Copyright | (c) Daan Leijen 2002 (c) Joachim Breitner 2011 |
---|---|
License | BSD-style |
Maintainer | libraries@haskell.org |
Portability | portable |
Safe Haskell | Safe |
Language | Haskell2010 |
Finite Int Sets
The
type represents a set of elements of type IntSet
Int
.
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
Performance information
Many operations 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).
Implementation
The implementation is based on big-endian patricia trees. This data
structure performs especially well on binary operations like union
and intersection
. However, my 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, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.37.5452
- D.R. Morrison, "PATRICIA -- Practical Algorithm To Retrieve Information Coded In Alphanumeric", Journal of the ACM, 15(4), October 1968, pages 514-534.
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.
Synopsis
- data IntSet
- type Key = Int
- empty :: IntSet
- singleton :: Key -> IntSet
- fromList :: [Key] -> IntSet
- fromRange :: (Key, Key) -> IntSet
- fromAscList :: [Key] -> IntSet
- fromDistinctAscList :: [Key] -> IntSet
- insert :: Key -> IntSet -> IntSet
- delete :: Key -> IntSet -> IntSet
- alterF :: Functor f => (Bool -> f Bool) -> Key -> IntSet -> f IntSet
- member :: Key -> IntSet -> Bool
- notMember :: Key -> IntSet -> Bool
- lookupLT :: Key -> IntSet -> Maybe Key
- lookupGT :: Key -> IntSet -> Maybe Key
- lookupLE :: Key -> IntSet -> Maybe Key
- lookupGE :: Key -> IntSet -> Maybe Key
- null :: IntSet -> Bool
- size :: IntSet -> Int
- isSubsetOf :: IntSet -> IntSet -> Bool
- isProperSubsetOf :: IntSet -> IntSet -> Bool
- disjoint :: IntSet -> IntSet -> Bool
- union :: IntSet -> IntSet -> IntSet
- unions :: Foldable f => f IntSet -> IntSet
- difference :: IntSet -> IntSet -> IntSet
- (\\) :: IntSet -> IntSet -> IntSet
- intersection :: IntSet -> IntSet -> IntSet
- filter :: (Key -> Bool) -> IntSet -> IntSet
- partition :: (Key -> Bool) -> IntSet -> (IntSet, IntSet)
- takeWhileAntitone :: (Key -> Bool) -> IntSet -> IntSet
- dropWhileAntitone :: (Key -> Bool) -> IntSet -> IntSet
- spanAntitone :: (Key -> Bool) -> IntSet -> (IntSet, IntSet)
- split :: Key -> IntSet -> (IntSet, IntSet)
- splitMember :: Key -> IntSet -> (IntSet, Bool, IntSet)
- splitRoot :: IntSet -> [IntSet]
- map :: (Key -> Key) -> IntSet -> IntSet
- mapMonotonic :: (Key -> Key) -> IntSet -> IntSet
- foldr :: (Key -> b -> b) -> b -> IntSet -> b
- foldl :: (a -> Key -> a) -> a -> IntSet -> a
- foldr' :: (Key -> b -> b) -> b -> IntSet -> b
- foldl' :: (a -> Key -> a) -> a -> IntSet -> a
- fold :: (Key -> b -> b) -> b -> IntSet -> b
- findMin :: IntSet -> Key
- findMax :: IntSet -> Key
- deleteMin :: IntSet -> IntSet
- deleteMax :: IntSet -> IntSet
- deleteFindMin :: IntSet -> (Key, IntSet)
- deleteFindMax :: IntSet -> (Key, IntSet)
- maxView :: IntSet -> Maybe (Key, IntSet)
- minView :: IntSet -> Maybe (Key, IntSet)
- elems :: IntSet -> [Key]
- toList :: IntSet -> [Key]
- toAscList :: IntSet -> [Key]
- toDescList :: IntSet -> [Key]
- showTree :: IntSet -> String
- showTreeWith :: Bool -> Bool -> IntSet -> String
Strictness properties
This module satisfies the following strictness property:
- Key arguments are evaluated to WHNF
Here are some examples that illustrate the property:
delete undefined s == undefined
Set type
A set of integers.
Instances
NFData IntSet Source # | |
Defined in Data.IntSet.Internal | |
Monoid IntSet Source # | |
Semigroup IntSet Source # | Since: containers-0.5.7 |
Data IntSet Source # | |
Defined in Data.IntSet.Internal 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 # | |
IsList IntSet Source # | Since: containers-0.5.6.2 |
Read IntSet Source # | |
Show IntSet Source # | |
Eq IntSet Source # | |
Ord IntSet Source # | |
Lift IntSet Source # | Since: containers-0.6.6 |
type Item IntSet Source # | |
Defined in Data.IntSet.Internal |
Construction
fromRange :: (Key, Key) -> IntSet Source #
\(O(n / W)\). Create a set from a range of integers.
fromRange (low, high) == fromList [low..high]
Since: containers-0.7
fromAscList :: [Key] -> IntSet Source #
\(O(n)\). Build a set from an ascending list of elements. The precondition (input list is ascending) is not checked.
fromDistinctAscList :: [Key] -> IntSet Source #
\(O(n)\). Build a set from an ascending list of distinct elements. The precondition (input list is strictly ascending) is not checked.
Insertion
insert :: Key -> IntSet -> IntSet Source #
\(O(\min(n,W))\). Add a value to the set. There is no left- or right bias for IntSets.
Deletion
delete :: Key -> IntSet -> IntSet Source #
\(O(\min(n,W))\). Delete a value in the set. Returns the original set when the value was not present.
Generalized insertion/deletion
Query
lookupLT :: Key -> IntSet -> Maybe Key Source #
\(O(\min(n,W))\). 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 Source #
\(O(\min(n,W))\). 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 Source #
\(O(\min(n,W))\). 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 Source #
\(O(\min(n,W))\). 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
isSubsetOf :: IntSet -> IntSet -> Bool Source #
\(O(n+m)\). Is this a subset?
(s1 `isSubsetOf` s2)
tells whether s1
is a subset of s2
.
isProperSubsetOf :: IntSet -> IntSet -> Bool Source #
\(O(n+m)\). Is this a proper subset? (ie. a subset but not equal).
disjoint :: IntSet -> IntSet -> Bool Source #
\(O(n+m)\). 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: containers-0.5.11
Combine
Filter
filter :: (Key -> Bool) -> IntSet -> IntSet Source #
\(O(n)\). Filter all elements that satisfy some predicate.
partition :: (Key -> Bool) -> IntSet -> (IntSet, IntSet) Source #
\(O(n)\). partition the set according to some predicate.
takeWhileAntitone :: (Key -> Bool) -> IntSet -> IntSet Source #
\(O(\min(n,W))\). Take while a predicate on the elements holds.
The user is responsible for ensuring that for all Int
s, j < k ==> p j >= p k
.
See note at spanAntitone
.
takeWhileAntitone p =fromDistinctAscList
.takeWhile
p .toList
takeWhileAntitone p =filter
p
Since: containers-0.6.7
dropWhileAntitone :: (Key -> Bool) -> IntSet -> IntSet Source #
\(O(\min(n,W))\). Drop while a predicate on the elements holds.
The user is responsible for ensuring that for all Int
s, j < k ==> p j >= p k
.
See note at spanAntitone
.
dropWhileAntitone p =fromDistinctAscList
.dropWhile
p .toList
dropWhileAntitone p =filter
(not . p)
Since: containers-0.6.7
spanAntitone :: (Key -> Bool) -> IntSet -> (IntSet, IntSet) Source #
\(O(\min(n,W))\). Divide a set at the point where a predicate on the elements stops holding.
The user is responsible for ensuring that for all Int
s, 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: containers-0.6.7
split :: Key -> IntSet -> (IntSet, IntSet) Source #
\(O(\min(n,W))\). 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
.
split 3 (fromList [1..5]) == (fromList [1,2], fromList [4,5])
splitMember :: Key -> IntSet -> (IntSet, Bool, IntSet) Source #
\(O(\min(n,W))\). Performs a split
but also returns whether the pivot
element was found in the original set.
splitRoot :: IntSet -> [IntSet] 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 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 Source #
\(O(n \min(n,W))\).
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 :: (Key -> Key) -> IntSet -> IntSet 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
Since: containers-0.6.3.1
Folds
Strict folds
foldr' :: (Key -> b -> b) -> b -> IntSet -> 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 -> Key -> a) -> a -> IntSet -> 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 :: (Key -> b -> b) -> b -> IntSet -> 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
deleteFindMin :: IntSet -> (Key, IntSet) Source #
\(O(\min(n,W))\). Delete and find the minimal element.
deleteFindMin set = (findMin set, deleteMin set)
deleteFindMax :: IntSet -> (Key, IntSet) Source #
\(O(\min(n,W))\). Delete and find the maximal element.
deleteFindMax set = (findMax set, deleteMax set)
maxView :: IntSet -> Maybe (Key, IntSet) Source #
\(O(\min(n,W))\). 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) Source #
\(O(\min(n,W))\). 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] Source #
\(O(n)\). An alias of toAscList
. The elements of a set in ascending order.
Subject to list fusion.
toList :: IntSet -> [Key] Source #
\(O(n)\). Convert the set to a list of elements. Subject to list fusion.
toAscList :: IntSet -> [Key] Source #
\(O(n)\). Convert the set to an ascending list of elements. Subject to list fusion.
toDescList :: IntSet -> [Key] Source #
\(O(n)\). Convert the set to a descending list of elements. Subject to list fusion.
Debugging
showTree :: IntSet -> String Source #
\(O(n \min(n,W))\). Show the tree that implements the set. The tree is shown in a compressed, hanging format.
showTreeWith :: Bool -> Bool -> IntSet -> String Source #
\(O(n \min(n,W))\). The expression (
) shows
the tree that implements the set. If showTreeWith
hang wide maphang
is
True
, a hanging tree is shown otherwise a rotated tree is shown. If
wide
is True
, an extra wide version is shown.