Data.TrieMap

Map type

class (Repr k, TrieKey (Rep k)) => TKey k

TKey k is a handy alias for (Repr k, TrieKey (Rep k)). To make a type an instance of TKey, create a Repr instance that will satisfy TrieKey (Rep k), possibly using the Template Haskell methods provided by Data.TrieMap.Representation.

Instances

(Repr k, TrieKey (Rep k)) => TKey k

data TMap k a

A map from keys k to values a, backed by a trie.

Instances

TKey k => Functor (TMap k)

TKey k => Foldable (TMap k)

TKey k => Traversable (TMap k)

(Eq k, TKey k, Eq a) => Eq (TMap k a)

(Ord k, TKey k, Ord a) => Ord (TMap k a)

(Show k, Show a, TKey k) => Show (TMap k a)

TKey k => Monoid (TMap k a)

Location type

data TLocation k a

A TLocation represents a TMap with a "hole" at a particular key position.

TLocations are used for element-wise operations on maps (insertion, deletion and update) in a two-stage process:

1. A TLocation (and the value at that position, if any) is obtained from a TMap by searching or indexing. 2. A new TMap is made from a TLocation by either filling the hole with a value (assign) or erasing it (clear).

Components

key :: TKey k => TLocation k a -> k

O(1). The key marking the position of the "hole" in the map.

before :: TKey k => TLocation k a -> TMap k a

before loc is the submap with keys less than key loc.

after :: TKey k => TLocation k a -> TMap k a

after loc is the submap with keys greater than key loc.

Locations in maps

search :: TKey k => k -> TMap k a -> (Maybe a, TLocation k a)

Search the map for the given key, returning the corresponding value (if any) and an updatable location for that key.

Properties:

 case search k m of
     (Nothing, loc) -> key loc == k && clear loc == m
     (Just v,  loc) -> key loc == k && assign v loc == m

lookup k m == fst (search k m)

index :: TKey k => Int -> TMap k a -> (a, TLocation k a)

Return the value and an updatable location for the ith key in the map. Calls error if i is out of range.

Properties:

 0 <= i && i < size m ==>
     let (v, loc) = index i m in
         size (before loc) == i && assign v loc == m

elemAt i m == let (v, loc) = index i m in (key loc, v)

minLocation :: TKey k => TMap k a -> Maybe (a, TLocation k a)

O(log n). Return the value and an updatable location for the least key in the map, or Nothing if the map is empty.

Properties:

 size m > 0 ==>
     let Just (v, loc) = minLocation i m in
         size (before loc) == 0 && assign v loc == m

findMin m == let Just (v, loc) = minLocation i m in (key loc, v)

maxLocation :: TKey k => TMap k a -> Maybe (a, TLocation k a)

Return the value and an updatable location for the greatest key in the map, or Nothing if the map is empty.

Properties:

 size m > 0 ==>
     let Just (v, loc) = maxLocation i m in
         size (after loc) == 0 && assign v loc == m

findMax m == let Just (v, loc) = maxLocation i m in (key loc, v)

Building maps

assign :: TKey k => a -> TLocation k a -> TMap k a

Return a map obtained by placing the given value at the location (replacing an existing value, if any).

assign v loc == before loc union singleton (key loc) v union after loc

clear :: TKey k => TLocation k a -> TMap k a

Return a map obtained by erasing the location.

clear loc == before loc union after loc

Operators

(!) :: TKey k => TMap k a -> k -> a

Find the value at a key. Calls error when the element can not be found.

(\\) :: TKey k => TMap k a -> TMap k b -> TMap k a

Same as difference.

Query

null :: TKey k => TMap k a -> Bool

O(1). Is the map empty?

size :: TKey k => TMap k a -> Int

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

member :: TKey k => k -> TMap k a -> Bool

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

notMember :: TKey k => k -> TMap k a -> Bool

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

lookup :: TKey k => k -> TMap k a -> Maybe a

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.

findWithDefault :: TKey k => a -> k -> TMap k a -> a

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.

Construction

empty :: TKey k => TMap k a

O(1). The empty map.

singleton :: TKey k => k -> a -> TMap k a

O(1). A map with a single element.

Insertion

insert :: TKey k => k -> a -> TMap k a -> TMap k a

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'

insertWith :: TKey k => (a -> a -> a) -> k -> a -> TMap k a -> TMap k a

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"

insertWithKey :: TKey k => (k -> a -> a -> a) -> k -> a -> TMap k a -> TMap k a

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"

insertLookupWithKey :: TKey k => (k -> a -> a -> a) -> k -> a -> TMap k a -> (Maybe a, TMap k a)

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")

Delete/Update

delete :: TKey k => k -> TMap k a -> TMap k a

Delete a key and its value from the map. When the key is not a member of the map, the original map is returned.

 delete 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
 delete 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
 delete 5 empty                         == empty

adjust :: TKey k => (a -> a) -> k -> TMap k a -> TMap k a

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

adjustWithKey :: TKey k => (k -> a -> a) -> k -> TMap k a -> TMap k a

Adjust a value at a specific key. When the key is not a member of the map, the original map is returned.

 let f key x = (show key) ++ ":new " ++ x
 adjustWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
 adjustWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
 adjustWithKey f 7 empty                         == empty

update :: TKey k => (a -> Maybe a) -> k -> TMap k a -> TMap k a

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

 let f x = if x == "a" then Just "new a" else Nothing
 update f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
 update f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
 update f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"

updateWithKey :: TKey k => (k -> a -> Maybe a) -> k -> TMap k a -> TMap k a

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"

alter :: TKey k => (Maybe a -> Maybe a) -> k -> TMap k a -> TMap k a

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 TMap. In short: lookup k (alter f k m) = f (lookup k m).

Combine

Union

union :: TKey k => TMap k a -> TMap k a -> TMap k a

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")]

unionWith :: TKey k => (a -> a -> a) -> TMap k a -> TMap k a -> TMap k a

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")]

unionWithKey :: TKey k => (k -> a -> a -> a) -> TMap k a -> TMap k a -> TMap k a

unionMaybeWith :: TKey k => (a -> a -> Maybe a) -> TMap k a -> TMap k a -> TMap k a

Union with a combining function. The implementation uses the efficient hedge-union algorithm.

unionMaybeWithKey :: TKey k => (k -> a -> a -> Maybe a) -> TMap k a -> TMap k a -> TMap k a

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 left_value right_value = Just ((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")]

symmetricDifference :: TKey k => TMap k a -> TMap k a -> TMap k a

symmetricDifference is equivalent to unionMaybeWith ( _ _ -> Nothing).

Difference

difference :: TKey k => TMap k a -> TMap k b -> TMap k a

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"

differenceWith :: TKey k => (a -> b -> Maybe a) -> TMap k a -> TMap k b -> TMap k a

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"

differenceWithKey :: TKey k => (k -> a -> b -> Maybe a) -> TMap k a -> TMap k b -> TMap k a

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"

Intersection

intersection :: TKey k => TMap k a -> TMap k b -> TMap k a

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"

intersectionWith :: TKey k => (a -> b -> c) -> TMap k a -> TMap k b -> TMap k c

Intersection with a combining function.

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

intersectionWithKey :: TKey k => (k -> a -> b -> c) -> TMap k a -> TMap k b -> TMap k c

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"

intersectionMaybeWith :: TKey k => (a -> b -> Maybe c) -> TMap k a -> TMap k b -> TMap k c

intersectionMaybeWith f m1 m2 is equivalent to mapMaybe id (intersectionWith f m1 m2).

intersectionMaybeWithKey :: TKey k => (k -> a -> b -> Maybe c) -> TMap k a -> TMap k b -> TMap k c

intersectionMaybeWithKey f m1 m2 is equivalent to mapMaybe id (intersectionWithKey f m1 m2).

Traversal

Map

map :: TKey k => (a -> b) -> TMap k a -> TMap k b

Map a function over all values in the map.

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

mapWithKey :: TKey k => (k -> a -> b) -> TMap k a -> TMap k b

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")]

mapKeys :: (TKey k, TKey k') => (k -> k') -> TMap k a -> TMap k' a

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"

mapKeysWith :: (TKey k, TKey k') => (a -> a -> a) -> (k -> k') -> TMap k a -> TMap k' a

mapKeysWith c f s is the map obtained by applying f to each key of s.

The size of the result may be smaller if f maps two or more distinct keys to the same new key. In this case the associated values will be combined using c.

 mapKeysWith (++) (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab"
 mapKeysWith (++) (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab"

mapKeysMonotonic :: (TKey k, TKey k') => (k -> k') -> TMap k a -> TMap k' a

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")]

Traverse

traverseWithKey :: (TKey k, Applicative f) => (k -> a -> f b) -> TMap k a -> f (TMap k b)

Map each key/element pair to an action, evaluate these actions from left to right, and collect the results.

Fold

foldrWithKey :: TKey k => (k -> a -> b -> b) -> b -> TMap k a -> b

Post-order fold. The function will be applied from the lowest value to the highest.

foldlWithKey :: TKey k => (b -> k -> a -> b) -> b -> TMap k a -> b

Pre-order fold. The function will be applied from the highest value to the lowest.

Conversion

elems :: TKey k => TMap k a -> [a]

Return all elements of the map in the ascending order of their keys.

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

keys :: TKey k => TMap k a -> [k]

Return all keys of the map in ascending order.

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

keysSet :: TKey k => TMap k a -> TSet k

The set of all keys of the map.

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

assocs :: TKey k => TMap k a -> [(k, a)]

Return all key/value pairs in the map in ascending key order.

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

Lists

fromList :: TKey k => [(k, a)] -> TMap k a

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")]

fromListWith :: TKey k => (a -> a -> a) -> [(k, a)] -> TMap k a

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

fromListWithKey :: TKey k => (k -> a -> a -> a) -> [(k, a)] -> TMap k a

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

Ordered lists

fromAscList :: TKey k => [(k, a)] -> TMap k a

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")]

fromAscListWith :: TKey k => (a -> a -> a) -> [(k, a)] -> TMap k a

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")]

fromAscListWithKey :: TKey k => (k -> a -> a -> a) -> [(k, a)] -> TMap k a

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")]

fromDistinctAscList :: TKey k => [(k, a)] -> TMap k a

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")]

Filter

filter :: TKey k => (a -> Bool) -> TMap k a -> TMap k a

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

filterWithKey :: TKey k => (k -> a -> Bool) -> TMap k a -> TMap k a

Filter all keys/values that satisfy the predicate.

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

partition :: TKey k => (a -> Bool) -> TMap k a -> (TMap k a, TMap k a)

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")])

partitionWithKey :: TKey k => (k -> a -> Bool) -> TMap k a -> (TMap k a, TMap k a)

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")])

mapMaybe :: TKey k => (a -> Maybe b) -> TMap k a -> TMap k b

O(n). Map values and collect the Just results.

 let f x = if x == "a" then Just "new a" else Nothing
 mapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a"

mapMaybeWithKey :: TKey k => (k -> a -> Maybe b) -> TMap k a -> TMap k b

Map keys/values and collect the Just results.

 let f k _ = if k < 5 then Just ("key : " ++ (show k)) else Nothing
 mapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key : 3"

mapEither :: TKey k => (a -> Either b c) -> TMap k a -> (TMap k b, TMap k c)

Map values and separate the Left and Right results.

 let f a = if a < "c" then Left a else Right a
 mapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
     == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")])

 mapEither (\ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
     == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])

mapEitherWithKey :: TKey k => (k -> a -> Either b c) -> TMap k a -> (TMap k b, TMap k c)

Map keys/values and separate the Left and Right results.

 let f k a = if k < 5 then Left (k * 2) else Right (a ++ a)
 mapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
     == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")])

 mapEitherWithKey (\_ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
     == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")])

split :: TKey k => k -> TMap k a -> (TMap k a, TMap k a)

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)

splitLookup :: TKey k => k -> TMap k a -> (TMap k a, Maybe a, TMap k a)

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)

Submap

isSubmapOf :: (TKey k, Eq a) => TMap k a -> TMap k a -> Bool

This function is defined as (isSubmapOf = isSubmapOfBy (==)).

isSubmapOfBy :: TKey k => (a -> b -> Bool) -> TMap k a -> TMap k b -> Bool

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)])

Indexed

lookupIndex :: TKey k => k -> TMap k a -> Maybe Int

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

 lookupIndex 2 (fromList [(5,"a"), (3,"b")]) == Nothing
 lookupIndex 3 (fromList [(5,"a"), (3,"b")]) == Just 0
 lookupIndex 5 (fromList [(5,"a"), (3,"b")]) == Just 1
 lookupIndex 6 (fromList [(5,"a"), (3,"b")]) == Nothing

findIndex :: TKey k => k -> TMap k a -> Int

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

elemAt :: TKey k => Int -> TMap k a -> (k, a)

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

updateAt :: TKey k => (k -> a -> Maybe a) -> Int -> TMap k a -> TMap k a

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

deleteAt :: TKey k => Int -> TMap k a -> TMap k a

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

Min/Max

findMin :: TKey k => TMap k a -> (k, a)

The minimal key of the map. Calls error if the map is empty.

 findMin (fromList [(5,"a"), (3,"b")]) == (3,"b")
 findMin empty                            Error: empty map has no minimal element

findMax :: TKey k => TMap k a -> (k, a)

The maximal key of the map. Calls error if the map is empty.

 findMax (fromList [(5,"a"), (3,"b")]) == (5,"a")
 findMax empty                            Error: empty map has no maximal element

deleteMin :: TKey k => TMap k a -> TMap k a

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

deleteMax :: TKey k => TMap k a -> TMap k a

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

deleteFindMin :: TKey k => TMap k a -> ((k, a), TMap k a)

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

deleteFindMax :: TKey k => TMap k a -> ((k, a), TMap k a)

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

updateMin :: TKey k => (a -> Maybe a) -> TMap k a -> TMap k a

Update the value at the minimal key.

 updateMin (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "Xb"), (5, "a")]
 updateMin (\ _ -> Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"

updateMax :: TKey k => (a -> Maybe a) -> TMap k a -> TMap k a

Update the value at the maximal key.

 updateMax (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "Xa")]
 updateMax (\ _ -> Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"

updateMinWithKey :: TKey k => (k -> a -> Maybe a) -> TMap k a -> TMap k a

Update the value at the minimal key.

 updateMinWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"3:b"), (5,"a")]
 updateMinWithKey (\ _ _ -> Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"

updateMaxWithKey :: TKey k => (k -> a -> Maybe a) -> TMap k a -> TMap k a

Update the value at the maximal key.

 updateMaxWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"b"), (5,"5:a")]
 updateMaxWithKey (\ _ _ -> Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"

minView :: TKey k => TMap k a -> Maybe (a, TMap k a)

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

maxView :: TKey k => TMap k a -> Maybe (a, TMap k a)

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

minViewWithKey :: TKey k => TMap k a -> Maybe ((k, a), TMap k a)

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

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

maxViewWithKey :: TKey k => TMap k a -> Maybe ((k, a), TMap k a)

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