containers-0.5.0.0: Assorted concrete container types

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

Data.Set

Contents

- Strictness properties
- Set type
- Operators
- Query
- Construction
- Combine
- Filter
- Map
- Folds
  - Strict folds
  - Legacy folds
- Min/Max
- Conversion
  - List
  - Ordered list
- Debugging

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.

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.

Synopsis

- 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
- 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)
- map :: (Ord a, 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
- 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
- showTree :: Show a => Set a -> String
- showTreeWith :: Show a => Bool -> Bool -> Set a -> String
- valid :: Ord a => Set a -> Bool

# Strictness properties

This module satisfies the following strictness property:

- Key arguments are evaluated to WHNF

Here are some examples that illustrate the property:

     delete undefined s  ==  undefined

# Set type

data Set a

A set of values `a`.

Instances

Typeable1 Set |
Foldable Set |
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 => Monoid (Set a) |
NFData a => NFData (Set a) |

# Operators

(\\) :: Ord a => Set a -> Set a -> Set a

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

# Query

null :: Set a -> Bool

O(1). Is this the empty set?

size :: Set a -> Int

O(1). The number of elements in the set.

member :: Ord a => a -> Set a -> Bool

O(log n). Is the element in the set?

notMember :: Ord a => a -> Set a -> Bool

O(log n). Is the element not in the set?

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 `isSubsetOf` s2)` tells whether `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

empty :: Set a

O(1). The empty set.

singleton :: a -> Set a

O(1). Create a singleton set.

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.

delete :: Ord a => a -> Set a -> Set a

O(log n). Delete an element from a set.

# Combine

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

O(n+m). The union of two sets, preferring the first set when equal elements are encountered. The implementation uses the efficient hedge-union algorithm. Hedge-union is more efficient on (bigset `union` smallset).

unions :: Ord a => [Set a] -> Set a

The union of a list of sets: (``unions` == `foldl` `union` `empty``).

difference :: Ord a => Set a -> Set a -> Set a

O(n+m). Difference of two sets. The implementation uses an efficient hedge algorithm comparable with hedge-union.

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

O(n+m). 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

filter :: (a -> Bool) -> Set a -> Set a

O(n). Filter all elements that satisfy the predicate.

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 (``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`.

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.

# Map

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

O(n*log n). ``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`

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

O(n). The

``mapMonotonic` f s == `map` f s`, but works only when `f` is monotonic. 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

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

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 -> b -> a) -> a -> Set b -> a

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' :: (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

findMin :: Set a -> a

O(log n). The minimal element of a set.

findMax :: Set a -> a

O(log n). The maximal element of a set.

deleteMin :: Set a -> Set a

O(log n). Delete the minimal element.

deleteMax :: Set a -> Set a

O(log n). Delete the maximal element.

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

elems :: Set a -> [a]

O(n). An alias of `toAscList`. The elements of a set in ascending order. Subject to list fusion.

toList :: Set a -> [a]

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

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

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

## Ordered list

toAscList :: Set a -> [a]

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.

# 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

valid :: Ord a => Set a -> Bool

O(n). Test if the internal set structure is valid.