Copyright | (c) Daan Leijen 2002 |
---|---|
License | BSD-style |
Maintainer | libraries@haskell.org |
Portability | portable |
Safe Haskell | Safe |
Language | Haskell2010 |
Finite Sets
The
type represents a set of elements of type Set
ee
. 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
- Guy Blelloch, Daniel Ferizovic, and Yihan Sun, "Just Join for Parallel Ordered Sets", https://arxiv.org/abs/1602.02120v3.
Synopsis
- data Set a
- empty :: Set a
- singleton :: a -> Set a
- fromList :: Ord a => [a] -> Set a
- fromAscList :: Eq a => [a] -> Set a
- fromDescList :: Eq a => [a] -> Set a
- fromDistinctAscList :: [a] -> Set a
- fromDistinctDescList :: [a] -> Set a
- powerSet :: Set a -> Set (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)
- 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
- null :: Set a -> Bool
- size :: Set a -> Int
- isSubsetOf :: Ord a => Set a -> Set a -> Bool
- isProperSubsetOf :: Ord a => Set a -> Set a -> Bool
- disjoint :: Ord a => Set a -> Set a -> Bool
- 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
- (\\) :: Ord a => Set a -> Set a -> Set a
- intersection :: Ord a => Set a -> Set a -> Set a
- cartesianProduct :: Set a -> Set b -> Set (a, b)
- disjointUnion :: Set a -> Set b -> Set (Either a b)
- 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]
- toAscList :: Set a -> [a]
- toDescList :: Set a -> [a]
- showTree :: Show a => Set a -> String
- showTreeWith :: Show a => Bool -> Bool -> Set a -> String
- valid :: Ord a => Set a -> Bool
Set type
A set of values a
.
Instances
Foldable Set Source # | Folds in order of increasing key. |
Defined in Data.Set.Internal fold :: Monoid m => Set m -> m # foldMap :: Monoid m => (a -> m) -> Set a -> m # foldMap' :: Monoid m => (a -> m) -> Set a -> m # foldr :: (a -> b -> b) -> b -> Set a -> b # foldr' :: (a -> b -> b) -> b -> Set a -> b # foldl :: (b -> a -> b) -> b -> Set a -> b # foldl' :: (b -> a -> b) -> b -> Set a -> b # foldr1 :: (a -> a -> a) -> Set a -> a # foldl1 :: (a -> a -> a) -> Set a -> a # elem :: Eq a => a -> Set a -> Bool # maximum :: Ord a => Set a -> a # | |
Eq1 Set Source # | Since: 0.5.9 |
Ord1 Set Source # | Since: 0.5.9 |
Defined in Data.Set.Internal | |
Show1 Set Source # | Since: 0.5.9 |
Lift a => Lift (Set a :: Type) Source # | Since: 0.6.6 |
(Data a, Ord a) => Data (Set a) Source # | |
Defined in Data.Set.Internal gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Set a -> c (Set a) # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Set a) # dataTypeOf :: Set a -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Set a)) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Set a)) # gmapT :: (forall b. Data b => b -> b) -> Set a -> Set a # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Set a -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Set a -> r # gmapQ :: (forall d. Data d => d -> u) -> Set a -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Set a -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Set a -> m (Set a) # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Set a -> m (Set a) # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Set a -> m (Set a) # | |
Ord a => Monoid (Set a) Source # | |
Ord a => Semigroup (Set a) Source # | Since: 0.5.7 |
Ord a => IsList (Set a) Source # | Since: 0.5.6.2 |
(Read a, Ord a) => Read (Set a) Source # | |
Show a => Show (Set a) Source # | |
NFData a => NFData (Set a) Source # | |
Defined in Data.Set.Internal | |
Eq a => Eq (Set a) Source # | |
Ord a => Ord (Set a) Source # | |
type Item (Set a) Source # | |
Defined in Data.Set.Internal |
Construction
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: 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 #
\(O(2^n \log n)\). 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: 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
Generalized insertion/deletion
alterF :: (Ord a, Functor f) => (Bool -> f Bool) -> a -> Set a -> f (Set a) Source #
\(O(\log n)\) (
can delete or insert alterF
f x s)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: 0.6.3.1
Query
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: 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.
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
(\\) :: 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
.
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])
.
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: 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: 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: 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: 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: 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 (
) is a pair split
x set(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: 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: 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: 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: 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: 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: 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: 0.5.8
Map
map :: Ord b => (a -> b) -> Set a -> Set b Source #
\(O(n \log n)\).
is the set obtained by applying map
f sf
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
, but works only when mapMonotonic
f s == map
f sf
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
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
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 #
\(O(n \log n)\). 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