O(1). The empty multimap.

size empty === 0

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

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

O(k). A key is retained only if it is associated with a non-empty set of values.

O(n*log n) where n is the length of the input list. Build a multimap from a list of key/value pairs.

fromList ([] :: [(Int, Char)]) === empty
fromList [(1, 'b'), (2, 'a'), (1, 'b')] === fromList [(1, 'b'), (2, 'a')]

O(log m * log k). If the key exists in the multimap, the new value will be inserted into the set of values for the key. It is a no-op if the value already exists in the set.

insert 1 'a' empty === singleton 1 'a'
insert 1 'a' (fromList [(1, 'b'), (2, 'a')]) === fromList [(1, 'a'), (1, 'b'), (2, 'a')]
insert 1 'a' (fromList [(1, 'a'), (2, 'c')]) === fromList [(1, 'a'), (2, 'c')]

O(log k). Delete a key and all its values from the map.

delete 1 (fromList [(1,'a'),(1,'b'),(2,'c')]) === singleton 2 'c'

O(log m * log k). Remove the first occurrence of the value associated with the key, if exists.

deleteWithValue 1 'c' (fromList [(1,'a'),(1,'b'),(2,'c')]) === fromList [(1,'a'),(1,'b'),(2,'c')]
deleteWithValue 1 'c' (fromList [(2,'c'),(1,'c')]) === singleton 2 'c'

O(log m * log k). Remove the maximal value associated with the key, if exists.

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

O(log m * log k). Remove the minimal value associated with the key, if exists.

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

O(m * log m * log k), assuming the function a -> a takes O(1). Update values at a specific key, if exists.

Since values are sets, the result may be smaller than the original multimap.

adjust ("new " ++) 1 (fromList [(1,"a"),(1,"b"),(2,"c")]) === fromList [(1,"new a"),(1,"new b"),(2,"c")]
adjust (const "z") 1 (fromList [(1,"a"),(1,"b"),(2,"c")]) === fromList [(1,"z"),(2,"c")]

O(m * log m * log k), assuming the function k -> a -> a takes O(1). Update values at a specific key, if exists.

Since values are sets, the result may be smaller than the original multimap.

adjustWithKey (\k x -> show k ++ ":new " ++ x) 1 (fromList [(1,"a"),(1,"b"),(2,"c")])
  === fromList [(1,"1:new a"),(1,"1:new b"),(2,"c")]

O(m * log m * log k), assuming the function a -> Maybe a takes O(1). The expression (update f k map) updates the values at key k, if exists. If f returns Nothing for a value, the value is deleted.

let f x = if x == "a" then Just "new a" else Nothing in do
  update f 1 (fromList [(1,"a"),(1,"b"),(2,"c")]) === fromList [(1,"new a"),(2, "c")]
  update f 1 (fromList [(1,"b"),(1,"c"),(2,"c")]) === singleton 2 "c"

O(m * log m * log k), assuming the function k -> a -> Maybe a takes O(1). The expression (updateWithKey f k map) updates the values at key k, if exists. If f returns Nothing for a value, the value is deleted.

let f k x = if x == "a" then Just (show k ++ ":new a") else Nothing in do
  updateWithKey f 1 (fromList [(1,"a"),(1,"b"),(2,"c")]) === fromList [(1,"1:new a"),(2,"c")]
  updateWithKey f 1 (fromList [(1,"b"),(1,"c"),(2,"c")]) === singleton 2 "c"

O(log k), assuming the function Set a -> Set a takes O(1). The expression (alter f k map) alters the values at k, if exists.

let (f, g) = (const Set.empty, Set.insert 'c') in do
  alter f 1 (fromList [(1, 'a'), (2, 'b')]) === singleton 2 'b'
  alter f 3 (fromList [(1, 'a'), (2, 'b')]) === fromList [(1, 'a'), (2, 'b')]
  alter g 1 (fromList [(1, 'a'), (2, 'b')]) === fromList [(1, 'c'), (1, 'a'), (2, 'b')]
  alter g 1 (fromList [(1, 'c'), (2, 'b')]) === fromList [(1, 'c'), (2, 'b')]
  alter g 3 (fromList [(1, 'a'), (2, 'b')]) === fromList [(1, 'a'), (2, 'b'), (3, 'c')]

O(log k), assuming the function k -> Set a -> Set a takes O(1). The expression (alterWithKey f k map) alters the values at k, if exists.

let (f, g) = (const (const Set.empty), Set.insert . show) in do
  alterWithKey f 1 (fromList [(1, "a"), (2, "b")]) === singleton 2 "b"
  alterWithKey f 3 (fromList [(1, "a"), (2, "b")]) === fromList [(1, "a"), (2, "b")]
  alterWithKey g 1 (fromList [(1, "a"), (2, "b")]) === fromList [(1, "1"), (1, "a"), (2, "b")]
  alterWithKey g 3 (fromList [(1, "a"), (2, "b")]) === fromList [(1, "a"), (2, "b"), (3, "3")]

O(log k). Lookup the values at a key in the map. It returns an empty set if the key is not in the map.

O(log k). Lookup the values at a key in the map. It returns an empty set if the key is not in the map.

fromList [(3, 'a'), (5, 'b'), (3, 'c')] ! 3 === Set.fromList "ac"
fromList [(3, 'a'), (5, 'b'), (3, 'c')] ! 2 === Set.empty

O(log k). Is the key a member of the map?

A key is a member of the map if and only if there is at least one value associated with it.

member 1 (fromList [(1, 'a'), (2, 'b'), (2, 'c')]) === True
member 1 (deleteMax 1 (fromList [(2, 'c'), (1, 'c')])) === False

O(log k). Is the key not a member of the map?

A key is a member of the map if and only if there is at least one value associated with it.

notMember 1 (fromList [(1, 'a'), (2, 'b'), (2, 'c')]) === False
notMember 1 (deleteMin 1 (fromList [(2, 'c'), (1, 'c')])) === True

O(1). Is the multimap empty?

Data.Multimap.Set.null empty === True
Data.Multimap.Set.null (singleton 1 'a') === False

O(1). Is the multimap non-empty?

notNull empty === False
notNull (singleton 1 'a') === True

The total number of values for all keys.

size is evaluated lazily. Forcing the size for the first time takes up to O(k) and subsequent forces take O(1).

size empty === 0
size (singleton 1 'a') === 1
size (fromList [(1, 'a'), (2, 'b'), (2, 'c'), (2, 'b')]) === 3

Union two multimaps, unioning values for duplicate keys.

union (fromList [(1,'a'),(2,'b'),(2,'c')]) (fromList [(1,'d'),(2,'b')])
  === fromList [(1,'a'),(1,'d'),(2,'b'),(2,'c')]

Union a number of multimaps, unioning values for duplicate keys.

unions [fromList [(1,'a'),(2,'b'),(2,'c')], fromList [(1,'d'),(2,'b')]]
  === fromList [(1,'a'),(1,'d'),(2,'b'),(2,'c')]

Difference of two multimaps.

If a key exists in the first multimap but not the second, it remains unchanged in the result. If a key exists in both multimaps, a set difference is performed on their values.

difference (fromList [(1,'a'),(2,'b'),(2,'c')]) (fromList [(1,'d'),(2,'b'),(2,'a')])
  === fromList [(1,'a'),(2,'c')]

O(n * log m), assuming the function a -> b takes O(1). Map a function over all values in the map.

Since values are sets, the result may be smaller than the original multimap.

Data.Multimap.Set.map (++ "x") (fromList [(1,"a"),(2,"b")]) === fromList [(1,"ax"),(2,"bx")]
Data.Multimap.Set.map (const "c") (fromList [(1,"a"),(1,"b"),(2,"b")]) === fromList [(1,"c"),(2,"c")]

O(n * log m), assuming the function k -> a -> b takes O(1). Map a function over all values in the map.

Since values are sets, the result may be smaller than the original multimap.

mapWithKey (\k x -> show k ++ ":" ++ x) (fromList [(1,"a"),(2,"b")]) === fromList [(1,"1:a"),(2,"2:b")]

O(n). Fold the values in the map using the given right-associative binary operator.

Data.Multimap.Set.foldr ((+) . length) 0 (fromList [(1, "hello"), (1, "world"), (2, "!")]) === 11

O(n). Fold the values in the map using the given left-associative binary operator.

Data.Multimap.Set.foldl (\len -> (+ len) . length) 0 (fromList [(1, "hello"), (1, "world"), (2, "!")]) === 11

O(n). Fold the key/value pairs in the map using the given right-associative binary operator.

foldrWithKey (\k a len -> length (show k) + length a + len) 0 (fromList [(1, "hello"), (1, "world"), (20, "!")]) === 15

O(n). Fold the key/value pairs in the map using the given left-associative binary operator.

foldlWithKey (\len k a -> length (show k) + length a + len) 0 (fromList [(1, "hello"), (1, "world"), (20, "!")]) === 15

O(n). Fold the key/value pairs in the map using the given monoid.

foldMapWithKey (\k x -> show k ++ ":" ++ x) (fromList [(1, "a"), (1, "c"), (2, "b")]) === "1:a1:c2:b"

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.

Data.Multimap.Set.foldr' ((+) . length) 0 (fromList [(1, "hello"), (1, "world"), (2, "!")]) === 11

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.

Data.Multimap.Set.foldl' (\len -> (+ len) . length) 0 (fromList [(1, "hello"), (1, "world"), (2, "!")]) === 11

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.

foldrWithKey' (\k a len -> length (show k) + length a + len) 0 (fromList [(1, "hello"), (1, "world"), (20, "!")]) === 15

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.

foldlWithKey' (\len k a -> length (show k) + length a + len) 0 (fromList [(1, "hello"), (1, "world"), (20, "!")]) === 15

O(n). Return all elements of the multimap in ascending order of their keys. Elements of each key appear in ascending order.

elems (fromList [(2,'a'),(1,'b'),(3,'d'),(3,'c'),(1,'b')]) === "bacd"
elems (empty :: SetMultimap Int Char) === []

O(k). Return all keys of the multimap in ascending order.

keys (fromList [(2,'a'),(1,'b'),(3,'c'),(1,'b')]) === [1,2,3]
keys (empty :: SetMultimap Int Char) === []

An alias for toAscList.

O(k). The set of all keys of the multimap.

keysSet (fromList [(2,'a'),(1,'b'),(3,'c'),(1,'b')]) === Set.fromList [1,2,3]
keysSet (empty :: SetMultimap Int Char) === Set.empty

Convert the multimap into a list of key/value pairs.

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

Convert the multimap into a list of key/value pairs in ascending order of keys. Elements of each key appear in ascending order.

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

Convert the multimap into a list of key/value pairs in descending order of keys. Elements of each key appear in descending order.

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

O(1). Convert the multimap into a regular map.

O(n), assuming the predicate function takes O(1). Retain all values that satisfy the predicate. A key is removed if none of its values satisfies the predicate.

Data.Multimap.Set.filter (> 'a') (fromList [(1,'a'),(1,'b'),(2,'a')]) === singleton 1 'b'
Data.Multimap.Set.filter (< 'a') (fromList [(1,'a'),(1,'b'),(2,'a')]) === empty

O(n), assuming the predicate function takes O(1). Retain all key/value pairs that satisfy the predicate. A key is removed if none of its values satisfies the predicate.

filterWithKey (\k a -> even k && a > 'a') (fromList [(1,'a'),(1,'b'),(2,'a'),(2,'b')]) === singleton 2 'b'

O(k), assuming the predicate function takes O(1). Retain all keys that satisfy the predicate.

Generalized filter.

let f a | a > 'b' = Just True
        | a < 'b' = Just False
        | a == 'b' = Nothing
 in do
   filterM f (fromList [(1,'a'),(1,'b'),(2,'a'),(2,'c')]) === Nothing
   filterM f (fromList [(1,'a'),(1,'c'),(2,'a'),(2,'c')]) === Just (fromList [(1,'c'),(2,'c')])

Generalized filterWithKey. | Generalized filterWithKey.

let f k a | even k && a > 'b' = Just True
          | odd k && a < 'b' = Just False
          | otherwise = Nothing
 in do
   filterWithKeyM f (fromList [(1,'a'),(1,'c'),(2,'a'),(2,'c')]) === Nothing
   filterWithKeyM f (fromList [(1,'a'),(3,'a'),(2,'c'),(4,'c')]) === Just (fromList [(2,'c'),(4,'c')])

O(n * log m), assuming the function a -> Maybe b takes O(1). Map values and collect the Just results.

mapMaybe (\a -> if a == "a" then Just "new a" else Nothing) (fromList [(1,"a"),(1,"b"),(2,"a"),(2,"c")])
  === fromList [(1,"new a"),(2,"new a")]

O(n * log m), assuming the function k -> a -> Maybe b takes O(1). Map key/value pairs and collect the Just results.

mapMaybeWithKey (\k a -> if k > 1 && a == "a" then Just "new a" else Nothing) (fromList [(1,"a"),(1,"b"),(2,"a"),(2,"c")])
  === singleton 2 "new a"

O(n * log m), assuming the function a -> Either b c takes O(1). Map values and separate the Left and Right results.

mapEither (\a -> if a < 'b' then Left a else Right a) (fromList [(1,'a'),(1,'c'),(2,'a'),(2,'c')])
  === (fromList [(1,'a'),(2,'a')],fromList [(1,'c'),(2,'c')])

O(n * log m), assuming the function k -> a -> Either b c takes O(1). Map key/value pairs and separate the Left and Right results.

mapEitherWithKey (\k a -> if even k && a < 'b' then Left a else Right a) (fromList [(1,'a'),(1,'c'),(2,'a'),(2,'c')])
  === (fromList [(2,'a')],fromList [(1,'a'),(1,'c'),(2,'c')])

O(log n). Return the smallest key and the associated values. Returns Nothing if the map is empty.

lookupMin (fromList [(1,'a'),(1,'c'),(2,'c')]) === Just (1, Set.fromList "ac")
lookupMin (empty :: SetMultimap Int Char) === Nothing

O(log n). Return the largest key and the associated values. Returns Nothing if the map is empty.

lookupMax (fromList [(1,'a'),(1,'c'),(2,'c')]) === Just (2, Set.fromList "c")
lookupMax (empty :: SetMultimap Int Char) === Nothing

O(log n). Return the largest key smaller than the given one, and the associated values, if exist.

lookupLT 1 (fromList [(1,'a'),(3,'b'),(3,'c'),(5,'c')]) === Nothing
lookupLT 4 (fromList [(1,'a'),(3,'b'),(3,'c'),(5,'c')]) === Just (3, Set.fromList "bc")

O(log n). Return the smallest key larger than the given one, and the associated values, if exist.

lookupGT 5 (fromList [(1,'a'),(3,'b'),(3,'c'),(5,'c')]) === Nothing
lookupGT 2 (fromList [(1,'a'),(3,'b'),(3,'c'),(5,'c')]) === Just (3, Set.fromList "bc")

O(log n). Return the largest key smaller than or equal to the given one, and the associated values, if exist.

lookupLE 0 (fromList [(1,'a'),(3,'b'),(3,'c'),(5,'c')]) === Nothing
lookupLE 1 (fromList [(1,'a'),(3,'b'),(3,'c'),(5,'c')]) === Just (1, Set.fromList "a")
lookupLE 4 (fromList [(1,'a'),(3,'b'),(3,'c'),(5,'c')]) === Just (3, Set.fromList "bc")

O(log n). Return the smallest key larger than or equal to the given one, and the associated values, if exist.

lookupGE 6 (fromList [(1,'a'),(3,'b'),(3,'c'),(5,'c')]) === Nothing
lookupGE 5 (fromList [(1,'a'),(3,'b'),(3,'c'),(5,'c')]) === Just (5, Set.fromList "c")
lookupGE 2 (fromList [(1,'a'),(3,'b'),(3,'c'),(5,'c')]) === Just (3, Set.fromList "bc")