Data.Set.AVL

This module provides an AVL tree based clone of the base package Data.Set.

There are some differences though..

• size is O(n), not O(1)
• The showTree and showTreeWith functions are not implemented.
• The complexities of isSubsetOf,isProperSubsetOf,union,intersection,difference are unknown (because my maths isn't good enough to figure it out), but are probably no worse than the originals.
• Conversion functions toTree, unsafeFromTree, toStdSet, fromStdSet. have been added.

Set type

data Set a

A set of values a.

Instances

Typeable1 Set

Eq a => Eq (Set a)

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

Ord a => Ord (Set a)

Show a => Show (Set a)

Ord a => Monoid (Set a)

Foldable (Set a) a

Ord a => Set (Set a) a

Ord a => SortingCollection (Set a) a

Ord a => Unfoldable (Set a) a

Ord a => Collection (Set a) a

Ord a => Map (Set a) a ()

Operators

(\\) :: Ord a => Set a -> Set a -> Set a

O(?). See difference.

Query

null :: Set a -> Bool

O(1). Is this the empty set?

size :: Set a -> Int

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

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

O(?). 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. 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

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

Combine

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

O(?). The union of two sets, preferring the first set when equal elements are encountered.

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(?). Difference of two sets.

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

O(?). The intersection of two sets.

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 identity

mapMonotonic f s == map f s, 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.

To/From Data.Set.Set

toStdSet :: Set a -> Set a

O(n). Convert an AVL tree based Set (as provided by this module) to a Data.Set.Set.

fromStdSet :: Set a -> Set a

O(n). Convert a Data.Set.Set to an AVL tree based Set (as provided by this module).

To/From raw AVL trees.

These conversions allow you to use the functions provided by Data.Tree.AVL.

toTree :: Set a -> AVL a

O(1). Convert an AVL tree based Set (as provided by this module) to a sorted AVL tree.

unsafeFromTree :: AVL a -> Set a

O(1). Convert a sorted AVL tree to an AVL tree based Set (as provided by this module). This function does not check the input AVL tree is sorted.

Debugging

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.