containers-0.3.0.0: Assorted concrete container types

Data.IntSet

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://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).

Synopsis

# Set type

data IntSet Source

A set of integers.

Instances

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

# Operators

O(n+m). See `difference`.

# Query

O(1). Is the set empty?

O(n). Cardinality of the set.

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

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

O(n+m). Is this a subset? `(s1 isSubsetOf s2)` tells whether `s1` is a subset of `s2`.

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

# Construction

O(1). The empty set.

O(1). A set of one element.

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.

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

# Combine

O(n+m). The union of two sets.

unions :: [IntSet] -> IntSetSource

The union of a list of sets.

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

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 (`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 :: 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

O(min(n,W)). The minimal element of the set.

O(min(n,W)). The maximal element of a set.

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

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

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

``` deleteFindMin set = (findMin set, deleteMin set)
```

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

``` deleteFindMax set = (findMax set, deleteMax set)
```

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.

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)). `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 -> 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

elems :: IntSet -> [Int]Source

O(n). The elements of a set. (For sets, this is equivalent to toList)

toList :: IntSet -> [Int]Source

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

fromList :: [Int] -> IntSetSource

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

## Ordered list

toAscList :: IntSet -> [Int]Source

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

fromAscList :: [Int] -> IntSetSource

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

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

# Debugging

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

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.