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Data.IntSet

Portability | portable
Stability | provisional
Maintainer | libraries@haskell.org

Contents

Set type
Operators
Query
Construction
Combine
Filter
Map
Fold
Conversion
  List
  Ordered list
Debugging

Description

An efficient implementation of integer sets.

Since many function names (but not the type name) clash with Prelude names, this module is usually imported `qualified`, e.g.

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

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
notMember :: 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.
Instances

Data IntSet
Eq IntSet
Monoid IntSet
Ord IntSet
Read IntSet
Show IntSet
Typeable IntSet

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?
notMember :: Int -> IntSet -> Bool
O(log n). Is the element not in 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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