Data.IntSet

Finite Int Sets

The IntSet type represents a set of elements of type Int. An IntSet is strict in its elements.

For a walkthrough of the most commonly used functions see their sets introduction.

These modules are intended to be imported qualified, to avoid name clashes with Prelude functions, e.g.

 import Data.IntSet (IntSet)
 import qualified Data.IntSet as IntSet

Implementation

The implementation is based on big-endian patricia trees. This data structure performs especially well on binary operations like union and intersection. Additionally, benchmarks show that it is also (much) faster on insertions and deletions when compared to a generic size-balanced set implementation (see Data.Set).

• Chris Okasaki and Andy Gill, "Fast Mergeable Integer Maps", Workshop on ML, September 1998, pages 77-86, https://web.archive.org/web/20150417234429/https://ittc.ku.edu/~andygill/papers/IntMap98.pdf.
• D.R. Morrison, "PATRICIA -- Practical Algorithm To Retrieve Information Coded In Alphanumeric", Journal of the ACM, 15(4), October 1968, pages 514-534, https://doi.org/10.1145/321479.321481.

Additionally, this implementation places bitmaps in the leaves of the tree. Their size is the natural size of a machine word (32 or 64 bits) and greatly reduces the memory footprint and execution times for dense sets, e.g. sets where it is likely that many values lie close to each other. The asymptotics are not affected by this optimization.

Performance information

The time complexity is given for each operation in big-O notation, with \(n\) referring to the number of entries in the map and \(W\) referring to the number of bits in an Int (32 or 64).

Operations like member, insert, and delete have a worst-case complexity of \(O(\min(n,W))\). This means that the operation can become linear in the number of elements with a maximum of \(W\) -- the number of bits in an Int (32 or 64). These peculiar asymptotics are determined by the depth of the Patricia trees:

• even for an extremely unbalanced tree, the depth cannot be larger than the number of elements \(n\),
• each level of a Patricia tree determines at least one more bit shared by all subelements, so there could not be more than \(W\) levels.

If all \(n\) elements in the tree are between 0 and \(N\) (or, say, between \(-N\) and \(N\)), the estimate can be refined to \(O(\min(n, \log N))\). If the set is sufficiently "dense", this becomes \(O(\min(n, \log n))\) or simply the familiar \(O(\log n)\), matching balanced binary trees.

The most performant scenario for IntSet are elements from a contiguous subset, in which case the complexity is proportional to \(\log n\), capped by \(W\). The worst scenario are exponentially growing elements (1,2,4, ldots,2^n), for which complexity grows as fast as \(n\) but again is capped by \(W\).

Binary set operations like union and intersection take \(O(\min(n, m \log \frac{2^W}{m}))\) time, where \(m\) and \(n\) are the sizes of the smaller and larger input sets respectively.

Set type

data IntSet

A set of integers.

Instances

Data IntSet

Defined in Data.IntSet.Internal

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> IntSet -> c IntSet

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c IntSet

toConstr :: IntSet -> Constr

dataTypeOf :: IntSet -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c IntSet)

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c IntSet)

gmapT :: (forall b. Data b => b -> b) -> IntSet -> IntSet

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> IntSet -> r

gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> IntSet -> r

gmapQ :: (forall d. Data d => d -> u) -> IntSet -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> IntSet -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> IntSet -> m IntSet

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> IntSet -> m IntSet

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> IntSet -> m IntSet

Monoid IntSet

mempty = empty

Defined in Data.IntSet.Internal

Methods

mempty :: IntSet

mappend :: IntSet -> IntSet -> IntSet

mconcat :: [IntSet] -> IntSet

Semigroup IntSet

(<>) = union

Since: 0.5.7

Defined in Data.IntSet.Internal

Methods

(<>) :: IntSet -> IntSet -> IntSet

sconcat :: NonEmpty IntSet -> IntSet

stimes :: Integral b => b -> IntSet -> IntSet

IsList IntSet

Since: 0.5.6.2

Defined in Data.IntSet.Internal

Associated Types

type Item IntSet

Methods

fromList :: [Item IntSet] -> IntSet

fromListN :: Int -> [Item IntSet] -> IntSet

toList :: IntSet -> [Item IntSet]

Read IntSet

Defined in Data.IntSet.Internal

Methods

readsPrec :: Int -> ReadS IntSet

readList :: ReadS [IntSet]

readPrec :: ReadPrec IntSet

readListPrec :: ReadPrec [IntSet]

Show IntSet

Defined in Data.IntSet.Internal

Methods

showsPrec :: Int -> IntSet -> ShowS

show :: IntSet -> String

showList :: [IntSet] -> ShowS

NFData IntSet

Defined in Data.IntSet.Internal

Methods

rnf :: IntSet -> ()

Eq IntSet

Defined in Data.IntSet.Internal

Methods

(==) :: IntSet -> IntSet -> Bool

(/=) :: IntSet -> IntSet -> Bool

Ord IntSet

Defined in Data.IntSet.Internal

Methods

compare :: IntSet -> IntSet -> Ordering

(<) :: IntSet -> IntSet -> Bool

(<=) :: IntSet -> IntSet -> Bool

(>) :: IntSet -> IntSet -> Bool

(>=) :: IntSet -> IntSet -> Bool

max :: IntSet -> IntSet -> IntSet

min :: IntSet -> IntSet -> IntSet

Lift IntSet

Since: 0.6.6

Defined in Data.IntSet.Internal

Methods

lift :: Quote m => IntSet -> m Exp

liftTyped :: forall (m :: Type -> Type). Quote m => IntSet -> Code m IntSet

type Item IntSet

Defined in Data.IntSet.Internal

type Item IntSet = Key

type Key = Int

Construction

empty :: IntSet

\(O(1)\). The empty set.

singleton :: Key -> IntSet

\(O(1)\). A set of one element.

fromList :: [Key] -> IntSet

\(O(n \min(n,W))\). Create a set from a list of integers.

fromRange :: (Key, Key) -> IntSet

\(O(n / W)\). Create a set from a range of integers.

fromRange (low, high) == fromList [low..high]

Since: 0.7

fromAscList :: [Key] -> IntSet

\(O(n)\). Build a set from an ascending list of elements.

Warning: This function should be used only if the elements are in non-decreasing order. This precondition is not checked. Use fromList if the precondition may not hold.

fromDistinctAscList :: [Key] -> IntSet

\(O(n)\). Build a set from an ascending list of distinct elements.

Warning: This function should be used only if the elements are in strictly increasing order. This precondition is not checked. Use fromList if the precondition may not hold.

Insertion

insert :: Key -> IntSet -> IntSet

\(O(\min(n,W))\). Add a value to the set. There is no left- or right bias for IntSets.

Deletion

delete :: Key -> IntSet -> IntSet

\(O(\min(n,W))\). Delete a value in the set. Returns the original set when the value was not present.

Generalized insertion/deletion

alterF :: Functor f => (Bool -> f Bool) -> Key -> IntSet -> f IntSet

\(O(\min(n,W))\). (alterF f x s) can delete or insert x in s depending on whether it is already present in s.

In short:

member x <$> alterF f x s = f (member x s)

Note: alterF is a variant of the at combinator from Control.Lens.At.

Since: 0.6.3.1

Query

member :: Key -> IntSet -> Bool

\(O(\min(n,W))\). Is the value a member of the set?

notMember :: Key -> IntSet -> Bool

\(O(\min(n,W))\). Is the element not in the set?

lookupLT :: Key -> IntSet -> Maybe Key

\(O(\min(n,W))\). Find largest element smaller than the given one.

lookupLT 3 (fromList [3, 5]) == Nothing
lookupLT 5 (fromList [3, 5]) == Just 3

lookupGT :: Key -> IntSet -> Maybe Key

\(O(\min(n,W))\). Find smallest element greater than the given one.

lookupGT 4 (fromList [3, 5]) == Just 5
lookupGT 5 (fromList [3, 5]) == Nothing

lookupLE :: Key -> IntSet -> Maybe Key

\(O(\min(n,W))\). 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 :: Key -> IntSet -> Maybe Key

\(O(\min(n,W))\). 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

null :: IntSet -> Bool

\(O(1)\). Is the set empty?

size :: IntSet -> Int

\(O(n)\). Cardinality of the set.

isSubsetOf :: IntSet -> IntSet -> Bool

\(O(\min(n, m \log \frac{2^W}{m})), m \leq n\). Is this a subset? (s1 `isSubsetOf` s2) tells whether s1 is a subset of s2.

isProperSubsetOf :: IntSet -> IntSet -> Bool

\(O(\min(n, m \log \frac{2^W}{m})), m \leq n\). Is this a proper subset? (ie. a subset but not equal).

disjoint :: IntSet -> IntSet -> Bool

\(O(\min(n, m \log \frac{2^W}{m})), m \leq n\). 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

Since: 0.5.11

Combine

union :: IntSet -> IntSet -> IntSet

\(O(\min(n, m \log \frac{2^W}{m})), m \leq n\). The union of two sets.

unions :: Foldable f => f IntSet -> IntSet

The union of a list of sets.

difference :: IntSet -> IntSet -> IntSet

\(O(\min(n, m \log \frac{2^W}{m})), m \leq n\). Difference between two sets.

(\\) :: IntSet -> IntSet -> IntSet infixl 9

\(O(\min(n, m \log \frac{2^W}{m})), m \leq n\). See difference.

intersection :: IntSet -> IntSet -> IntSet

\(O(\min(n, m \log \frac{2^W}{m})), m \leq n\). The intersection of two sets.

intersections :: NonEmpty IntSet -> IntSet

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

Since: 0.8

symmetricDifference :: IntSet -> IntSet -> IntSet

\(O(\min(n, m \log \frac{2^W}{m})), m \leq n\). The symmetric difference of two sets.

The result contains elements that appear in exactly one of the two sets.

symmetricDifference (fromList [0,2,4,6]) (fromList [0,3,6,9]) == fromList [2,3,4,9]

Since: 0.8

newtype Intersection

IntSets form a Semigroup under intersection.

A Monoid instance is not defined because it would be impractical to construct mempty, the IntSet containing all Ints.

Since: 0.8

Constructors

Intersection

Fields

• getIntersection :: IntSet

Instances

Semigroup Intersection

Defined in Data.IntSet.Internal

Methods

(<>) :: Intersection -> Intersection -> Intersection

sconcat :: NonEmpty Intersection -> Intersection

stimes :: Integral b => b -> Intersection -> Intersection

Show Intersection

Defined in Data.IntSet.Internal

Methods

showsPrec :: Int -> Intersection -> ShowS

show :: Intersection -> String

showList :: [Intersection] -> ShowS

Eq Intersection

Defined in Data.IntSet.Internal

Methods

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

(/=) :: Intersection -> Intersection -> Bool

Ord Intersection

Defined in Data.IntSet.Internal

Methods

compare :: Intersection -> Intersection -> Ordering

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

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

(>) :: Intersection -> Intersection -> Bool

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

max :: Intersection -> Intersection -> Intersection

min :: Intersection -> Intersection -> Intersection

Filter

filter :: (Key -> Bool) -> IntSet -> IntSet

\(O(n)\). Filter all elements that satisfy some predicate.

partition :: (Key -> Bool) -> IntSet -> (IntSet, IntSet)

\(O(n)\). partition the set according to some predicate.

takeWhileAntitone :: (Key -> Bool) -> IntSet -> IntSet

\(O(\min(n,W))\). Take while a predicate on the elements holds. The user is responsible for ensuring that for all Ints, j < k ==> p j >= p k. See note at spanAntitone.

takeWhileAntitone p = fromDistinctAscList . takeWhile p . toList
takeWhileAntitone p = filter p

Since: 0.6.7

dropWhileAntitone :: (Key -> Bool) -> IntSet -> IntSet

\(O(\min(n,W))\). Drop while a predicate on the elements holds. The user is responsible for ensuring that for all Ints, j < k ==> p j >= p k. See note at spanAntitone.

dropWhileAntitone p = fromDistinctAscList . dropWhile p . toList
dropWhileAntitone p = filter (not . p)

Since: 0.6.7

spanAntitone :: (Key -> Bool) -> IntSet -> (IntSet, IntSet)

\(O(\min(n,W))\). Divide a set at the point where a predicate on the elements stops holding. The user is responsible for ensuring that for all Ints, 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.

Since: 0.6.7

split :: Key -> IntSet -> (IntSet, IntSet)

\(O(\min(n,W))\). 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.

split 3 (fromList [1..5]) == (fromList [1,2], fromList [4,5])

splitMember :: Key -> IntSet -> (IntSet, Bool, IntSet)

\(O(\min(n,W))\). Performs a split but also returns whether the pivot element was found in the original set.

splitRoot :: IntSet -> [IntSet]

\(O(1)\). 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 submap less than all elements in the second, and so on).

Examples:

splitRoot (fromList [1..120]) == [fromList [1..63],fromList [64..120]]
splitRoot empty == []

Note that the current implementation does not return more than two subsets, but you should not depend on this behaviour because it can change in the future without notice. Also, the current version does not continue splitting all the way to individual singleton sets -- it stops at some point.

Map

map :: (Key -> Key) -> IntSet -> IntSet

\(O(n \min(n,W))\). 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 :: (Key -> Key) -> IntSet -> IntSet

\(O(n)\). The

mapMonotonic f s == map f s, but works only when f is strictly increasing. 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

Warning: This function should be used only if f is monotonically strictly increasing. This precondition is not checked. Use map if the precondition may not hold.

Since: 0.6.3.1

Folds

foldr :: (Key -> b -> b) -> b -> IntSet -> b

\(O(n)\). 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 -> Key -> a) -> a -> IntSet -> a

\(O(n)\). 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

foldMap :: Monoid a => (Key -> a) -> IntSet -> a

\(O(n)\). Map the elements in the set to a monoid and combine with (<>).

Since: 0.8

Strict folds

foldr' :: (Key -> b -> b) -> b -> IntSet -> b

\(O(n)\). 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 -> Key -> a) -> a -> IntSet -> a

\(O(n)\). 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 :: (Key -> b -> b) -> b -> IntSet -> b

Deprecated: Use Data.IntSet.foldr instead

\(O(n)\). Fold the elements in the set using the given right-associative binary operator.

Min/Max

lookupMin :: IntSet -> Maybe Key

\(O(\min(n,W))\). The minimal element of the set. Returns Nothing if the set is empty.

Since: 0.8

lookupMax :: IntSet -> Maybe Key

\(O(\min(n,W))\). The maximal element of the set. Returns Nothing if the set is empty.

Since: 0.8

findMin :: IntSet -> Key

\(O(\min(n,W))\). The minimal element of the set. Calls error if the set is empty.

findMax :: IntSet -> Key

\(O(\min(n,W))\). The maximal element of the set. Calls error if the set is empty.

deleteMin :: IntSet -> IntSet

\(O(\min(n,W))\). Delete the minimal element. Returns an empty set if the set is empty.

Note that this is a change of behaviour for consistency with Set – versions prior to 0.5 threw an error if the IntSet was already empty.

deleteMax :: IntSet -> IntSet

\(O(\min(n,W))\). Delete the maximal element. Returns an empty set if the set is empty.

Note that this is a change of behaviour for consistency with Set – versions prior to 0.5 threw an error if the IntSet was already empty.

deleteFindMin :: IntSet -> (Key, IntSet)

\(O(\min(n,W))\). Delete and find the minimal element.

deleteFindMin set = (findMin set, deleteMin set)

deleteFindMax :: IntSet -> (Key, IntSet)

\(O(\min(n,W))\). Delete and find the maximal element.

deleteFindMax set = (findMax set, deleteMax set)

maxView :: IntSet -> Maybe (Key, IntSet)

\(O(\min(n,W))\). Retrieves the maximal key of the set, and the set stripped of that element, or Nothing if passed an empty set.

minView :: IntSet -> Maybe (Key, IntSet)

\(O(\min(n,W))\). Retrieves the minimal key of the set, and the set stripped of that element, or Nothing if passed an empty set.

Conversion

List

elems :: IntSet -> [Key]

\(O(n)\). An alias of toAscList. The elements of a set in ascending order. Subject to list fusion.

toList :: IntSet -> [Key]

\(O(n)\). Convert the set to a list of elements. Subject to list fusion.

toAscList :: IntSet -> [Key]

\(O(n)\). Convert the set to an ascending list of elements. Subject to list fusion.

toDescList :: IntSet -> [Key]

\(O(n)\). Convert the set to a descending list of elements. Subject to list fusion.

Debugging

showTree :: IntSet -> String

\(O(n \min(n,W))\). Show the tree that implements the set. The tree is shown in a compressed, hanging format.

showTreeWith :: Bool -> Bool -> IntSet -> String

\(O(n \min(n,W))\). 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.