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 Old interface, DEPRECATED
Description

An efficient implementation of sets.

This module is intended to be imported qualified, to avoid name clashes with Prelude functions. eg.

```  import 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 perferred 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 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) 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 emptySet :: Set a mkSet :: Ord a => [a] -> Set a setToList :: Set a -> [a] unitSet :: a -> Set a elementOf :: Ord a => a -> Set a -> Bool isEmptySet :: Set a -> Bool cardinality :: Set a -> Int unionManySets :: Ord a => [Set a] -> Set a minusSet :: Ord a => Set a -> Set a -> Set a mapSet :: (Ord a, Ord b) => (b -> a) -> Set b -> Set a intersect :: Ord a => Set a -> Set a -> Set a addToSet :: Ord a => Set a -> a -> Set a delFromSet :: Ord a => Set a -> a -> Set a
Set type
data Set a
A set of values a. Instances
 Typeable1 Set (Data a, Ord a) => Data (Set a) Eq a => Eq (Set a) Ord a => Monoid (Set a) Ord a => Ord (Set a) Show a => Show (Set a)
Operators
(\\) :: Ord a => Set a -> Set a -> Set a
O(n+m). See difference.
Query
null :: Set a -> Bool
O(1). Is this the empty set?
size :: Set a -> Int
O(1). The number of elements in the set.
member :: Ord a => a -> Set a -> Bool
O(log n). Is the element in the set?
isSubsetOf :: Ord a => Set a -> Set a -> Bool
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
O(n+m). Is this a proper subset? (ie. a subset but not equal).
Construction
empty :: Set a
O(1). The empty set.
singleton :: a -> Set a
O(1). Create a singleton set.
insert :: Ord a => a -> Set a -> Set a
O(log n). Insert an element in a set.
delete :: Ord a => a -> Set a -> Set a
O(log n). Delete an element from a set.
Combine
union :: Ord a => Set a -> Set a -> Set a
O(n+m). The union of two sets. Uses the efficient hedge-union algorithm. Hedge-union is more efficient on (bigset union smallset).
unions :: Ord a => [Set a] -> Set a
The union of a list of sets: (unions == foldl union empty).
difference :: Ord a => Set a -> Set a -> Set a
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
O(n+m). The intersection of two sets. Intersection is more efficient on (bigset intersection smallset).
Filter
filter :: Ord a => (a -> Bool) -> Set a -> Set a
O(n). Filter all elements that satisfy the predicate.
partition :: Ord a => (a -> Bool) -> Set a -> (Set a, Set a)
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)
O(log n). The expression (split x set) is a pair (set1,set2) where all elements in set1 are lower than x and all elements in set2 larger than x. x is not found in neither set1 nor set2.
splitMember :: Ord a => a -> Set a -> (Set a, Bool, Set a)
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

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

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
O(n). Fold over the elements of a set in an unspecified order.
Min/Max
findMin :: Set a -> a
O(log n). The minimal element of a set.
findMax :: Set a -> a
O(log n). The maximal element of a set.
deleteMin :: Set a -> Set a
O(log n). Delete the minimal element.
deleteMax :: Set a -> Set a
O(log n). Delete the maximal element.
deleteFindMin :: Set a -> (a, Set a)

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

``` deleteFindMin set = (findMin set, deleteMin set)
```
deleteFindMax :: Set a -> (a, Set a)

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

``` deleteFindMax set = (findMax set, deleteMax set)
```
Conversion
List
elems :: Set a -> [a]
O(n). The elements of a set.
toList :: Set a -> [a]
O(n). Convert the set to a list of elements.
fromList :: Ord a => [a] -> Set a
O(n*log n). Create a set from a list of elements.
Ordered list
toAscList :: Set a -> [a]
O(n). Convert the set to an ascending list of elements.
fromAscList :: Eq a => [a] -> Set a
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
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
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

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
O(n). Test if the internal set structure is valid.
Old interface, DEPRECATED
emptySet :: Set a
Obsolete equivalent of empty.
mkSet :: Ord a => [a] -> Set a
Obsolete equivalent of fromList.
setToList :: Set a -> [a]
Obsolete equivalent of elems.
unitSet :: a -> Set a
Obsolete equivalent of singleton.
elementOf :: Ord a => a -> Set a -> Bool
Obsolete equivalent of member.
isEmptySet :: Set a -> Bool
Obsolete equivalent of null.
cardinality :: Set a -> Int
Obsolete equivalent of size.
unionManySets :: Ord a => [Set a] -> Set a
Obsolete equivalent of unions.
minusSet :: Ord a => Set a -> Set a -> Set a
Obsolete equivalent of difference.
mapSet :: (Ord a, Ord b) => (b -> a) -> Set b -> Set a
Obsolete equivalent of map.
intersect :: Ord a => Set a -> Set a -> Set a
Obsolete equivalent of intersection.
addToSet :: Ord a => Set a -> a -> Set a
Obsolete equivalent of flip insert.
delFromSet :: Ord a => Set a -> a -> Set a
Obsolete equivalent of flip delete.