containers-0.6.7: Assorted concrete container types

Copyright | (c) Daan Leijen 2002
License | BSD-style
Maintainer | libraries@haskell.org
Portability | portable
Safe Haskell | Trustworthy
Language | Haskell2010

Data.Set.Internal

Contents

- Set type
- Operators
- Query
- Construction
- Combine
- Filter
- Indexed
- Map
- Folds
  - Strict folds
  - Legacy folds
- Min/Max
- Conversion
  - List
  - Ordered list
- Debugging

Description

# WARNING

This module is considered internal.

The Package Versioning Policy does not apply.

The contents of this module may change in any way whatsoever and without any warning between minor versions of this package.

Authors importing this module are expected to track development closely.

# Description

An efficient implementation of sets.

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

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

- Guy Blelloch, Daniel Ferizovic, and Yihan Sun, "Just Join for Parallel Ordered Sets", https://arxiv.org/abs/1602.02120v3.

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.

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, the behavior of the set is completely undefined.

Since: containers-0.5.9

Synopsis

- data Set a
  - = Bin !Size !a !(Set a) !(Set a)
  - | Tip
- type Size = Int
- (\\) :: 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
- lookupLT :: Ord a => a -> Set a -> Maybe a
- lookupGT :: Ord a => a -> Set a -> Maybe a
- lookupLE :: Ord a => a -> Set a -> Maybe a
- lookupGE :: Ord a => a -> Set a -> Maybe a
- isSubsetOf :: Ord a => Set a -> Set a -> Bool
- isProperSubsetOf :: Ord a => Set a -> Set a -> Bool
- disjoint :: 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
- alterF :: (Ord a, Functor f) => (Bool -> f Bool) -> a -> Set a -> f (Set a)
- powerSet :: Set a -> Set (Set a)
- union :: Ord a => Set a -> Set a -> Set a
- unions :: (Foldable f, Ord a) => f (Set a) -> Set a
- difference :: Ord a => Set a -> Set a -> Set a
- intersection :: Ord a => Set a -> Set a -> Set a
- intersections :: Ord a => NonEmpty (Set a) -> Set a
- cartesianProduct :: Set a -> Set b -> Set (a, b)
- disjointUnion :: Set a -> Set b -> Set (Either a b)
- newtype Intersection a = Intersection {
  - getIntersection :: Set a
  }
- filter :: (a -> Bool) -> Set a -> Set a
- takeWhileAntitone :: (a -> Bool) -> Set a -> Set a
- dropWhileAntitone :: (a -> Bool) -> Set a -> Set a
- spanAntitone :: (a -> Bool) -> Set a -> (Set a, Set a)
- partition :: (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)
- splitRoot :: Set a -> [Set a]
- lookupIndex :: Ord a => a -> Set a -> Maybe Int
- findIndex :: Ord a => a -> Set a -> Int
- elemAt :: Int -> Set a -> a
- deleteAt :: Int -> Set a -> Set a
- take :: Int -> Set a -> Set a
- drop :: Int -> Set a -> Set a
- splitAt :: Int -> Set a -> (Set a, Set a)
- map :: Ord b => (a -> b) -> Set a -> Set b
- mapMonotonic :: (a -> b) -> Set a -> Set b
- foldr :: (a -> b -> b) -> b -> Set a -> b
- foldl :: (a -> b -> a) -> a -> Set b -> a
- foldr' :: (a -> b -> b) -> b -> Set a -> b
- foldl' :: (a -> b -> a) -> a -> Set b -> a
- fold :: (a -> b -> b) -> b -> Set a -> b
- lookupMin :: Set a -> Maybe a
- lookupMax :: Set a -> Maybe a
- 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]
- toDescList :: Set a -> [a]
- fromAscList :: Eq a => [a] -> Set a
- fromDistinctAscList :: [a] -> Set a
- fromDescList :: Eq a => [a] -> Set a
- fromDistinctDescList :: [a] -> Set a
- showTree :: Show a => Set a -> String
- showTreeWith :: Show a => Bool -> Bool -> Set a -> String
- valid :: Ord a => Set a -> Bool
- bin :: a -> Set a -> Set a -> Set a
- balanced :: Set a -> Bool
- link :: a -> Set a -> Set a -> Set a
- merge :: Set a -> Set a -> Set a

# Set type

data Set a Source #

A set of values `a`.

Constructors

Bin !Size !a !(Set a) !(Set a) |
Tip |

#### Instances

Instances details

Foldable Set Source # | Folds in order of increasing key.

Instance details

Defined in Data.Set.Internal

Methods

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

foldMap :: Monoid m => (a -> m) -> Set a -> 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 Source # | 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 Source # | 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 Source # | 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 #
Lift a => Lift (Set a :: Type) Source # | Since: containers-0.6.6
Instance details Defined in Data.Set.Internal Methods lift :: Quote m => Set a -> m Exp Source # liftTyped :: forall (m :: Type -> Type). Quote m => Set a -> Code m (Set a) Source #
(Data a, Ord a) => Data (Set a) Source # |

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 #

dataTypeOf :: Set a -> DataType 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 :: forall r r'. (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 => Monoid (Set a) Source # |
Instance details Defined in Data.Set.Internal Methods mempty :: Set a Source # mappend :: Set a -> Set a -> Set a Source # mconcat :: [Set a] -> Set a Source #
Ord a => Semigroup (Set a) Source # | 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 => IsList (Set a) Source # | Since: containers-0.5.6.2
Instance details Defined in Data.Set.Internal Associated Types type Item (Set a) Source # Methods fromList :: [Item (Set a)] -> Set a Source # fromListN :: Int -> [Item (Set a)] -> Set a Source # toList :: Set a -> [Item (Set a)] Source #
(Read a, Ord a) => Read (Set a) Source # |
Instance details Defined in Data.Set.Internal Methods readsPrec :: Int -> ReadS (Set a) Source # readList :: ReadS [Set a] Source # readPrec :: ReadPrec (Set a) Source # readListPrec :: ReadPrec [Set a] Source #
Show a => Show (Set a) Source # |
Instance details Defined in Data.Set.Internal Methods showsPrec :: Int -> Set a -> ShowS Source # show :: Set a -> String Source # showList :: [Set a] -> ShowS Source #
NFData a => NFData (Set a) Source # |
Instance details Defined in Data.Set.Internal Methods rnf :: Set a -> () Source #
Eq a => Eq (Set a) Source # |
Instance details Defined in Data.Set.Internal Methods (==) :: Set a -> Set a -> Bool # (/=) :: Set a -> Set a -> Bool #
Ord a => Ord (Set a) Source # |
Instance details Defined in Data.Set.Internal Methods compare :: Set a -> Set a -> Ordering # (<) :: Set a -> Set a -> Bool # (<=) :: Set a -> Set a -> Bool # (>) :: Set a -> Set a -> Bool # (>=) :: Set a -> Set a -> Bool # max :: Set a -> Set a -> Set a # min :: Set a -> Set a -> Set a #
type Item (Set a) Source # |
Instance details Defined in Data.Set.Internal type Item (Set a) = a

type Size = Int Source #

# Operators

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

\(O\bigl(m \log\bigl(\frac{n+1}{m+1}\bigr)\bigr), \; m \leq n\). 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?

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

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

\(O\bigl(m \log\bigl(\frac{n+1}{m+1}\bigr)\bigr), \; m \leq n\). `(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

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

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

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

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

\(O\bigl(m \log\bigl(\frac{n+1}{m+1}\bigr)\bigr), \; m \leq n\). 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: containers-0.5.11

# 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.

alterF :: (Ord a, Functor f) => (Bool -> f Bool) -> a -> Set a -> f (Set a) Source #

\(O(\log n)\) `(`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.

Since: containers-0.6.3.1

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

\(O(2^n \log n)\). 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: containers-0.5.11

# Combine

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

\(O\bigl(m \log\bigl(\frac{n+1}{m+1}\bigr)\bigr), \; m \leq n\). 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 the sets in a Foldable structure : (``unions` == `foldl` `union` `empty``).

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

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

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

\(O\bigl(m \log\bigl(\frac{n+1}{m+1}\bigr)\bigr), \; m \leq n\). 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])`.

intersections :: Ord a => NonEmpty (Set a) -> Set a Source #

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

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

\(O(nm)\). 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: containers-0.5.11

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

\(O(n+m)\). 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: containers-0.5.11

newtype Intersection a Source #

Sets form a `Semigroup` under `intersection`.

Constructors

Intersection |

Fields

- getIntersection :: Set a

#### Instances

Instances details

Ord a => Semigroup (Intersection a) Source # |
Instance details Defined in Data.Set.Internal Methods (<>) :: Intersection a -> Intersection a -> Intersection a Source # sconcat :: NonEmpty (Intersection a) -> Intersection a Source # stimes :: Integral b => b -> Intersection a -> Intersection a Source #
Show a => Show (Intersection a) Source # |
Instance details Defined in Data.Set.Internal Methods showsPrec :: Int -> Intersection a -> ShowS Source # show :: Intersection a -> String Source # showList :: [Intersection a] -> ShowS Source #
Eq a => Eq (Intersection a) Source # |
Instance details Defined in Data.Set.Internal Methods (==) :: Intersection a -> Intersection a -> Bool # (/=) :: Intersection a -> Intersection a -> Bool #
Ord a => Ord (Intersection a) Source # |

Instance details

Defined in Data.Set.Internal

Methods

compare :: Intersection a -> Intersection a -> Ordering #

(<) :: Intersection a -> Intersection a -> Bool #

(<=) :: Intersection a -> Intersection a -> Bool #

(>) :: Intersection a -> Intersection a -> Bool #

(>=) :: Intersection a -> Intersection a -> Bool #

max :: Intersection a -> Intersection a -> Intersection a #

min :: Intersection a -> Intersection a -> Intersection a #

# 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 [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: 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 #

\(O(\log n)\). 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 #

\(O(\log n)\). 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.

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`.

## Ordered list

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.

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.

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

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.

# Debugging

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

\(O(n \log n)\). 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 #

\(O(n \log n)\). 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

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

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

bin :: a -> Set a -> Set a -> Set a Source #

balanced :: Set a -> Bool Source #

link :: a -> Set a -> Set a -> Set a Source #

merge :: Set a -> Set a -> Set a Source #