Haskell Hierarchical Libraries (base package)ContentsIndex
Data.IntSet
Portabilityportable
Stabilityprovisional
Maintainerlibraries@haskell.org
Contents
Set type
Operators
Query
Construction
Combine
Filter
Map
Fold
Conversion
List
Ordered list
Debugging
Description

An efficient implementation of integer sets.

This module is intended to be imported qualified, to avoid name clashes with Prelude functions. eg. 

  import Data.IntSet as Set

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://www.cse.ogi.edu/~andy/pub/finite.htm
  • D.R. Morrison, "PATRICIA -- Practical Algorithm To Retrieve Information Coded In Alphanumeric", Journal of the ACM, 15(4), October 1968, pages 514-534.

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).

Synopsis
data IntSet
(\\) :: IntSet -> IntSet -> IntSet
null :: IntSet -> Bool
size :: IntSet -> Int
member :: Int -> IntSet -> Bool
isSubsetOf :: IntSet -> IntSet -> Bool
isProperSubsetOf :: IntSet -> IntSet -> Bool
empty :: IntSet
singleton :: Int -> IntSet
insert :: Int -> IntSet -> IntSet
delete :: Int -> IntSet -> IntSet
union :: IntSet -> IntSet -> IntSet
unions :: [IntSet] -> IntSet
difference :: IntSet -> IntSet -> IntSet
intersection :: IntSet -> IntSet -> IntSet
filter :: (Int -> Bool) -> IntSet -> IntSet
partition :: (Int -> Bool) -> IntSet -> (IntSet, IntSet)
split :: Int -> IntSet -> (IntSet, IntSet)
splitMember :: Int -> IntSet -> (IntSet, Bool, IntSet)
map :: (Int -> Int) -> IntSet -> IntSet
fold :: (Int -> b -> b) -> b -> IntSet -> b
elems :: IntSet -> [Int]
toList :: IntSet -> [Int]
fromList :: [Int] -> IntSet
toAscList :: IntSet -> [Int]
fromAscList :: [Int] -> IntSet
fromDistinctAscList :: [Int] -> IntSet
showTree :: IntSet -> String
showTreeWith :: Bool -> Bool -> IntSet -> String
Set type
data IntSet
A set of integers.
show/hide Instances
Operators
(\\) :: IntSet -> IntSet -> IntSet
O(n+m). See difference.
Query
null :: IntSet -> Bool
O(1). Is the set empty?
size :: IntSet -> Int
O(n). Cardinality of the set.
member :: Int -> IntSet -> Bool
O(min(n,W)). Is the value a member of the set?
isSubsetOf :: IntSet -> IntSet -> Bool
O(n+m). Is this a subset? (s1 isSubsetOf s2) tells whether s1 is a subset of s2.
isProperSubsetOf :: IntSet -> IntSet -> Bool
O(n+m). Is this a proper subset? (ie. a subset but not equal).
Construction
empty :: IntSet
O(1). The empty set.
singleton :: Int -> IntSet
O(1). A set of one element.
insert :: Int -> IntSet -> IntSet
O(min(n,W)). Add a value to the set. When the value is already an element of the set, it is replaced by the new one, ie. insert is left-biased.
delete :: Int -> IntSet -> IntSet
O(min(n,W)). Delete a value in the set. Returns the original set when the value was not present.
Combine
union :: IntSet -> IntSet -> IntSet
O(n+m). The union of two sets.
unions :: [IntSet] -> IntSet
The union of a list of sets.
difference :: IntSet -> IntSet -> IntSet
O(n+m). Difference between two sets.
intersection :: IntSet -> IntSet -> IntSet
O(n+m). The intersection of two sets.
Filter
filter :: (Int -> Bool) -> IntSet -> IntSet
O(n). Filter all elements that satisfy some predicate.
partition :: (Int -> Bool) -> IntSet -> (IntSet, IntSet)
O(n). partition the set according to some predicate.
split :: Int -> IntSet -> (IntSet, IntSet)

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. 

 split 3 (fromList [1..5]) == (fromList [1,2], fromList [3,4])
splitMember :: Int -> IntSet -> (IntSet, Bool, IntSet)
O(log n). Performs a split but also returns whether the pivot element was found in the original set.
Map
map :: (Int -> Int) -> 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

Fold
fold :: (Int -> b -> b) -> b -> IntSet -> b

O(n). Fold over the elements of a set in an unspecified order. 

 sum set   == fold (+) 0 set
 elems set == fold (:) [] set
Conversion
List
elems :: IntSet -> [Int]
O(n). The elements of a set. (For sets, this is equivalent to toList)
toList :: IntSet -> [Int]
O(n). Convert the set to a list of elements.
fromList :: [Int] -> IntSet
O(n*min(n,W)). Create a set from a list of integers.
Ordered list
toAscList :: IntSet -> [Int]
O(n). Convert the set to an ascending list of elements.
fromAscList :: [Int] -> IntSet
O(n*min(n,W)). Build a set from an ascending list of elements.
fromDistinctAscList :: [Int] -> IntSet
O(n*min(n,W)). Build a set from an ascending list of distinct elements.
Debugging
showTree :: IntSet -> String
O(n). Show the tree that implements the set. The tree is shown in a compressed, hanging format.
showTreeWith :: Bool -> Bool -> IntSet -> String
O(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.
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