A set of values a.

\(O\bigl(m \log\bigl(\frac{n}{m}+1\bigr)\bigr), \; 0 < m \leq n\). See difference.

\(O(1)\). Is this the empty set?

\(O(1)\). The number of elements in the set.

\(O(\log n)\). Is the element in the set?

\(O(\log n)\). Is the element not in the set?

\(O(\log n)\). Find largest element smaller than the given one.

lookupLT 3 (fromList [3, 5]) == Nothing
lookupLT 5 (fromList [3, 5]) == Just 3

\(O(\log n)\). Find smallest element greater than the given one.

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

\(O(\log n)\). Find largest element smaller or equal to the given one.

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

\(O(\log n)\). Find smallest element greater or equal to the given one.

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

\(O\bigl(m \log\bigl(\frac{n}{m}+1\bigr)\bigr), \; 0 < m \leq n\). (s1 `isSubsetOf` s2) indicates whether s1 is a subset of s2.

s1 `isSubsetOf` s2 = all (`member` s2) s1
s1 `isSubsetOf` s2 = null (s1 `difference` s2)
s1 `isSubsetOf` s2 = s1 `union` s2 == s2
s1 `isSubsetOf` s2 = s1 `intersection` s2 == s1

\(O\bigl(m \log\bigl(\frac{n}{m}+1\bigr)\bigr), \; 0 < m \leq n\). (s1 `isProperSubsetOf` s2) indicates whether s1 is a proper subset of s2.

s1 `isProperSubsetOf` s2 = s1 `isSubsetOf` s2 && s1 /= s2

\(O\bigl(m \log\bigl(\frac{n}{m}+1\bigr)\bigr), \; 0 < m \leq n\). Check whether two sets are disjoint (i.e., their intersection is empty).

disjoint (fromList [2,4,6])   (fromList [1,3])     == True
disjoint (fromList [2,4,6,8]) (fromList [2,3,5,7]) == False
disjoint (fromList [1,2])     (fromList [1,2,3,4]) == False
disjoint (fromList [])        (fromList [])        == True

xs `disjoint` ys = null (xs `intersection` ys)

Since: 0.5.11

\(O(1)\). The empty set.

\(O(1)\). Create a singleton set.

\(O(\log n)\). Insert an element in a set. If the set already contains an element equal to the given value, it is replaced with the new value.

\(O(\log n)\). Delete an element from a set.

\(O(\log n)\). Pop an element from the set.

Returns Nothing if the element is not a member of the set. Otherwise returns Just the set with the element removed.

pop 1 (fromList [0,2,4]) == Nothing
pop 2 (fromList [0,2,4]) == Just (fromList [0,4])

Since: 0.8.1

\(O(\log n)\) (alterF f x s) can delete or insert x in s depending on whether an equal element is found in s.

In short:

member x <$> alterF f x s = f (member x s)

Note that unlike insert, alterF will not replace an element equal to the given value.

Note: alterF is a variant of the at combinator from Control.Lens.At.

Examples

-- Get whether the element is a member, and also insert or remove it.
getAndSet :: Ord a => a -> Bool -> Set a -> (Bool, Set a)
getAndSet x new = alterF (\old -> (old, new)) x

-- Delete the element. If it is absent the result is Nothing.
mustDelete :: Ord a => a -> Set a -> Maybe (Set a)
mustDelete = alterF (\b -> if b then Just False else Nothing)

Since: 0.6.3.1

\(O(2^n \log n)\). Calculate the power set of a set: the set of all its subsets.

t `member` powerSet s == t `isSubsetOf` s

Example:

powerSet (fromList [1,2,3]) =
  fromList $ map fromList [[],[1],[1,2],[1,2,3],[1,3],[2],[2,3],[3]]

Since: 0.5.11

\(O\bigl(m \log\bigl(\frac{n}{m}+1\bigr)\bigr), \; 0 < m \leq n\). The union of two sets, preferring the first set when equal elements are encountered.

The union of the sets in a Foldable structure : (unions == foldl union empty).

\(O\bigl(m \log\bigl(\frac{n}{m}+1\bigr)\bigr), \; 0 < m \leq n\). Difference of two sets.

Return elements of the first set not existing in the second set.

difference (fromList [5, 3]) (fromList [5, 7]) == singleton 3

\(O\bigl(m \log\bigl(\frac{n}{m}+1\bigr)\bigr), \; 0 < m \leq n\). The intersection of two sets. Elements of the result come from the first set, so for example

import qualified Data.Set as S
data AB = A | B deriving Show
instance Ord AB where compare _ _ = EQ
instance Eq AB where _ == _ = True
main = print (S.singleton A `S.intersection` S.singleton B,
              S.singleton B `S.intersection` S.singleton A)

prints (fromList [A],fromList [B]).

The intersection of a series of sets. Intersections are performed left-to-right.

Since: 0.8

\(O\bigl(m \log\bigl(\frac{n}{m}+1\bigr)\bigr), \; 0 < m \leq n\). The symmetric difference of two sets.

The result contains elements that appear in exactly one of the two sets.

symmetricDifference (fromList [0,2,4,6]) (fromList [0,3,6,9]) == fromList [2,3,4,9]

Since: 0.8

\(O(nm)\). Calculate the Cartesian product of two sets.

cartesianProduct xs ys = fromList $ liftA2 (,) (toList xs) (toList ys)

Example:

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

Since: 0.5.11

\(O(n+m)\). Calculate the disjoint union of two sets.

 disjointUnion xs ys = map Left xs `union` map Right ys

Example:

disjointUnion (fromList [1,2]) (fromList ["hi", "bye"]) =
  fromList [Left 1, Left 2, Right "hi", Right "bye"]

Since: 0.5.11

Sets form a Semigroup under intersection.

Since: 0.8

\(O(n)\). Keep all elements that satisfy the predicate.

\(O(n)\). Keep all elements that satisfy the Applicative predicate.

Since: 0.8.1

\(O(\log n)\). Take while a predicate on the elements holds. The user is responsible for ensuring that for all elements j and k in the set, j < k ==> p j >= p k. See note at spanAntitone.

takeWhileAntitone p = fromDistinctAscList . takeWhile p . toList
takeWhileAntitone p = filter p

Since: 0.5.8

\(O(\log n)\). Drop while a predicate on the elements holds. The user is responsible for ensuring that for all elements j and k in the set, j < k ==> p j >= p k. See note at spanAntitone.

dropWhileAntitone p = fromDistinctAscList . dropWhile p . toList
dropWhileAntitone p = filter (not . p)

Since: 0.5.8

\(O(\log n)\). Divide a set at the point where a predicate on the elements stops holding. The user is responsible for ensuring that for all elements j and k in the set, j < k ==> p j >= p k.

spanAntitone p xs = (takeWhileAntitone p xs, dropWhileAntitone p xs)
spanAntitone p xs = partition p xs

Note: if p is not actually antitone, then spanAntitone will split the set at some unspecified point where the predicate switches from holding to not holding (where the predicate is seen to hold before the first element and to fail after the last element).

Since: 0.5.8

\(O(n \log n)\). Map elements and collect the Just results.

If the function is monotonically non-decreasing, this function takes \(O(n)\) time.

Since: 0.8.1

\(O(n)\). Partition the set into two sets, one with all elements that satisfy the predicate and one with all elements that don't satisfy the predicate. See also split.

\(O(\log n)\). The expression (split x set) is a pair (set1,set2) where set1 comprises the elements of set less than x and set2 comprises the elements of set greater than x.

\(O(\log n)\). Performs a split but also returns whether the pivot element was found in the original set.

\(O(1)\). Decompose a set into pieces based on the structure of the underlying tree. This function is useful for consuming a set in parallel.

No guarantee is made as to the sizes of the pieces; an internal, but deterministic process determines this. However, it is guaranteed that the pieces returned will be in ascending order (all elements in the first subset less than all elements in the second, and so on).

Examples:

splitRoot (fromList [1..6]) ==
  [fromList [1,2,3],fromList [4],fromList [5,6]]

splitRoot empty == []

Note that the current implementation does not return more than three subsets, but you should not depend on this behaviour because it can change in the future without notice.

Since: 0.5.4

\(O(\log n)\). Look up the index of an element, which is its zero-based index in the sorted sequence of elements. The index is a number from 0 up to, but not including, the size of the set.

isJust   (lookupIndex 2 (fromList [5,3])) == False
fromJust (lookupIndex 3 (fromList [5,3])) == 0
fromJust (lookupIndex 5 (fromList [5,3])) == 1
isJust   (lookupIndex 6 (fromList [5,3])) == False

Since: 0.5.4

\(O(\log n)\). Return the index of an element, which is its zero-based index in the sorted sequence of elements. The index is a number from 0 up to, but not including, the size of the set. Calls error when the element is not a member of the set.

Note: This function is partial. Prefer lookupIndex.

findIndex 2 (fromList [5,3])    Error: element is not in the set
findIndex 3 (fromList [5,3]) == 0
findIndex 5 (fromList [5,3]) == 1
findIndex 6 (fromList [5,3])    Error: element is not in the set

Since: 0.5.4

\(O(\log n)\). Retrieve an element by its index, i.e. by its zero-based index in the sorted sequence of elements. If the index is out of range (less than zero, greater or equal to size of the set), error is called.

Note: This function is partial.

elemAt 0 (fromList [5,3]) == 3
elemAt 1 (fromList [5,3]) == 5
elemAt 2 (fromList [5,3])    Error: index out of range

Since: 0.5.4

\(O(\log n)\). Delete the element at index, i.e. by its zero-based index in the sorted sequence of elements. If the index is out of range (less than zero, greater or equal to size of the set), error is called.

Note: This function is partial.

deleteAt 0    (fromList [5,3]) == singleton 5
deleteAt 1    (fromList [5,3]) == singleton 3
deleteAt 2    (fromList [5,3])    Error: index out of range
deleteAt (-1) (fromList [5,3])    Error: index out of range

Since: 0.5.4

\(O(\log n)\). Take a given number of elements in order, beginning with the smallest ones.

take n = fromDistinctAscList . take n . toAscList

Since: 0.5.8

\(O(\log n)\). Drop a given number of elements in order, beginning with the smallest ones.

drop n = fromDistinctAscList . drop n . toAscList

Since: 0.5.8

\(O(\log n)\). Split a set at a particular index.

splitAt !n !xs = (take n xs, drop n xs)

\(O(n \log n)\). map f s is the set obtained by applying f to each element of s.

If f is monotonically non-decreasing, this function takes \(O(n)\) time.

It's worth noting that the size of the result may be smaller if, for some (x,y), x /= y && f x == f y

\(O(n)\). mapMonotonic f s == map f s, but works only when f is strictly increasing. Semi-formally, we have:

and [x < y ==> f x < f y | x <- ls, y <- ls]
                    ==> mapMonotonic f s == map f s
    where ls = toList s

Warning: This function should be used only if f is monotonically strictly increasing. This precondition is not checked. Use map if the precondition may not hold.

\(O(n)\). Fold the elements in the set using the given right-associative binary operator, such that foldr f z == foldr f z . toAscList.

For example,

toAscList set = foldr (:) [] set

\(O(n)\). Fold the elements in the set using the given left-associative binary operator, such that foldl f z == foldl f z . toAscList.

For example,

toDescList set = foldl (flip (:)) [] set

\(O(n)\). 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.

\(O(n)\). 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.

\(O(n)\). Fold the elements in the set using the given right-associative binary operator.

\(O(\log n)\). The minimal element of the set. Returns Nothing if the set is empty.

Since: 0.5.9

\(O(\log n)\). The maximal element of the set. Returns Nothing if the set is empty.

Since: 0.5.9

\(O(\log n)\). The minimal element of the set. Calls error if the set is empty.

Note: This function is partial. Prefer lookupMin.

\(O(\log n)\). The maximal element of the set. Calls error if the set is empty.

Note: This function is partial. Prefer lookupMax.

\(O(\log n)\). Delete the minimal element. Returns an empty set if the set is empty.

\(O(\log n)\). Delete the maximal element. Returns an empty set if the set is empty.

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

Calls error if the set is empty.

Note: This function is partial. Prefer minView.

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

Calls error if the set is empty.

Note: This function is partial. Prefer maxView.

\(O(\log n)\). Retrieves the maximal key of the set, and the set stripped of that element, or Nothing if passed an empty set.

\(O(\log n)\). Retrieves the minimal key of the set, and the set stripped of that element, or Nothing if passed an empty set.

\(O(n)\). An alias of toAscList. The elements of a set in ascending order. Subject to list fusion.

\(O(n)\). Convert the set to a list of elements. Subject to list fusion.

\(O(n \log n)\). Create a set from a list of elements.

If the elements are in non-decreasing order, this function takes \(O(n)\) time.

\(O(n)\). Convert the set to an ascending list of elements. Subject to list fusion.

\(O(n)\). Convert the set to a descending list of elements. Subject to list fusion.

\(O(n)\). Build a set from an ascending list in linear time.

Warning: This function should be used only if the elements are in non-decreasing order. This precondition is not checked. Use fromList if the precondition may not hold.

\(O(n)\). Build a set from an ascending list of distinct elements in linear time.

Warning: This function should be used only if the elements are in strictly increasing order. This precondition is not checked. Use fromList if the precondition may not hold.

\(O(n)\). Build a set from a descending list in linear time.

Warning: This function should be used only if the elements are in non-increasing order. This precondition is not checked. Use fromList if the precondition may not hold.

Since: 0.5.8

\(O(n)\). Build a set from a descending list of distinct elements in linear time.

Warning: This function should be used only if the elements are in strictly decreasing order. This precondition is not checked. Use fromList if the precondition may not hold.

Since: 0.5.8

\(O(n \log n)\). Show the tree that implements the set. The tree is shown in a compressed, hanging format.

\(O(n \log n)\). The expression (showTreeWith hang wide map) shows the tree that implements the set. 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.

Set> putStrLn $ showTreeWith True False $ fromDistinctAscList [1..5]
4
+--2
|  +--1
|  +--3
+--5

Set> putStrLn $ showTreeWith True True $ fromDistinctAscList [1..5]
4
|
+--2
|  |
|  +--1
|  |
|  +--3
|
+--5

Set> putStrLn $ showTreeWith False True $ fromDistinctAscList [1..5]
+--5
|
4
|
|  +--3
|  |
+--2
   |
   +--1

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

A tactic for dealing with elements present in one set but not the other in merge or mergeA.

A tactic of type WhenMissing f a is an abstract representation of a function of type a -> f Bool.

Since: 0.8.1

A tactic for dealing with elements present in one set but not the other in merge.

A tactic of type SimpleWhenMissing a is an abstract representation of a function of type a -> Bool.

Since: 0.8.1

Drop all the elements that are missing from the other set.

dropMissing :: SimpleWhenMissing a

dropMissing = filterMissing (\_ -> False)

but dropMissing is more efficient.

Since: 0.8.1

Preserve the elements that are missing from the other set.

preserveMissing :: SimpleWhenMissing a

preserveMissing = filterMissing (\_ -> True)

but preserveMissing is more efficient.

Since: 0.8.1

Filter the elements that are missing from the other set.

filterMissing :: (a -> Bool) -> SimpleWhenMissing a

Since: 0.8.1

Filter the elements that are missing from the other set using some Applicative action.

Since: 0.8.1

Create a WhenMissing from two functions.

whenMissing must be called with two functions f and g such that g = filterA f. g may be a more efficient way of applying f to all elements in a Set.

Warning: It is the caller's responsibility to ensure the above property.

Examples

preserveMissing :: Applicative f => WhenMissing f a
preserveMissing = whenMissing f g
  where
    f _x = pure True
    g s = pure s
    -- Note that this satisfies g = filterA f

import Data.Functor.Const (Const(..))
import Data.Monoid (All(..))

-- For a usage of this, see examples on mergeA
isEmpty :: WhenMissing (Const All) a
isEmpty = whenMissing f g
  where
    f _x = Const (All False)
    g s = Const (All (null s))
    -- Note that this satisfies g = filterA f

Since: 0.8.1

Along with filterAMissing, witnesses the isomorphism between WhenMissing f a and a -> f Bool.

Since: 0.8.1

A tactic for dealing with elements present in both sets in merge or mergeA.

A tactic of type WhenMatched f a is an abstract representation of a function of type a -> f Bool.

Since: 0.8.1

A tactic for dealing with elements present in both sets in merge.

A tactic of type SimpleWhenMatched a is an abstract representation of a function of type a -> Bool.

Since: 0.8.1

When an element is found in both sets, drop the element.

Since: 0.8.1

When an element is found in both sets, keep the element.

Since: 0.8.1

When an element is found in both sets, choose whether to keep the element in the merged set.

Since: 0.8.1

When an element is found in both sets, choose whether to keep the element in the merged set.

Since: 0.8.1

Along with filterAMatched, witnesses the isomorphism between WhenMatched f a and a -> f Bool.

Since: 0.8.1

Merge two sets.

merge takes two SimpleWhenMissing tactics, a SimpleWhenMatched tactic, and two sets. It uses the tactics to merge the sets.

Consider

merge (filterMissing g1) (filterMissing g2) (filterMatched f) s1 s2

g1 = (==2)
g2 = (==3)
f  = (==6)
s1 = [2, 4, 6, 8, 10, 12]
s2 = [3, 6, 9, 12]

merge will pass the elements to g1, g2, or f as appropriate, producing a Bool for each element.

m1:      [   2,            4,     6,     8,           10     12]
m2:      [          3,            6,            9,           12]
result:  [g1 2,  g2 3,  g1 4,   f 6,  g1 8,  g2 9, g1 10,  f 12]
       = [True,  True, False,  True, False, False, False, False]

The result set contains the element for which we have True.

>>> merge (filterMissing g1) (filterMissing g2) (filterMatched f) s1 s2
fromList [2,3,6]

The other tactics below are optimizations or simplifications of filterMissing for special cases. Most importantly,

• dropMissing drops all elements.
• preserveMissing leaves all elements alone.

When merge is given three arguments, it is inlined at the call site. To prevent excessive inlining, you should typically use merge to define your custom combining functions.

Since: 0.8.1

An applicative version of merge.

mergeA takes two WhenMissing tactics, a WhenMatched tactic, and two sets. It uses the tactics to merge the sets.

Behaves just like merge while allowing Applicative effects. Effects are performed in increasing order of elements.

Consider,

mergeA (filterAMissing g1) (filterAMissing g2) (filterAMatched f) s1 s2

g1 x = (x == 2) <$ putStrLn ("g1 " ++ show x)
g2 x = (x == 3) <$ putStrLn ("g2 " ++ show x)
f x = (x == 6) <$ putStrLn ("f " ++ show x)
s1 = [2, 4, 6, 8, 10, 12]
s2 = [3, 6, 9, 12]

As with merge, the result set is [2,3,6]. Additionally, g1, g2, and f perform IO effects, printing the elements in increasing order.

>>> mergeA (filterAMissing g1) (filterAMissing g2) (filterAMatched f) s1 s2
g1 2
g2 3
g1 4
f 6
g1 8
g2 9
g1 10
f 12
fromList [2,3,6]

When mergeA is given three arguments, it is inlined at the call site. To prevent excessive inlining, you should generally only use mergeA to define custom combining functions.

Examples

data Pair a = Pair !a !a deriving Functor

instance Applicative Pair where
   pure x = Pair x x
   liftA2 f (Pair x1 y1) (Pair x2 y2) = Pair (f x1 x2) (f y1 y2)

-- | Calculate the union and intersection of two sets.
unionIntersection :: Ord a => Set a -> Set a -> (Set a, Set a)
unionIntersection m1 m2 =
  case mergeA preserveAndDropMissing preserveAndDropMissing preserveMatched m1 m2 of
    Pair mu mi -> (mu, mi)
  where
    -- use Pair to build the union and intersection together
    preserveAndDropMissing = whenMissing (\_x -> Pair True False) (\s -> Pair s empty)

import Data.Functor.Const (Const(..))
import Data.Monoid (All(..))

-- | Whether the first set is a subset of the second set.
isSubsetOf :: Ord a => Set a -> Set a -> Bool
isSubsetOf m1 m2 =
  getAll (getConst (mergeA isEmpty dropMissing dropMatched m1 m2))
  where
    isEmpty = whenMissing (\_x -> Const (All False)) (\s -> Const (All (null s)))

Since: 0.8.1