Portability | portable |
---|---|

Stability | provisional |

Maintainer | libraries@haskell.org |

Safe Haskell | Safe |

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

API of this module is strict in the keys, but lazy in the values.
If you need value-strict maps, use `Strict`

instead.
The `Map`

type itself is shared between the lazy and strict modules,
meaning that the same `Map`

value can be passed to functions in
both modules (although that is rarely needed).

These modules are intended to be imported qualified, to avoid name clashes with Prelude functions, e.g.

import qualified Data.Map.Lazy 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).

- data Map k a
- (!) :: Ord k => Map k a -> k -> a
- (\\) :: Ord k => Map k a -> Map k b -> Map k a
- null :: Map k a -> Bool
- size :: Map k a -> Int
- member :: Ord k => k -> Map k a -> Bool
- notMember :: Ord k => k -> Map k a -> Bool
- lookup :: Ord k => k -> Map k a -> Maybe a
- findWithDefault :: Ord k => a -> k -> Map k a -> a
- lookupLT :: Ord k => k -> Map k v -> Maybe (k, v)
- lookupGT :: Ord k => k -> Map k v -> Maybe (k, v)
- lookupLE :: Ord k => k -> Map k v -> Maybe (k, v)
- lookupGE :: Ord k => k -> Map k v -> Maybe (k, v)
- empty :: Map k a
- singleton :: k -> a -> Map k a
- insert :: Ord k => k -> a -> Map k a -> Map k a
- insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a
- insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
- insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)
- delete :: Ord k => k -> Map k a -> Map k a
- adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k a
- adjustWithKey :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k a
- update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k a
- updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a
- updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a, Map k a)
- alter :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a
- union :: Ord k => Map k a -> Map k a -> Map k a
- unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a
- unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a
- unions :: Ord k => [Map k a] -> Map k a
- unionsWith :: Ord k => (a -> a -> a) -> [Map k a] -> Map k a
- difference :: Ord k => Map k a -> Map k b -> Map k a
- differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a
- differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a
- intersection :: Ord k => Map k a -> Map k b -> Map k a
- intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k c
- intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c
- mergeWithKey :: Ord k => (k -> a -> b -> Maybe c) -> (Map k a -> Map k c) -> (Map k b -> Map k c) -> Map k a -> Map k b -> Map k c
- map :: (a -> b) -> Map k a -> Map k b
- mapWithKey :: (k -> a -> b) -> Map k a -> Map k b
- traverseWithKey :: Applicative t => (k -> a -> t b) -> Map k a -> t (Map k b)
- mapAccum :: (a -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)
- mapAccumWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)
- mapAccumRWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)
- mapKeys :: Ord k2 => (k1 -> k2) -> Map k1 a -> Map k2 a
- mapKeysWith :: Ord k2 => (a -> a -> a) -> (k1 -> k2) -> Map k1 a -> Map k2 a
- mapKeysMonotonic :: (k1 -> k2) -> Map k1 a -> Map k2 a
- foldr :: (a -> b -> b) -> b -> Map k a -> b
- foldl :: (a -> b -> a) -> a -> Map k b -> a
- foldrWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b
- foldlWithKey :: (a -> k -> b -> a) -> a -> Map k b -> a
- foldr' :: (a -> b -> b) -> b -> Map k a -> b
- foldl' :: (a -> b -> a) -> a -> Map k b -> a
- foldrWithKey' :: (k -> a -> b -> b) -> b -> Map k a -> b
- foldlWithKey' :: (a -> k -> b -> a) -> a -> Map k b -> a
- elems :: Map k a -> [a]
- keys :: Map k a -> [k]
- assocs :: Map k a -> [(k, a)]
- keysSet :: Map k a -> Set k
- fromSet :: (k -> a) -> Set k -> Map k a
- toList :: Map k a -> [(k, a)]
- fromList :: Ord k => [(k, a)] -> Map k a
- fromListWith :: Ord k => (a -> a -> a) -> [(k, a)] -> Map k a
- fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k, a)] -> Map k a
- toAscList :: Map k a -> [(k, a)]
- toDescList :: Map k a -> [(k, a)]
- fromAscList :: Eq k => [(k, a)] -> Map k a
- fromAscListWith :: Eq k => (a -> a -> a) -> [(k, a)] -> Map k a
- fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(k, a)] -> Map k a
- fromDistinctAscList :: [(k, a)] -> Map k a
- filter :: (a -> Bool) -> Map k a -> Map k a
- filterWithKey :: (k -> a -> Bool) -> Map k a -> Map k a
- partition :: (a -> Bool) -> Map k a -> (Map k a, Map k a)
- partitionWithKey :: (k -> a -> Bool) -> Map k a -> (Map k a, Map k a)
- mapMaybe :: (a -> Maybe b) -> Map k a -> Map k b
- mapMaybeWithKey :: (k -> a -> Maybe b) -> Map k a -> Map k b
- mapEither :: (a -> Either b c) -> Map k a -> (Map k b, Map k c)
- mapEitherWithKey :: (k -> a -> Either b c) -> Map k a -> (Map k b, Map k c)
- split :: Ord k => k -> Map k a -> (Map k a, Map k a)
- splitLookup :: Ord k => k -> Map k a -> (Map k a, Maybe a, Map k a)
- isSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool
- isSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool
- isProperSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool
- isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool
- lookupIndex :: Ord k => k -> Map k a -> Maybe Int
- findIndex :: Ord k => k -> Map k a -> Int
- elemAt :: Int -> Map k a -> (k, a)
- updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a
- deleteAt :: Int -> Map k a -> Map k a
- findMin :: Map k a -> (k, a)
- findMax :: Map k a -> (k, a)
- deleteMin :: Map k a -> Map k a
- deleteMax :: Map k a -> Map k a
- deleteFindMin :: Map k a -> ((k, a), Map k a)
- deleteFindMax :: Map k a -> ((k, a), Map k a)
- updateMin :: (a -> Maybe a) -> Map k a -> Map k a
- updateMax :: (a -> Maybe a) -> Map k a -> Map k a
- updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
- updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
- minView :: Map k a -> Maybe (a, Map k a)
- maxView :: Map k a -> Maybe (a, Map k a)
- minViewWithKey :: Map k a -> Maybe ((k, a), Map k a)
- maxViewWithKey :: Map k a -> Maybe ((k, a), Map k a)
- showTree :: (Show k, Show a) => Map k a -> String
- showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> String
- valid :: Ord k => Map k a -> Bool

# Strictness properties

This module satisfies the following strictness property:

- Key arguments are evaluated to WHNF

Here are some examples that illustrate the property:

insertWith (\ new old -> old) undefined v m == undefined insertWith (\ new old -> old) k undefined m == OK delete undefined m == undefined

# Map type

A Map from keys `k`

to values `a`

.

Typeable2 Map | |

Functor (Map k) | |

Foldable (Map k) | |

(Functor (Map k), Foldable (Map k)) => Traversable (Map k) | |

(Eq k, Eq a) => Eq (Map k a) | |

(Typeable (Map k a), Data k, Data a, Ord k) => Data (Map k a) | |

(Eq (Map k v), 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'

# Query

*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

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 `(`

,
or `Just`

value)`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 `(`

returns
the value at key `findWithDefault`

def k map)`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'

lookupLT :: Ord k => k -> Map k v -> Maybe (k, v)Source

*O(log n)*. Find largest key smaller than the given one and return the
corresponding (key, value) pair.

lookupLT 3 (fromList [(3,'a'), (5,'b')]) == Nothing lookupLT 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')

lookupGT :: Ord k => k -> Map k v -> Maybe (k, v)Source

*O(log n)*. Find smallest key greater than the given one and return the
corresponding (key, value) pair.

lookupGT 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b') lookupGT 5 (fromList [(3,'a'), (5,'b')]) == Nothing

lookupLE :: Ord k => k -> Map k v -> Maybe (k, v)Source

*O(log n)*. Find largest key smaller or equal to the given one and return
the corresponding (key, value) pair.

lookupLE 2 (fromList [(3,'a'), (5,'b')]) == Nothing lookupLE 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a') lookupLE 5 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')

lookupGE :: Ord k => k -> Map k v -> Maybe (k, v)Source

*O(log n)*. Find smallest key greater or equal to the given one and return
the corresponding (key, value) pair.

lookupGE 3 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a') lookupGE 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b') lookupGE 6 (fromList [(3,'a'), (5,'b')]) == Nothing

# Construction

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.

will insert the pair (key, value) into `insertWith`

f key value mp`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 :: 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.

will insert the pair (key, value) into `insertWithKey`

f key value mp`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 :: 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 (

)
is a pair where the first element is equal to (`insertLookupWithKey`

f k x map

)
and the second element equal to (`lookup`

k map

).
`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")])

## 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 (

) updates the value `update`

f k map`x`

at `k`

(if it is in the map). If (`f x`

) is `Nothing`

, the element is
deleted. If it is (

), the key `Just`

y`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 (

) updates the
value `updateWithKey`

f k map`x`

at `k`

(if it is in the map). If (`f k x`

) is `Nothing`

,
the element is deleted. If it is (

), the key `Just`

y`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 (

) alters the value `alter`

f k map`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 (

) takes the left-biased union of `union`

t1 t2`t1`

and `t2`

.
It prefers `t1`

when duplicate keys are encountered,
i.e. (

).
The implementation uses the efficient `union`

== `unionWith`

`const`

*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 (

), the element is updated with a new value `Just`

y`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 (

), the element is updated with a new value `Just`

y`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"

## Universal combining function

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

*O(n+m)*. A high-performance universal combining function. This function
is used to define `unionWith`

, `unionWithKey`

, `differenceWith`

,
`differenceWithKey`

, `intersectionWith`

, `intersectionWithKey`

and can be
used to define other custom combine functions.

Please make sure you know what is going on when using `mergeWithKey`

,
otherwise you can be surprised by unexpected code growth or even
corruption of the data structure.

When `mergeWithKey`

is given three arguments, it is inlined to the call
site. You should therefore use `mergeWithKey`

only to define your custom
combining functions. For example, you could define `unionWithKey`

,
`differenceWithKey`

and `intersectionWithKey`

as

myUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) id id m1 m2 myDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2 myIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) (const empty) (const empty) m1 m2

When calling

, a function combining two
`mergeWithKey`

combine only1 only2`IntMap`

s is created, such that

- if a key is present in both maps, it is passed with both corresponding
values to the
`combine`

function. Depending on the result, the key is either present in the result with specified value, or is left out; - a nonempty subtree present only in the first map is passed to
`only1`

and the output is added to the result; - a nonempty subtree present only in the second map is passed to
`only2`

and the output is added to the result.

The `only1`

and `only2`

methods *must return a map with a subset (possibly empty) of the keys of the given map*.
The values can be modified arbitrarily. Most common variants of `only1`

and
`only2`

are `id`

and

, but for example `const`

`empty`

or
`map`

f

could be used for any `filterWithKey`

f`f`

.

# 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")]

traverseWithKey :: Applicative t => (k -> a -> t b) -> Map k a -> t (Map k b)Source

*O(n)*.

That is, behaves exactly like a regular `traverseWithKey`

f s == `fromList`

$ `traverse`

((k, v) -> (,) k $ f k v) (`toList`

m)`traverse`

except that the traversing
function also has access to the key associated with a value.

traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')]) traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')]) == Nothing

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)*.

is the map obtained by applying `mapKeys`

f s`f`

to each key of `s`

.

The size of the result may be smaller if `f`

maps two or more distinct
keys to the same new key. In this case the value at the greatest of the
original keys is retained.

mapKeys (+ 1) (fromList [(5,"a"), (3,"b")]) == fromList [(4, "b"), (6, "a")] mapKeys (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c" mapKeys (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"

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

*O(n*log n)*.

is the map obtained by applying `mapKeysWith`

c f s`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)*.

, but works only when `mapKeysMonotonic`

f s == `mapKeys`

f s`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

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.

# Conversion

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

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

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

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

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

*O(n)*. An alias for `toAscList`

. Return all key/value pairs in the map
in ascending key order. Subject to list fusion.

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

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

fromSet :: (k -> a) -> Set k -> Map k aSource

*O(n)*. Build a map from a set of keys and a function which for each key
computes its value.

fromSet (\k -> replicate k 'a') (Data.Set.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")] fromSet undefined Data.Set.empty == empty

## Lists

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

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

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

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 the map to a list of key/value pairs where the keys are
in ascending order. Subject to list fusion.

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

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

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

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

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 :: (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 :: (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 :: (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 :: (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 :: (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 :: (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 :: (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 :: (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 (

) is a pair `split`

k map`(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 (

) splits a map just
like `splitLookup`

k map`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 (

) returns `isSubmapOfBy`

f t1 t2`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 (

) returns `isProperSubmapOfBy`

f m1 m2`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 (

) shows
the tree that implements the map. Elements are shown using the `showTreeWith`

showelem hang wide map`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,())