containers-0.4.0.0: Assorted concrete container types

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

Data.Set

Contents

- Set type
- Operators
- Query
- Construction
- Combine
- Filter
- Map
- Fold
- Min/Max
- Conversion
  - List
  - Ordered list
- Debugging

Description

An efficient implementation of sets.

Since many function names (but not the type name) clash with Prelude names, this module is usually imported `qualified`, 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
- 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 :: Ord a => (a -> Bool) -> Set a -> Set a
- partition :: Ord a => (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
- 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]
- 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

# Set type

data Set a Source

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) |

# Operators

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

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

# Query

null :: Set a -> Bool Source

O(1). Is this the empty set?

size :: Set a -> Int Source

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

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

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

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

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

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

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 Source

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

# Construction

empty :: Set aSource

O(1). The empty set.

singleton :: a -> Set aSource

O(1). Create a singleton set.

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

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 aSource

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

# Combine

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

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 aSource

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

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

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 aSource

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 :: Ord a => (a -> Bool) -> Set a -> Set aSource

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

partition :: Ord a => (a -> Bool) -> Set a -> (Set a, Set a)Source

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)Source

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)Source

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 bSource

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 bSource

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

# Fold

fold :: (a -> b -> b) -> b -> Set a -> bSource

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

# Min/Max

findMin :: Set a -> aSource

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

findMax :: Set a -> aSource

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

deleteMin :: Set a -> Set aSource

O(log n). Delete the minimal element.

deleteMax :: Set a -> Set aSource

O(log n). Delete the maximal element.

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

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

     deleteFindMin set = (findMin set, deleteMin set)

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

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

     deleteFindMax set = (findMax set, deleteMax set)

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

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)Source

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]Source

O(n). The elements of a set.

toList :: Set a -> [a]Source

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

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

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

## Ordered list

toAscList :: Set a -> [a]Source

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

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

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

fromDistinctAscList :: [a] -> Set aSource

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 Source

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 Source

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 Source

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