Portability | non-portable (TypeFamilies, MagicHash) |
---|---|

Stability | provisional |

Maintainer | libraries@haskell.org |

Safe Haskell | Safe-Infered |

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://citeseer.ist.psu.edu/okasaki98fast.html - 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).

Unlike the reference implementation in Data.IntSet, Data.Interned.IntSet uses hash consing to ensure that there is only ever one copy of any given IntSet in memory. This is enabled by the normal form of the PATRICIA trie.

This can mean a drastic reduction in the memory footprint of a program in exchange for much more costly set manipulation.

- 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)
- findMin :: IntSet -> Int
- findMax :: IntSet -> Int
- deleteMin :: IntSet -> IntSet
- deleteMax :: IntSet -> IntSet
- deleteFindMin :: IntSet -> (Int, IntSet)
- deleteFindMax :: IntSet -> (Int, IntSet)
- maxView :: IntSet -> Maybe (Int, IntSet)
- minView :: IntSet -> Maybe (Int, 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

A set of integers.

# Operators

# Query

isSubsetOf :: IntSet -> IntSet -> BoolSource

*O(n+m)*. Is this a subset?
`(s1 `

tells whether `isSubsetOf`

s2)`s1`

is a subset of `s2`

.

isProperSubsetOf :: IntSet -> IntSet -> BoolSource

*O(n+m)*. Is this a proper subset? (ie. a subset but not equal).

# Construction

insert :: Int -> IntSet -> IntSetSource

*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 -> IntSetSource

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

# Combine

difference :: IntSet -> IntSet -> IntSetSource

*O(n+m)*. Difference between two sets.

intersection :: IntSet -> IntSet -> IntSetSource

*O(n+m)*. The intersection of two sets.

# Filter

filter :: (Int -> Bool) -> IntSet -> IntSetSource

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

partition :: (Int -> Bool) -> IntSet -> (IntSet, IntSet)Source

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

split :: Int -> 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 :: Int -> 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.

# Min/Max

deleteFindMin :: IntSet -> (Int, IntSet)Source

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

deleteFindMin set = (findMin set, deleteMin set)

deleteFindMax :: IntSet -> (Int, IntSet)Source

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

deleteFindMax set = (findMax set, deleteMax set)

maxView :: IntSet -> Maybe (Int, 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 (Int, 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.

# Map

map :: (Int -> Int) -> IntSet -> IntSetSource

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

is the set obtained by applying `map`

f s`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 -> bSource

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

sum set == fold (+) 0 set elems set == fold (:) [] set

# Conversion

## List

## Ordered list

fromAscList :: [Int] -> IntSetSource

*O(n)*. Build a set from an ascending list of elements.
*The precondition (input list is ascending) is not checked.*

fromDistinctAscList :: [Int] -> IntSetSource

*O(n)*. Build a set from an ascending list of distinct elements.
*The precondition (input list is strictly ascending) is not checked.*

# Debugging

showTree :: IntSet -> StringSource

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

showTreeWith :: Bool -> Bool -> IntSet -> StringSource

*O(n)*. The expression (

) shows
the tree that implements the set. If `showTreeWith`

hang wide map`hang`

is
`True`

, a *hanging* tree is shown otherwise a rotated tree is shown. If
`wide`

is `True`

, an extra wide version is shown.