containers-0.4.2.1: Assorted concrete container types

Data.Map

Description

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.

Synopsis

# Map type

data Map k a Source

A Map from keys `k` to values `a`.

Instances

 Typeable2 Map Functor (Map k) Foldable (Map k) Traversable (Map k) (Eq k, Eq a) => Eq (Map k a) (Data k, Data a, Ord k) => Data (Map k a) (Ord k, Ord v) => Ord (Map k v) (Ord k, Read k, Read e) => Read (Map k e) (Show k, Show a) => Show (Map k a) Ord k => Monoid (Map k v) (NFData k, NFData a) => NFData (Map k a)

# Operators

(!) :: Ord k => Map k a -> k -> aSource

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'
```

(\\) :: Ord k => Map k a -> Map k b -> Map k aSource

Same as `difference`.

# Query

null :: Map k a -> BoolSource

O(1). Is the map empty?

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

size :: Map k a -> IntSource

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 :: Ord k => k -> Map k a -> BoolSource

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
```

notMember :: Ord k => k -> Map k a -> BoolSource

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
```

lookup :: Ord k => k -> Map k a -> Maybe aSource

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
```

findWithDefault :: Ord k => a -> k -> Map k a -> aSource

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'
```

# Construction

empty :: Map k aSource

O(1). The empty map.

``` empty      == fromList []
size empty == 0
```

singleton :: k -> a -> Map k aSource

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

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

## Insertion

insert :: Ord k => k -> a -> Map k a -> Map k aSource

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'
```

insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k aSource

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"
```

insertWith' :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k aSource

Same as `insertWith`, but the combining function is applied strictly. This is often the most desirable behavior.

For example, to update a counter:

``` insertWith' (+) k 1 m
```

insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k aSource

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"
```

insertWithKey' :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k aSource

Same as `insertWithKey`, but the combining function is applied strictly.

insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)Source

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")])
```

insertLookupWithKey' :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)Source

O(log n). A strict version of `insertLookupWithKey`.

## Delete/Update

delete :: Ord k => k -> Map k a -> Map k aSource

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
```

adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k aSource

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
```

adjustWithKey :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k aSource

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
```

update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k aSource

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"
```

updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k aSource

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"
```

updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a, Map k a)Source

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")
```

alter :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k aSource

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")]
```

# Combine

## Union

union :: Ord k => Map k a -> Map k a -> Map k aSource

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")]
```

unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k aSource

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 :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k aSource

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 left_value right_value = (show key) ++ ":" ++ left_value ++ "|" ++ right_value
unionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")]
```

unions :: Ord k => [Map k a] -> Map k aSource

The union of a list of maps: (`unions == 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")]
```

unionsWith :: Ord k => (a -> a -> a) -> [Map k a] -> Map k aSource

The union of a list of maps, with a combining operation: (`unionsWith f == 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")]
```

## Difference

difference :: Ord k => Map k a -> Map k b -> Map k aSource

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"
```

differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k aSource

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"
```

differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k aSource

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"
```

## Intersection

intersection :: Ord k => Map k a -> Map k b -> Map k aSource

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"
```

intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k cSource

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

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

intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k cSource

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"
```

# Traversal

## Map

map :: (a -> b) -> Map k a -> Map k bSource

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

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

mapWithKey :: (k -> a -> b) -> Map k a -> Map k bSource

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")]
```

mapAccum :: (a -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)Source

O(n). The function `mapAccum` threads an accumulating argument through the map in ascending order of keys.

``` let f a b = (a ++ b, b ++ "X")
mapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")])
```

mapAccumWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)Source

O(n). The function `mapAccumWithKey` threads an accumulating argument through the map in ascending order of keys.

``` let f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X")
mapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")])
```

mapAccumRWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)Source

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

mapKeys :: Ord k2 => (k1 -> k2) -> Map k1 a -> Map k2 aSource

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"
```

mapKeysWith :: Ord k2 => (a -> a -> a) -> (k1 -> k2) -> Map k1 a -> Map k2 aSource

O(n*log n). `mapKeysWith c f s` is the map obtained by applying `f` to each key of `s`.

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

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

mapKeysMonotonic :: (k1 -> k2) -> Map k1 a -> Map k2 aSource

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
```

# Folds

foldr :: (a -> b -> b) -> b -> Map k a -> bSource

O(n). Fold the values in the map using the given right-associative binary operator, such that `foldr f z == foldr f z . elems`.

For example,

``` elems map = foldr (:) [] map
```
``` let f a len = len + (length a)
foldr f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
```

foldl :: (a -> b -> a) -> a -> Map k b -> aSource

O(n). Fold the values in the map using the given left-associative binary operator, such that `foldl f z == foldl f z . elems`.

For example,

``` elems = reverse . foldl (flip (:)) []
```
``` let f len a = len + (length a)
foldl f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
```

foldrWithKey :: (k -> a -> b -> b) -> b -> Map k a -> bSource

O(n). Fold the keys and values in the map using the given right-associative binary operator, such that `foldrWithKey f z == foldr (uncurry f) z . toAscList`.

For example,

``` keys map = foldrWithKey (\k x ks -> k:ks) [] map
```
``` let f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
foldrWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)"
```

foldlWithKey :: (a -> k -> b -> a) -> a -> Map k b -> aSource

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)"
```

## Strict folds

foldr' :: (a -> b -> b) -> b -> Map k a -> bSource

O(n). A strict version of `foldr`. Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value.

foldl' :: (a -> b -> a) -> a -> Map k b -> aSource

O(n). A strict version of `foldl`. Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value.

foldrWithKey' :: (k -> a -> b -> b) -> b -> Map k a -> bSource

O(n). A strict version of `foldrWithKey`. Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value.

foldlWithKey' :: (a -> k -> b -> a) -> a -> Map k b -> aSource

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.

## Legacy folds

fold :: (a -> b -> b) -> b -> Map k a -> bSource

O(n). Fold the values in the map using the given right-associative binary operator. This function is an equivalent of `foldr` and is present for compatibility only.

Please note that fold will be deprecated in the future and removed.

foldWithKey :: (k -> a -> b -> b) -> b -> Map k a -> bSource

O(n). Fold the keys and values in the map using the given right-associative binary operator. This function is an equivalent of `foldrWithKey` and is present for compatibility only.

Please note that foldWithKey will be deprecated in the future and removed.

# Conversion

elems :: Map k a -> [a]Source

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 == []
```

keys :: Map k a -> [k]Source

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

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

keysSet :: Map k a -> Set kSource

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
```

assocs :: Map k a -> [(k, a)]Source

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 == []
```

## Lists

toList :: Map k a -> [(k, a)]Source

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

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

fromList :: Ord k => [(k, a)] -> Map k aSource

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")]
```

fromListWith :: Ord k => (a -> a -> a) -> [(k, a)] -> Map k aSource

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
```

fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k, a)] -> Map k aSource

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
```

## Ordered lists

toAscList :: Map k a -> [(k, a)]Source

O(n). Convert to an ascending list.

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

toDescList :: Map k a -> [(k, a)]Source

O(n). Convert to a descending list.

fromAscList :: Eq k => [(k, a)] -> Map k aSource

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
```

fromAscListWith :: Eq k => (a -> a -> a) -> [(k, a)] -> Map k aSource

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
```

fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(k, a)] -> Map k aSource

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
```

fromDistinctAscList :: [(k, a)] -> Map k aSource

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
```

# Filter

filter :: Ord k => (a -> Bool) -> Map k a -> Map k aSource

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
```

filterWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> Map k aSource

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

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

partition :: Ord k => (a -> Bool) -> Map k a -> (Map k a, Map k a)Source

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")])
```

partitionWithKey :: Ord k => (k -> a -> Bool) -> Map k a -> (Map k a, Map k a)Source

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")])
```

mapMaybe :: Ord k => (a -> Maybe b) -> Map k a -> Map k bSource

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 :: Ord k => (k -> a -> Maybe b) -> Map k a -> Map k bSource

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

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

mapEither :: Ord k => (a -> Either b c) -> Map k a -> (Map k b, Map k c)Source

O(n). Map values and separate the `Left` and `Right` results.

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

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

mapEitherWithKey :: Ord k => (k -> a -> Either b c) -> Map k a -> (Map k b, Map k c)Source

O(n). Map keys/values and separate the `Left` and `Right` results.

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

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

split :: Ord k => k -> Map k a -> (Map k a, Map k a)Source

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)
```

splitLookup :: Ord k => k -> Map k a -> (Map k a, Maybe a, Map k a)Source

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)
```

# Submap

isSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> BoolSource

O(n+m). This function is defined as (`isSubmapOf = isSubmapOfBy (==)`).

isSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> BoolSource

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)])
```

isProperSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> BoolSource

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

isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> BoolSource

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)])
```

# Indexed

lookupIndex :: Ord k => k -> Map k a -> Maybe IntSource

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
```

findIndex :: Ord k => k -> Map k a -> IntSource

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
```

elemAt :: Int -> Map k a -> (k, a)Source

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
```

updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k aSource

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
```

deleteAt :: Int -> Map k a -> Map k aSource

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
```

# Min/Max

findMin :: Map k a -> (k, a)Source

O(log n). 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 :: Map k a -> (k, a)Source

O(log n). 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 :: Map k a -> Map k aSource

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
```

deleteMax :: Map k a -> Map k aSource

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
```

deleteFindMin :: Map k a -> ((k, a), Map k a)Source

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
```

deleteFindMax :: Map k a -> ((k, a), Map k a)Source

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
```

updateMin :: (a -> Maybe a) -> Map k a -> Map k aSource

O(log n). Update the value at the minimal key.

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

updateMax :: (a -> Maybe a) -> Map k a -> Map k aSource

O(log n). Update the value at the maximal key.

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

updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k aSource

O(log n). Update the value at the minimal key.

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

updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k aSource

O(log n). Update the value at the maximal key.

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

minView :: Map k a -> Maybe (a, Map k a)Source

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
```

maxView :: Map k a -> Maybe (a, Map k a)Source

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
```

minViewWithKey :: Map k a -> Maybe ((k, a), Map k a)Source

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
```

maxViewWithKey :: Map k a -> Maybe ((k, a), Map k a)Source

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
```

# Debugging

showTree :: (Show k, Show a) => Map k a -> StringSource

O(n). Show the tree that implements the map. The tree is shown in a compressed, hanging format. See `showTreeWith`.

showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> StringSource

O(n). The expression (`showTreeWith showelem hang wide map`) shows the tree that implements the map. Elements are shown using the `showElem` function. 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.

```  Map> let t = fromDistinctAscList [(x,()) | x <- [1..5]]
Map> putStrLn \$ showTreeWith (\k x -> show (k,x)) True False t
(4,())
+--(2,())
|  +--(1,())
|  +--(3,())
+--(5,())

Map> putStrLn \$ showTreeWith (\k x -> show (k,x)) True True t
(4,())
|
+--(2,())
|  |
|  +--(1,())
|  |
|  +--(3,())
|
+--(5,())

Map> putStrLn \$ showTreeWith (\k x -> show (k,x)) False True t
+--(5,())
|
(4,())
|
|  +--(3,())
|  |
+--(2,())
|
+--(1,())
```

valid :: Ord k => Map k a -> BoolSource

O(n). Test if the internal map structure is valid.

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