containers-0.4.2.1: 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 NFData 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.

# 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

# Folds

foldr :: (Int -> b -> b) -> b -> IntSet -> bSource

O(n). Fold the elements in the set using the given right-associative binary operator, such that foldr f z == foldr f z . toAscList.

For example,

toAscList set = foldr (:) [] set

foldl :: (a -> Int -> a) -> a -> IntSet -> aSource

O(n). Fold the elements in the set using the given left-associative binary operator, such that foldl f z == foldl f z . toAscList.

For example,

toDescList set = foldl (flip (:)) [] set

## Strict folds

foldr' :: (Int -> b -> b) -> b -> IntSet -> bSource

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 -> Int -> a) -> a -> IntSet -> aSource

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 :: (Int -> b -> b) -> b -> IntSet -> bSource

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

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.

# 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.