# Data.Set

## Finite Sets

The `Set e` type represents a set of elements of type `e`. Most operations require that `e` be an instance of the `Ord` class. A `Set` is strict in its elements.

For a walkthrough of the most commonly used functions see the sets introduction.

Note that the implementation is generally left-biased. Functions that take two sets as arguments and combine them, such as `union` and `intersection`, prefer the entries in the first argument to those in the second. Of course, this bias can only be observed when equality is an equivalence relation instead of structural equality.

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

``` import Data.Set (Set)
import qualified Data.Set as Set```

### Warning

The size of the set must not exceed `maxBound::Int`. Violation of this condition is not detected and if the size limit is exceeded, its behaviour is undefined.

### Implementation

The implementation of `Set` 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.

Bounds for `union`, `intersection`, and `difference` are as given by

## Set type

data Set a Source

A set of values `a`.

Instances
Foldable Set
Instance details

Defined in Data.Set.Internal

#### Methods

fold :: Monoid m => Set m -> m Source

foldMap :: Monoid m => (a -> m) -> Set a -> m Source

foldr :: (a -> b -> b) -> b -> Set a -> b Source

foldr' :: (a -> b -> b) -> b -> Set a -> b Source

foldl :: (b -> a -> b) -> b -> Set a -> b Source

foldl' :: (b -> a -> b) -> b -> Set a -> b Source

foldr1 :: (a -> a -> a) -> Set a -> a Source

foldl1 :: (a -> a -> a) -> Set a -> a Source

toList :: Set a -> [a] Source

null :: Set a -> Bool Source

length :: Set a -> Int Source

elem :: Eq a => a -> Set a -> Bool Source

maximum :: Ord a => Set a -> a Source

minimum :: Ord a => Set a -> a Source

sum :: Num a => Set a -> a Source

product :: Num a => Set a -> a Source

Eq1 Set

Since: containers-0.5.9

Instance details

Defined in Data.Set.Internal

#### Methods

liftEq :: (a -> b -> Bool) -> Set a -> Set b -> Bool Source

Ord1 Set

Since: containers-0.5.9

Instance details

Defined in Data.Set.Internal

#### Methods

liftCompare :: (a -> b -> Ordering) -> Set a -> Set b -> Ordering Source

Show1 Set

Since: containers-0.5.9

Instance details

Defined in Data.Set.Internal

#### Methods

liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Set a -> ShowS Source

liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [Set a] -> ShowS Source

Ord a => IsList (Set a)

Since: containers-0.5.6.2

Instance details

Defined in Data.Set.Internal

#### Associated Types

type Item (Set a) :: Type Source

#### Methods

fromList :: [Item (Set a)] -> Set a Source

fromListN :: Int -> [Item (Set a)] -> Set a Source

toList :: Set a -> [Item (Set a)] Source

Eq a => Eq (Set a)
Instance details

Defined in Data.Set.Internal

#### Methods

(==) :: Set a -> Set a -> Bool Source

(/=) :: Set a -> Set a -> Bool Source

(Data a, Ord a) => Data (Set a)
Instance details

Defined in Data.Set.Internal

#### Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Set a -> c (Set a) Source

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Set a) Source

toConstr :: Set a -> Constr Source

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Set a)) Source

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Set a)) Source

gmapT :: (forall b. Data b => b -> b) -> Set a -> Set a Source

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Set a -> r Source

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Set a -> r Source

gmapQ :: (forall d. Data d => d -> u) -> Set a -> [u] Source

gmapQi :: Int -> (forall d. Data d => d -> u) -> Set a -> u Source

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Set a -> m (Set a) Source

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Set a -> m (Set a) Source

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Set a -> m (Set a) Source

Ord a => Ord (Set a)
Instance details

Defined in Data.Set.Internal

#### Methods

compare :: Set a -> Set a -> Ordering Source

(<) :: Set a -> Set a -> Bool Source

(<=) :: Set a -> Set a -> Bool Source

(>) :: Set a -> Set a -> Bool Source

(>=) :: Set a -> Set a -> Bool Source

max :: Set a -> Set a -> Set a Source

min :: Set a -> Set a -> Set a Source

Instance details

Defined in Data.Set.Internal

#### Methods

Show a => Show (Set a)
Instance details

Defined in Data.Set.Internal

#### Methods

showsPrec :: Int -> Set a -> ShowS Source

show :: Set a -> String Source

showList :: [Set a] -> ShowS Source

Ord a => Semigroup (Set a)

Since: containers-0.5.7

Instance details

Defined in Data.Set.Internal

#### Methods

(<>) :: Set a -> Set a -> Set a Source

sconcat :: NonEmpty (Set a) -> Set a Source

stimes :: Integral b => b -> Set a -> Set a Source

Ord a => Monoid (Set a)
Instance details

Defined in Data.Set.Internal

#### Methods

mappend :: Set a -> Set a -> Set a Source

mconcat :: [Set a] -> Set a Source

NFData a => NFData (Set a)
Instance details

Defined in Data.Set.Internal

#### Methods

rnf :: Set a -> () Source

type Item (Set a)
Instance details

Defined in Data.Set.Internal

type Item (Set a) = a

## Construction

O(1). The empty set.

singleton :: a -> Set a Source

O(1). Create a singleton set.

fromList :: Ord a => [a] -> Set a Source

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

If the elements are ordered, a linear-time implementation is used, with the performance equal to `fromDistinctAscList`.

fromAscList :: Eq a => [a] -> Set a Source

O(n). Build a set from an ascending list in linear time. The precondition (input list is ascending) is not checked.

fromDescList :: Eq a => [a] -> Set a Source

O(n). Build a set from a descending list in linear time. The precondition (input list is descending) is not checked.

Since: containers-0.5.8

fromDistinctAscList :: [a] -> Set a Source

O(n). Build a set from an ascending list of distinct elements in linear time. The precondition (input list is strictly ascending) is not checked.

fromDistinctDescList :: [a] -> Set a Source

O(n). Build a set from a descending list of distinct elements in linear time. The precondition (input list is strictly descending) is not checked.

powerSet :: Set a -> Set (Set a) Source

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 [[], , , , [1,2], [1,3], [2,3], [1,2,3]]
```

Since: containers-0.5.11

## Insertion

insert :: Ord a => a -> Set a -> Set a Source

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.

## Deletion

delete :: Ord a => a -> Set a -> Set a Source

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

## Query

member :: Ord a => a -> Set a -> Bool Source

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

notMember :: Ord a => a -> Set a -> Bool Source

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

lookupLT :: Ord a => a -> Set a -> Maybe a Source

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

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

lookupGT :: Ord a => a -> Set a -> Maybe a Source

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

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

lookupLE :: Ord a => a -> Set a -> Maybe a Source

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```

lookupGE :: Ord a => a -> Set a -> Maybe a Source

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```

null :: Set a -> Bool Source

O(1). Is this the empty set?

size :: Set a -> Int Source

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

isSubsetOf :: Ord a => Set a -> Set a -> Bool Source

O(n+m). Is this a subset? `(s1 `isSubsetOf` s2)` tells whether `s1` is a subset of `s2`.

isProperSubsetOf :: Ord a => Set a -> Set a -> Bool Source

O(n+m). Is this a proper subset? (ie. a subset but not equal).

disjoint :: Ord a => Set a -> Set a -> Bool Source

O(n+m). 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```

Since: containers-0.5.11

## Combine

union :: Ord a => Set a -> Set a -> Set a Source

O(m*log(n/m + 1)), m <= n. The union of two sets, preferring the first set when equal elements are encountered.

unions :: (Foldable f, Ord a) => f (Set a) -> Set a Source

The union of a list of sets: (`unions == foldl union empty`).

difference :: Ord a => Set a -> Set a -> Set a Source

O(m*log(n/m + 1)), m <= n. Difference of two sets.

(\\) :: Ord a => Set a -> Set a -> Set a infixl 9 Source

O(m*log(n/m+1)), m <= n. See `difference`.

intersection :: Ord a => Set a -> Set a -> Set a Source

O(m*log(n/m + 1)), m <= 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])`.

cartesianProduct :: Set a -> Set b -> Set (a, b) Source

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: containers-0.5.11

disjointUnion :: Set a -> Set b -> Set (Either a b) Source

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: containers-0.5.11

## Filter

filter :: (a -> Bool) -> Set a -> Set a Source

O(n). Filter all elements that satisfy the predicate.

takeWhileAntitone :: (a -> Bool) -> Set a -> Set a Source

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: containers-0.5.8

dropWhileAntitone :: (a -> Bool) -> Set a -> Set a Source

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: containers-0.5.8

spanAntitone :: (a -> Bool) -> Set a -> (Set a, Set a) Source

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: containers-0.5.8

partition :: (a -> Bool) -> Set a -> (Set a, Set a) Source

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`.

split :: Ord a => a -> Set a -> (Set a, Set a) Source

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`.

splitMember :: Ord a => a -> Set a -> (Set a, Bool, Set a) Source

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

splitRoot :: Set a -> [Set a] Source

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 ,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: containers-0.5.4

## Indexed

lookupIndex :: Ord a => a -> Set a -> Maybe Int Source

O(log n). Lookup 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: containers-0.5.4

findIndex :: Ord a => a -> Set a -> Int Source

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.

```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: containers-0.5.4

elemAt :: Int -> Set a -> a Source

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.

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

Since: containers-0.5.4

deleteAt :: Int -> Set a -> Set a Source

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.

```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: containers-0.5.4

take :: Int -> Set a -> Set a Source

Take a given number of elements in order, beginning with the smallest ones.

```take n = fromDistinctAscList . take n . toAscList
```

Since: containers-0.5.8

drop :: Int -> Set a -> Set a Source

Drop a given number of elements in order, beginning with the smallest ones.

```drop n = fromDistinctAscList . drop n . toAscList
```

Since: containers-0.5.8

splitAt :: Int -> Set a -> (Set a, Set a) Source

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

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

## Map

map :: Ord b => (a -> b) -> Set a -> Set b Source

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

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

mapMonotonic :: (a -> b) -> Set a -> Set b Source

O(n). The

`mapMonotonic f s == map f s`, but works only when `f` is strictly increasing. The precondition is not checked. 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```

## Folds

foldr :: (a -> b -> b) -> b -> Set a -> b Source

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`

foldl :: (a -> b -> a) -> a -> Set b -> a Source

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`

### Strict folds

foldr' :: (a -> b -> b) -> b -> Set a -> b Source

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 -> Set b -> a Source

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.

### Legacy folds

fold :: (a -> b -> b) -> b -> Set a -> b Source

O(n). Fold the elements in the set 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.

## Min/Max

lookupMin :: Set a -> Maybe a Source

O(log n). The minimal element of a set.

Since: containers-0.5.9

lookupMax :: Set a -> Maybe a Source

O(log n). The maximal element of a set.

Since: containers-0.5.9

findMin :: Set a -> a Source

O(log n). The minimal element of a set.

findMax :: Set a -> a Source

O(log n). The maximal element of a set.

deleteMin :: Set a -> Set a Source

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

deleteMax :: Set a -> Set a Source

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

deleteFindMin :: Set a -> (a, Set a) Source

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

`deleteFindMin set = (findMin set, deleteMin set)`

deleteFindMax :: Set a -> (a, Set a) Source

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

`deleteFindMax set = (findMax set, deleteMax set)`

maxView :: Set a -> Maybe (a, Set a) Source

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

minView :: Set a -> Maybe (a, Set a) Source

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

## Conversion

### List

elems :: Set a -> [a] Source

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

toList :: Set a -> [a] Source

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

toAscList :: Set a -> [a] Source

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

toDescList :: Set a -> [a] Source

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

## Debugging

showTree :: Show a => Set a -> String Source

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

showTreeWith :: Show a => Bool -> Bool -> Set a -> String Source

O(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```

valid :: Ord a => Set a -> Bool Source

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

© The University of Glasgow and others