficient hedge algorithm comparable with hedge-union.

 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. The implementation uses an efficient hedge algorithm comparable with hedge-union.

 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). Intersection of two maps. Return data in the first map for the keys existing in both maps. (intersection m1 m2 == intersectionWith const m1 m2).

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

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

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

O(n+m). Intersection with a combining function. Intersection is more efficient on (bigset `intersection` smallset).

 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*log n). mapKeys f s is the map obtained by applying f to each key of s.

The size of the result may be smaller if f maps two or more distinct keys to the same new key. In this case the value at the smallest of these keys is retained.

 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*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"

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 better performance than mapKeys.

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

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

 valid (mapKeysMonotonic (\ _ -> 1)     (fromList [(5,"a"), (3,"b")])) == False

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). The set of all keys of the map.

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

 keysSet empty == Data.Set.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 to a list of key/value pairs.

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

 toList empty == []

O(n*log n). Build a map from a list of key/value pairs. See also fromAscList. If the list contains more than one value for the same key, the last value for the key is retained.

 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*log n). Build 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*log n). Build a map from a list of key/value pairs with a combining function. See also fromAscListWithKey.

 let f k a1 a2 = (show k) ++ a1 ++ a2

 fromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "3ab"), (5, "5a5ba")]

 fromListWithKey f [] == empty

O(n). Convert to an ascending list.

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

O(n). Build a map from an ascending list in linear time. The precondition (input list is ascending) is not checked.

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

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

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

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

O(n). Build a map from an ascending list in linear time with a combining function for equal keys. The precondition (input list is ascending) is not checked.

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

 valid (fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")]) == True

 valid (fromAscListWith (++) [(5,"a"), (3,"b"), (5,"b")]) == False

O(n). Build a map from an ascending list in linear time with a combining function for equal keys. The precondition (input list is ascending) is not checked.

 let f k a1 a2 = (show k) ++ ":" ++ a1 ++ a2

 fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")] == fromList [(3, "b"), (5, "5:b5:ba")]

 valid (fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")]) == True

 valid (fromAscListWithKey f [(5,"a"), (3,"b"), (5,"b"), (5,"b")]) == False

O(n). Build a map from an ascending list of distinct elements in linear time. The precondition is not checked.

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

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

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

O(n). Filter all values that satisfy the 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 the predicate.

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

O(n). Partition the map according to a predicate. The first map contains all elements that satisfy the predicate, the second all elements that fail the predicate. See also split.

 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 a 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 the keys in map1 are smaller than k and the keys in map2 larger than k. Any key equal to k is found in neither map1 nor map2.

 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). The expression (splitLookup k map) splits a map just like split but also returns lookup k 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 t1 t2) returns True if all keys in t1 are in tree t2, and when f returns True when applied to their respective values. For example, the following expressions are all True:

 isSubmapOfBy (==) (fromList [('a',1)]) (fromList [('a',1),('b',2)])

 isSubmapOfBy (<=) (fromList [('a',1)]) (fromList [('a',1),('b',2)])

 isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1),('b',2)])

But the following are all False:

 isSubmapOfBy (==) (fromList [('a',2)]) (fromList [('a',1),('b',2)])

 isSubmapOfBy (<)  (fromList [('a',1)]) (fromList [('a',1),('b',2)])

 isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1)])

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). Lookup the index of a key. The index is a number from 0 up to, but not including, the size of the map.

 isJust (lookupIndex 2 (fromList [(5,"a"), (3,"b")]))   == False

 fromJust (lookupIndex 3 (fromList [(5,"a"), (3,"b")])) == 0

 fromJust (lookupIndex 5 (fromList [(5,"a"), (3,"b")])) == 1

 isJust (lookupIndex 6 (fromList [(5,"a"), (3,"b")]))   == False

O(log n). Return the index of a key. The index is a number from 0 up to, but not including, the size of the map. Calls error when the key is not a member of the map.

 findIndex 2 (fromList [(5,"a"), (3,"b")])    Error: element is not in the map

 findIndex 3 (fromList [(5,"a"), (3,"b")]) == 0

 findIndex 5 (fromList [(5,"a"), (3,"b")]) == 1

 findIndex 6 (fromList [(5,"a"), (3,"b")])    Error: element is not in the map

O(log n). Retrieve an element by index. Calls error when an invalid index is used.

 elemAt 0 (fromList [(5,"a"), (3,"b")]) == (3,"b")

 elemAt 1 (fromList [(5,"a"), (3,"b")]) == (5, "a")

 elemAt 2 (fromList [(5,"a"), (3,"b")])    Error: index out of range

O(log n). Update the element at index. Calls error when an invalid index is used.

 updateAt (\ _ _ -> Just "x") 0    (fromList [(5,"a"), (3,"b")]) == fromList [(3, "x"), (5, "a")]

 updateAt (\ _ _ -> Just "x") 1    (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "x")]

 updateAt (\ _ _ -> Just "x") 2    (fromList [(5,"a"), (3,"b")])    Error: index out of range

 updateAt (\ _ _ -> Just "x") (-1) (fromList [(5,"a"), (3,"b")])    Error: index out of range

 updateAt (\_ _  -> Nothing)  0    (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"

 updateAt (\_ _  -> Nothing)  1    (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"

 updateAt (\_ _  -> Nothing)  2    (fromList [(5,"a"), (3,"b")])    Error: index out of range

 updateAt (\_ _  -> Nothing)  (-1) (fromList [(5,"a"), (3,"b")])    Error: index out of range

O(log n). Delete the element at index. Defined as (deleteAt i map = updateAt (k x -> Nothing) i map).

 deleteAt 0  (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"

 deleteAt 1  (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"

 deleteAt 2 (fromList [(5,"a"), (3,"b")])     Error: index out of range

 deleteAt (-1) (fromList [(5,"a"), (3,"b")])  Error: index out of range

O(log n). The minimal key of the map. Calls error is the map is empty.

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

 findMin empty                            Error: empty map has no minimal element

O(log n). The maximal key of the map. Calls error is the map is empty.

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

 findMax empty                            Error: empty map has no maximal element

O(log n). Delete the minimal key. Returns an empty map if the map is empty.

 deleteMin (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(5,"a"), (7,"c")]

 deleteMin empty == empty

O(log n). Delete the maximal key. Returns an empty map if the map is empty.

 deleteMax (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(3,"b"), (5,"a")]

 deleteMax empty == empty

O(log n). Delete and find the minimal element.

 deleteFindMin (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((3,"b"), fromList[(5,"a"), (10,"c")]) 

 deleteFindMin                                            Error: can not return the minimal element of an empty map

O(log n). Delete and find the maximal element.

 deleteFindMax (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((10,"c"), fromList [(3,"b"), (5,"a")])

 deleteFindMax empty                                      Error: can not return the maximal element of an empty map

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 value associated with minimal key of the map, and the map stripped of that element, or Nothing if passed an empty map.

 minView (fromList [(5,"a"), (3,"b")]) == Just ("b", singleton 5 "a")

 minView empty == Nothing

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

 maxView (fromList [(5,"a"), (3,"b")]) == Just ("a", singleton 3 "b")

 maxView empty == Nothing