Copyright | (c) Daan Leijen 2002 |
---|---|

License | BSD-style |

Maintainer | libraries@haskell.org |

Portability | portable |

Safe Haskell | Trustworthy |

Language | Haskell98 |

# WARNING

This module is considered **internal**.

The Package Versioning Policy **does not apply**.

This contents of this module may change **in any way whatsoever**
and **without any warning** between minor versions of this package.

Authors importing this module are expected to track development closely.

# Description

An efficient implementation of sets.

These modules are intended to be imported qualified, to avoid name clashes with Prelude functions, e.g.

import Data.Set (Set) import qualified Data.Set as Set

The implementation of `Set`

is based on *size balanced* binary trees (or
trees of *bounded balance*) as described by:

- Stephen Adams, "
*Efficient sets: a balancing act*", Journal of Functional Programming 3(4):553-562, October 1993, http://www.swiss.ai.mit.edu/~adams/BB/. - J. Nievergelt and E.M. Reingold,
"
*Binary search trees of bounded balance*", SIAM journal of computing 2(1), March 1973.

Bounds for `union`

, `intersection`

, and `difference`

are as given
by

- Guy Blelloch, Daniel Ferizovic, and Yihan Sun,
"
*Just Join for Parallel Ordered Sets*", https://arxiv.org/abs/1602.02120v3.

Note that the implementation is *left-biased* -- the elements of a
first argument are always preferred to the second, for example in
`union`

or `insert`

. Of course, left-biasing can only be observed
when equality is an equivalence relation instead of structural
equality.

*Warning*: The size of the set must not exceed `maxBound::Int`

. Violation of
this condition is not detected and if the size limit is exceeded, the
behavior of the set is completely undefined.

- data Set a
- (\\) :: Ord a => Set a -> Set a -> Set a
- null :: Set a -> Bool
- size :: Set a -> Int
- member :: Ord a => a -> Set a -> Bool
- notMember :: Ord a => a -> Set a -> Bool
- lookupLT :: Ord a => a -> Set a -> Maybe a
- lookupGT :: Ord a => a -> Set a -> Maybe a
- lookupLE :: Ord a => a -> Set a -> Maybe a
- lookupGE :: Ord a => a -> Set a -> Maybe a
- isSubsetOf :: Ord a => Set a -> Set a -> Bool
- isProperSubsetOf :: Ord a => Set a -> Set a -> Bool
- empty :: Set a
- singleton :: a -> Set a
- insert :: Ord a => a -> Set a -> Set a
- delete :: Ord a => a -> Set a -> Set a
- union :: Ord a => Set a -> Set a -> Set a
- unions :: Ord a => [Set a] -> Set a
- difference :: Ord a => Set a -> Set a -> Set a
- intersection :: Ord a => Set a -> Set a -> Set a
- filter :: (a -> Bool) -> Set a -> Set a
- takeWhileAntitone :: (a -> Bool) -> Set a -> Set a
- dropWhileAntitone :: (a -> Bool) -> Set a -> Set a
- spanAntitone :: (a -> Bool) -> Set a -> (Set a, Set a)
- partition :: (a -> Bool) -> Set a -> (Set a, Set a)
- split :: Ord a => a -> Set a -> (Set a, Set a)
- splitMember :: Ord a => a -> Set a -> (Set a, Bool, Set a)
- splitRoot :: Set a -> [Set a]
- lookupIndex :: Ord a => a -> Set a -> Maybe Int
- findIndex :: Ord a => a -> Set a -> Int
- elemAt :: Int -> Set a -> a
- deleteAt :: Int -> Set a -> Set a
- take :: Int -> Set a -> Set a
- drop :: Int -> Set a -> Set a
- splitAt :: Int -> Set a -> (Set a, Set a)
- map :: Ord b => (a -> b) -> Set a -> Set b
- mapMonotonic :: (a -> b) -> Set a -> Set b
- foldr :: (a -> b -> b) -> b -> Set a -> b
- foldl :: (a -> b -> a) -> a -> Set b -> a
- foldr' :: (a -> b -> b) -> b -> Set a -> b
- foldl' :: (a -> b -> a) -> a -> Set b -> a
- fold :: (a -> b -> b) -> b -> Set a -> b
- lookupMin :: Set a -> Maybe a
- lookupMax :: Set a -> Maybe a
- findMin :: Set a -> a
- findMax :: Set a -> a
- deleteMin :: Set a -> Set a
- deleteMax :: Set a -> Set a
- deleteFindMin :: Set a -> (a, Set a)
- deleteFindMax :: Set a -> (a, Set a)
- maxView :: Set a -> Maybe (a, Set a)
- minView :: Set a -> Maybe (a, Set a)
- elems :: Set a -> [a]
- toList :: Set a -> [a]
- fromList :: Ord a => [a] -> Set a
- toAscList :: Set a -> [a]
- toDescList :: Set a -> [a]
- fromAscList :: Eq a => [a] -> Set a
- fromDistinctAscList :: [a] -> Set a
- fromDescList :: Eq a => [a] -> Set a
- fromDistinctDescList :: [a] -> Set a
- showTree :: Show a => Set a -> String
- showTreeWith :: Show a => Bool -> Bool -> Set a -> String
- valid :: Ord a => Set a -> Bool
- bin :: a -> Set a -> Set a -> Set a
- balanced :: Set a -> Bool
- link :: a -> Set a -> Set a -> Set a
- merge :: Set a -> Set a -> Set a

# Set type

A set of values `a`

.

Foldable Set # | |

Eq1 Set # | |

Ord1 Set # | |

Show1 Set # | |

Ord a => IsList (Set a) # | |

Eq a => Eq (Set a) # | |

(Data a, Ord a) => Data (Set a) # | |

Ord a => Ord (Set a) # | |

(Read a, Ord a) => Read (Set a) # | |

Show a => Show (Set a) # | |

Ord a => Semigroup (Set a) # | |

Ord a => Monoid (Set a) # | |

NFData a => NFData (Set a) # | |

type Item (Set a) # | |

# Operators

# Query

lookupLT :: Ord a => a -> Set a -> Maybe a #

*O(log n)*. Find largest element smaller than the given one.

lookupLT 3 (fromList [3, 5]) == Nothing lookupLT 5 (fromList [3, 5]) == Just 3

lookupGT :: Ord a => a -> Set a -> Maybe a #

*O(log n)*. Find smallest element greater than the given one.

lookupGT 4 (fromList [3, 5]) == Just 5 lookupGT 5 (fromList [3, 5]) == Nothing

lookupLE :: Ord a => a -> Set a -> Maybe a #

*O(log n)*. Find largest element smaller or equal to the given one.

lookupLE 2 (fromList [3, 5]) == Nothing lookupLE 4 (fromList [3, 5]) == Just 3 lookupLE 5 (fromList [3, 5]) == Just 5

lookupGE :: Ord a => a -> Set a -> Maybe a #

*O(log n)*. Find smallest element greater or equal to the given one.

lookupGE 3 (fromList [3, 5]) == Just 3 lookupGE 4 (fromList [3, 5]) == Just 5 lookupGE 6 (fromList [3, 5]) == Nothing

isSubsetOf :: Ord a => Set a -> Set a -> Bool #

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

tells whether `isSubsetOf`

s2)`s1`

is a subset of `s2`

.

isProperSubsetOf :: Ord a => Set a -> Set a -> Bool #

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

# Construction

insert :: Ord a => a -> Set a -> Set a #

*O(log n)*. Insert an element in a set.
If the set already contains an element equal to the given value,
it is replaced with the new value.

# Combine

union :: Ord a => Set a -> Set a -> Set a #

*O(m*log(n/m + 1)), m <= n*. The union of two sets, preferring the first set when
equal elements are encountered.

intersection :: Ord a => Set a -> Set a -> Set a #

*O(m*log(n/m + 1)), m <= n*. The intersection of two sets.
Elements of the result come from the first set, so for example

import qualified Data.Set as S data AB = A | B deriving Show instance Ord AB where compare _ _ = EQ instance Eq AB where _ == _ = True main = print (S.singleton A `S.intersection` S.singleton B, S.singleton B `S.intersection` S.singleton A)

prints `(fromList [A],fromList [B])`

.

# Filter

takeWhileAntitone :: (a -> Bool) -> Set a -> Set a #

*O(log n)*. Take while a predicate on the elements holds.
The user is responsible for ensuring that for all elements `j`

and `k`

in the set,
`j < k ==> p j >= p k`

. See note at `spanAntitone`

.

takeWhileAntitone p =`fromDistinctAscList`

.`takeWhile`

p .`toList`

takeWhileAntitone p =`filter`

p

dropWhileAntitone :: (a -> Bool) -> Set a -> Set a #

*O(log n)*. Drop while a predicate on the elements holds.
The user is responsible for ensuring that for all elements `j`

and `k`

in the set,
`j < k ==> p j >= p k`

. See note at `spanAntitone`

.

dropWhileAntitone p =`fromDistinctAscList`

.`dropWhile`

p .`toList`

dropWhileAntitone p =`filter`

(not . p)

spanAntitone :: (a -> Bool) -> Set a -> (Set a, Set a) #

*O(log n)*. Divide a set at the point where a predicate on the elements stops holding.
The user is responsible for ensuring that for all elements `j`

and `k`

in the set,
`j < k ==> p j >= p k`

.

spanAntitone p xs = (`takeWhileAntitone`

p xs,`dropWhileAntitone`

p xs) spanAntitone p xs = partition p xs

Note: if `p`

is not actually antitone, then `spanAntitone`

will split the set
at some *unspecified* point where the predicate switches from holding to not
holding (where the predicate is seen to hold before the first element and to fail
after the last element).

partition :: (a -> Bool) -> Set a -> (Set a, Set a) #

*O(n)*. Partition the set into two sets, one with all elements that satisfy
the predicate and one with all elements that don't satisfy the predicate.
See also `split`

.

split :: Ord a => a -> Set a -> (Set a, Set a) #

*O(log n)*. 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`

.

splitMember :: Ord a => a -> Set a -> (Set a, Bool, Set a) #

*O(log n)*. Performs a `split`

but also returns whether the pivot
element was found in the original set.

splitRoot :: Set a -> [Set a] #

*O(1)*. Decompose a set into pieces based on the structure of the underlying
tree. This function is useful for consuming a set in parallel.

No guarantee is made as to the sizes of the pieces; an internal, but deterministic process determines this. However, it is guaranteed that the pieces returned will be in ascending order (all elements in the first subset less than all elements in the second, and so on).

Examples:

splitRoot (fromList [1..6]) == [fromList [1,2,3],fromList [4],fromList [5,6]]

splitRoot empty == []

Note that the current implementation does not return more than three subsets, but you should not depend on this behaviour because it can change in the future without notice.

# Indexed

lookupIndex :: Ord a => a -> Set a -> Maybe Int #

*O(log n)*. Lookup the *index* of an element, which is its zero-based index in
the sorted sequence of elements. The index is a number from *0* up to, but not
including, the `size`

of the set.

isJust (lookupIndex 2 (fromList [5,3])) == False fromJust (lookupIndex 3 (fromList [5,3])) == 0 fromJust (lookupIndex 5 (fromList [5,3])) == 1 isJust (lookupIndex 6 (fromList [5,3])) == False

findIndex :: Ord a => a -> Set a -> Int #

*O(log n)*. Return the *index* of an element, which is its zero-based
index in the sorted sequence of elements. The index is a number from *0* up
to, but not including, the `size`

of the set. Calls `error`

when the element
is not a `member`

of the set.

findIndex 2 (fromList [5,3]) Error: element is not in the set findIndex 3 (fromList [5,3]) == 0 findIndex 5 (fromList [5,3]) == 1 findIndex 6 (fromList [5,3]) Error: element is not in the set

*O(log n)*. Retrieve an element by its *index*, i.e. by its zero-based
index in the sorted sequence of elements. If the *index* is out of range (less
than zero, greater or equal to `size`

of the set), `error`

is called.

elemAt 0 (fromList [5,3]) == 3 elemAt 1 (fromList [5,3]) == 5 elemAt 2 (fromList [5,3]) Error: index out of range

deleteAt :: Int -> Set a -> Set a #

*O(log n)*. Delete the element at *index*, i.e. by its zero-based index in
the sorted sequence of elements. If the *index* is out of range (less than zero,
greater or equal to `size`

of the set), `error`

is called.

deleteAt 0 (fromList [5,3]) == singleton 5 deleteAt 1 (fromList [5,3]) == singleton 3 deleteAt 2 (fromList [5,3]) Error: index out of range deleteAt (-1) (fromList [5,3]) Error: index out of range

take :: Int -> Set a -> Set a #

Take a given number of elements in order, beginning with the smallest ones.

take n =`fromDistinctAscList`

.`take`

n .`toAscList`

drop :: Int -> Set a -> Set a #

Drop a given number of elements in order, beginning with the smallest ones.

drop n =`fromDistinctAscList`

.`drop`

n .`toAscList`

# Map

map :: Ord b => (a -> b) -> Set a -> Set b #

*O(n*log n)*.

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`

mapMonotonic :: (a -> b) -> Set a -> Set b #

*O(n)*. The

, but works only when `mapMonotonic`

f s == `map`

f s`f`

is strictly increasing.
*The precondition is not checked.*
Semi-formally, we have:

and [x < y ==> f x < f y | x <- ls, y <- ls] ==> mapMonotonic f s == map f s where ls = toList s

# Folds

## Strict folds

foldr' :: (a -> b -> b) -> b -> Set a -> b #

*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 -> b -> a) -> a -> Set b -> a #

*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 :: (a -> b -> b) -> b -> Set a -> b #

*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(log n)*. Delete the minimal element. Returns an empty set if the set is empty.

*O(log n)*. Delete the maximal element. Returns an empty set if the set is empty.

deleteFindMin :: Set a -> (a, Set a) #

*O(log n)*. Delete and find the minimal element.

deleteFindMin set = (findMin set, deleteMin set)

deleteFindMax :: Set a -> (a, Set a) #

*O(log n)*. Delete and find the maximal element.

deleteFindMax set = (findMax set, deleteMax set)

maxView :: Set a -> Maybe (a, Set a) #

*O(log n)*. Retrieves the maximal key of the set, and the set
stripped of that element, or `Nothing`

if passed an empty set.

minView :: Set a -> Maybe (a, Set a) #

*O(log n)*. Retrieves the minimal key of the set, and the set
stripped of that element, or `Nothing`

if passed an empty set.

# Conversion

## List

*O(n)*. An alias of `toAscList`

. The elements of a set in ascending order.
Subject to list fusion.

fromList :: Ord a => [a] -> Set a #

*O(n*log n)*. Create a set from a list of elements.

If the elements are ordered, a linear-time implementation is used,
with the performance equal to `fromDistinctAscList`

.

## Ordered list

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

toDescList :: Set a -> [a] #

*O(n)*. Convert the set to a descending list of elements. Subject to list
fusion.

fromAscList :: Eq a => [a] -> Set a #

*O(n)*. Build a set from an ascending list in linear time.
*The precondition (input list is ascending) is not checked.*

fromDistinctAscList :: [a] -> Set a #

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

fromDescList :: Eq a => [a] -> Set a #

*O(n)*. Build a set from a descending list in linear time.
*The precondition (input list is descending) is not checked.*

fromDistinctDescList :: [a] -> Set a #

*O(n)*. Build a set from a descending list of distinct elements in linear time.
*The precondition (input list is strictly descending) is not checked.*

# Debugging

showTree :: Show a => Set a -> String #

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

showTreeWith :: Show a => Bool -> Bool -> Set a -> 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.

Set> putStrLn $ showTreeWith True False $ fromDistinctAscList [1..5] 4 +--2 | +--1 | +--3 +--5 Set> putStrLn $ showTreeWith True True $ fromDistinctAscList [1..5] 4 | +--2 | | | +--1 | | | +--3 | +--5 Set> putStrLn $ showTreeWith False True $ fromDistinctAscList [1..5] +--5 | 4 | | +--3 | | +--2 | +--1