Data.Set
 Portability portable Stability provisional Maintainer libraries@haskell.org
 Contents Set type Operators Query Construction Combine Filter Map Fold Min/Max Conversion List Ordered list Debugging
Description

An efficient implementation of sets.

Since many function names (but not the type name) clash with Prelude names, this module is usually imported qualified, e.g.

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

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.

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. Of course, left-biasing can only be observed when equality is an equivalence relation instead of structural equality.

Synopsis
 data Set a (\\) :: Ord a => Set a -> Set a -> Set a null :: Set a -> Bool size :: Set a -> Int member :: Ord a => a -> Set a -> Bool notMember :: Ord a => a -> Set a -> Bool isSubsetOf :: Ord a => Set a -> Set a -> Bool isProperSubsetOf :: Ord a => Set a -> Set a -> Bool empty :: Set a singleton :: a -> Set a insert :: Ord a => a -> Set a -> Set a delete :: Ord a => a -> Set a -> Set a union :: Ord a => Set a -> Set a -> Set a unions :: Ord a => [Set a] -> Set a difference :: Ord a => Set a -> Set a -> Set a intersection :: Ord a => Set a -> Set a -> Set a filter :: Ord a => (a -> Bool) -> Set a -> Set a partition :: Ord a => (a -> Bool) -> Set a -> (Set a, Set a) split :: Ord a => a -> Set a -> (Set a, Set a) splitMember :: Ord a => a -> Set a -> (Set a, Bool, Set a) map :: (Ord a, Ord b) => (a -> b) -> Set a -> Set b mapMonotonic :: (a -> b) -> Set a -> Set b fold :: (a -> b -> b) -> b -> Set a -> b findMin :: Set a -> a findMax :: Set a -> a deleteMin :: Set a -> Set a deleteMax :: Set a -> Set a deleteFindMin :: Set a -> (a, Set a) deleteFindMax :: Set a -> (a, Set a) maxView :: Set a -> Maybe (a, Set a) minView :: Set a -> Maybe (a, Set a) elems :: Set a -> [a] toList :: Set a -> [a] fromList :: Ord a => [a] -> Set a toAscList :: Set a -> [a] fromAscList :: Eq a => [a] -> Set a fromDistinctAscList :: [a] -> Set a showTree :: Show a => Set a -> String showTreeWith :: Show a => Bool -> Bool -> Set a -> String valid :: Ord a => Set a -> Bool
Set type
 data Set a Source
A set of values a. Instances
 Typeable1 Set Foldable Set Eq a => Eq (Set a) (Data a, Ord a) => Data (Set a) Ord a => Ord (Set a) (Read a, Ord a) => Read (Set a) Show a => Show (Set a) Ord a => Monoid (Set a)
Operators
 (\\) :: Ord a => Set a -> Set a -> Set a Source
O(n+m). See difference.
Query
 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.
 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?
 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).
Construction
 empty :: Set a Source
O(1). The empty set.
 singleton :: a -> Set a Source
O(1). Create a singleton set.
 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.
 delete :: Ord a => a -> Set a -> Set a Source
O(log n). Delete an element from a set.
Combine
 union :: Ord a => Set a -> Set a -> Set a Source
O(n+m). The union of two sets, preferring the first set when equal elements are encountered. The implementation uses the efficient hedge-union algorithm. Hedge-union is more efficient on (bigset union smallset).
 unions :: Ord a => [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(n+m). Difference of two sets. The implementation uses an efficient hedge algorithm comparable with hedge-union.
 intersection :: Ord a => Set a -> Set a -> Set a Source

O(n+m). 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]).

Filter
 filter :: Ord a => (a -> Bool) -> Set a -> Set a Source
O(n). Filter all elements that satisfy the predicate.
 partition :: Ord a => (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.
Map
 map :: (Ord a, 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 monotonic. 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
```
Fold
 fold :: (a -> b -> b) -> b -> Set a -> b Source
O(n). Fold over the elements of a set in an unspecified order.
Min/Max
 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.
 deleteMax :: Set a -> Set a Source
O(log n). Delete the maximal element.
 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). The elements of a set.
 toList :: Set a -> [a] Source
O(n). Convert the set to a list of elements.
 fromList :: Ord a => [a] -> Set a Source
O(n*log n). Create a set from a list of elements.
Ordered list
 toAscList :: Set a -> [a] Source
O(n). Convert the set to an ascending list of elements.
 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.
 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.
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.