nonempty-containers-0.3.5.0: Non-empty variants of containers data types, with full API

Copyright | (c) Justin Le 2018
License | BSD3
Maintainer | justin@jle.im
Stability | experimental
Portability | non-portable
Safe Haskell | None
Language | Haskell2010

Data.Set.NonEmpty

Contents

- Non-Empty Set Type
  - Conversions between empty and non-empty sets
- Construction
- Insertion
- Deletion
- Query
- Combine
- Filter
- Indexed
- Map
- Folds
  - Strict folds
- Min/Max
- Conversion
  - List
- Debugging

Description

# Non-Empty Finite Sets

The ``NESet` e` type represents a non-empty set of elements of type `e`. Most operations require that `e` be an instance of the `Ord` class. A `NESet` is strict in its elements.

See documentation for `NESet` for information on how to convert and manipulate such non-empty set.

This module essentially re-imports the API of Data.Set and its `Set` type, along with semantics and asymptotics. In most situations, asymptotics are different only by a constant factor. In some situations, asmyptotics are even better (constant-time instead of log-time). All typeclass constraints are identical to their Data.Set counterparts.

Because `NESet` is implemented using `Set`, all of the caveats of using `Set` apply (such as the limitation of the maximum size of sets).

All functions take non-empty sets as inputs. In situations where their results can be guarunteed to also be non-empty, they also return non-empty sets. In situations where their results could potentially be empty, `Set` is returned instead.

Some functions (`partition`, `spanAntitone`, `split`) have modified return types to account for possible configurations of non-emptiness.

This module is intended to be imported qualified, to avoid name clashes with Prelude and Data.Set functions:

    import qualified Data.Set.NonEmpty as NES

Synopsis

- data NESet a
- pattern IsNonEmpty :: NESet a -> Set a
- pattern IsEmpty :: Set a
- nonEmptySet :: Set a -> Maybe (NESet a)
- toSet :: NESet a -> Set a
- withNonEmpty :: r -> (NESet a -> r) -> Set a -> r
- insertSet :: Ord a => a -> Set a -> NESet a
- insertSetMin :: a -> Set a -> NESet a
- insertSetMax :: a -> Set a -> NESet a
- unsafeFromSet :: Set a -> NESet a
- singleton :: a -> NESet a
- fromList :: Ord a => NonEmpty a -> NESet a
- fromAscList :: Eq a => NonEmpty a -> NESet a
- fromDescList :: Eq a => NonEmpty a -> NESet a
- fromDistinctAscList :: NonEmpty a -> NESet a
- fromDistinctDescList :: NonEmpty a -> NESet a
- powerSet :: NESet a -> NESet (NESet a)
- insert :: Ord a => a -> NESet a -> NESet a
- delete :: Ord a => a -> NESet a -> Set a
- member :: Ord a => a -> NESet a -> Bool
- notMember :: Ord a => a -> NESet a -> Bool
- lookupLT :: Ord a => a -> NESet a -> Maybe a
- lookupGT :: Ord a => a -> NESet a -> Maybe a
- lookupLE :: Ord a => a -> NESet a -> Maybe a
- lookupGE :: Ord a => a -> NESet a -> Maybe a
- size :: NESet a -> Int
- isSubsetOf :: Ord a => NESet a -> NESet a -> Bool
- isProperSubsetOf :: Ord a => NESet a -> NESet a -> Bool
- disjoint :: Ord a => NESet a -> NESet a -> Bool
- union :: Ord a => NESet a -> NESet a -> NESet a
- unions :: (Foldable1 f, Ord a) => f (NESet a) -> NESet a
- difference :: Ord a => NESet a -> NESet a -> Set a
- (\\) :: Ord a => NESet a -> NESet a -> Set a
- intersection :: Ord a => NESet a -> NESet a -> Set a
- cartesianProduct :: NESet a -> NESet b -> NESet (a, b)
- disjointUnion :: NESet a -> NESet b -> NESet (Either a b)
- filter :: (a -> Bool) -> NESet a -> Set a
- takeWhileAntitone :: (a -> Bool) -> NESet a -> Set a
- dropWhileAntitone :: (a -> Bool) -> NESet a -> Set a
- spanAntitone :: (a -> Bool) -> NESet a -> These (NESet a) (NESet a)
- partition :: (a -> Bool) -> NESet a -> These (NESet a) (NESet a)
- split :: Ord a => a -> NESet a -> Maybe (These (NESet a) (NESet a))
- splitMember :: Ord a => a -> NESet a -> (Bool, Maybe (These (NESet a) (NESet a)))
- splitRoot :: NESet a -> NonEmpty (NESet a)
- lookupIndex :: Ord a => a -> NESet a -> Maybe Int
- findIndex :: Ord a => a -> NESet a -> Int
- elemAt :: Int -> NESet a -> a
- deleteAt :: Int -> NESet a -> Set a
- take :: Int -> NESet a -> Set a
- drop :: Int -> NESet a -> Set a
- splitAt :: Int -> NESet a -> These (NESet a) (NESet a)
- map :: Ord b => (a -> b) -> NESet a -> NESet b
- mapMonotonic :: (a -> b) -> NESet a -> NESet b
- foldr :: (a -> b -> b) -> b -> NESet a -> b
- foldl :: (a -> b -> a) -> a -> NESet b -> a
- foldr1 :: Foldable t => (a -> a -> a) -> t a -> a
- foldl1 :: Foldable t => (a -> a -> a) -> t a -> a
- foldr' :: (a -> b -> b) -> b -> NESet a -> b
- foldl' :: (a -> b -> a) -> a -> NESet b -> a
- foldr1' :: (a -> a -> a) -> NESet a -> a
- foldl1' :: (a -> a -> a) -> NESet a -> a
- findMin :: NESet a -> a
- findMax :: NESet a -> a
- deleteMin :: NESet a -> Set a
- deleteMax :: NESet a -> Set a
- deleteFindMin :: NESet a -> (a, Set a)
- deleteFindMax :: NESet a -> (a, Set a)
- elems :: NESet a -> NonEmpty a
- toList :: NESet a -> NonEmpty a
- toAscList :: NESet a -> NonEmpty a
- toDescList :: NESet a -> NonEmpty a
- valid :: Ord a => NESet a -> Bool

# Non-Empty Set Type

data NESet a Source #

A non-empty (by construction) set of values `a`. At least one value exists in an ``NESet` a` at all times.

Functions that take an `NESet` can safely operate on it with the assumption that it has at least one item.

Functions that return an `NESet` provide an assurance that the result has at least one item.

Data.Set.NonEmpty re-exports the API of Data.Set, faithfully reproducing asymptotics, typeclass constraints, and semantics. Functions that ensure that input and output sets are both non-empty (like `insert`) return `NESet`, but functions that might potentially return an empty map (like `delete`) return a `Set` instead.

You can directly construct an `NESet` with the API from Data.Set.NonEmpty; it's more or less the same as constructing a normal `Set`, except you don't have access to `empty`. There are also a few ways to construct an `NESet` from a `Set`:

1. The `nonEmptySet` smart constructor will convert a ``Set` a` into a ``Maybe` (`NESet` a)`, returning `Nothing` if the original `Set` was empty.
2. You can use the `insertSet` family of functions to insert a value into a `Set` to create a guaranteed `NESet`.
3. You can use the `IsNonEmpty` and `IsEmpty` patterns to "pattern match" on a `Set` to reveal it as either containing a `NESet` or an empty map.
4. `withNonEmpty` offers a continuation-based interface for deconstructing a `Set` and treating it as if it were an `NESet`.

You can convert an `NESet` into a `Set` with `toSet` or `IsNonEmpty`, essentially "obscuring" the non-empty property from the type.

#### Instances

Instances details

Foldable1 NESet Source # | Traverses elements in ascending order

Instance details

Defined in Data.Set.NonEmpty.Internal

Methods

fold1 :: Semigroup m => NESet m -> m Source #

foldMap1 :: Semigroup m => (a -> m) -> NESet a -> m Source #

foldMap1' :: Semigroup m => (a -> m) -> NESet a -> m Source #

toNonEmpty :: NESet a -> NonEmpty a Source #

maximum :: Ord a => NESet a -> a Source #

minimum :: Ord a => NESet a -> a Source #

head :: NESet a -> a Source #

last :: NESet a -> a Source #

foldrMap1 :: (a -> b) -> (a -> b -> b) -> NESet a -> b Source #

foldlMap1' :: (a -> b) -> (b -> a -> b) -> NESet a -> b Source #

foldlMap1 :: (a -> b) -> (b -> a -> b) -> NESet a -> b Source #

foldrMap1' :: (a -> b) -> (a -> b -> b) -> NESet a -> b Source #

Eq1 NESet Source # |
Instance details Defined in Data.Set.NonEmpty.Internal Methods liftEq :: (a -> b -> Bool) -> NESet a -> NESet b -> Bool Source #
Ord1 NESet Source # |
Instance details Defined in Data.Set.NonEmpty.Internal Methods liftCompare :: (a -> b -> Ordering) -> NESet a -> NESet b -> Ordering Source #
Show1 NESet Source # |
Instance details Defined in Data.Set.NonEmpty.Internal Methods liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> NESet a -> ShowS Source # liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [NESet a] -> ShowS Source #
Foldable NESet Source # | Traverses elements in ascending order `foldr1`, `foldl1`, `minimum`, `maximum` are all total.

Instance details

Defined in Data.Set.NonEmpty.Internal

Methods

fold :: Monoid m => NESet m -> m Source #

foldMap :: Monoid m => (a -> m) -> NESet a -> m Source #

foldMap' :: Monoid m => (a -> m) -> NESet a -> m Source #

foldr :: (a -> b -> b) -> b -> NESet a -> b Source #

foldr' :: (a -> b -> b) -> b -> NESet a -> b Source #

foldl :: (b -> a -> b) -> b -> NESet a -> b Source #

foldl' :: (b -> a -> b) -> b -> NESet a -> b Source #

foldr1 :: (a -> a -> a) -> NESet a -> a Source #

foldl1 :: (a -> a -> a) -> NESet a -> a Source #

toList :: NESet a -> [a] Source #

null :: NESet a -> Bool Source #

length :: NESet a -> Int Source #

elem :: Eq a => a -> NESet a -> Bool Source #

maximum :: Ord a => NESet a -> a Source #

minimum :: Ord a => NESet a -> a Source #

sum :: Num a => NESet a -> a Source #

product :: Num a => NESet a -> a Source #

(FromJSON a, Ord a) => FromJSON (NESet a) Source # |
Instance details Defined in Data.Set.NonEmpty.Internal Methods parseJSON :: Value -> Parser (NESet a) Source # parseJSONList :: Value -> Parser [NESet a] Source # omittedField :: Maybe (NESet a) Source #
ToJSON a => ToJSON (NESet a) Source # |
Instance details Defined in Data.Set.NonEmpty.Internal Methods toJSON :: NESet a -> Value Source # toEncoding :: NESet a -> Encoding Source # toJSONList :: [NESet a] -> Value Source # toEncodingList :: [NESet a] -> Encoding Source # omitField :: NESet a -> Bool Source #
NFData a => NFData (NESet a) Source # |
Instance details Defined in Data.Set.NonEmpty.Internal Methods rnf :: NESet a -> () Source #
Ord a => Semigroup (NESet a) Source # | Left-biased union
Instance details Defined in Data.Set.NonEmpty.Internal Methods (<>) :: NESet a -> NESet a -> NESet a Source # sconcat :: NonEmpty (NESet a) -> NESet a Source # stimes :: Integral b => b -> NESet a -> NESet a Source #
(Data a, Ord a) => Data (NESet a) Source # |

Instance details

Defined in Data.Set.NonEmpty.Internal

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> NESet a -> c (NESet a) Source #

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (NESet a) Source #

toConstr :: NESet a -> Constr Source #

dataTypeOf :: NESet a -> DataType Source #

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (NESet a)) Source #

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (NESet a)) Source #

gmapT :: (forall b. Data b => b -> b) -> NESet a -> NESet a Source #

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> NESet a -> r Source #

gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> NESet a -> r Source #

gmapQ :: (forall d. Data d => d -> u) -> NESet a -> [u] Source #

gmapQi :: Int -> (forall d. Data d => d -> u) -> NESet a -> u Source #

gmapM :: Monad m => (forall d. Data d => d -> m d) -> NESet a -> m (NESet a) Source #

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> NESet a -> m (NESet a) Source #

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> NESet a -> m (NESet a) Source #

(Read a, Ord a) => Read (NESet a) Source # |
Instance details Defined in Data.Set.NonEmpty.Internal Methods readsPrec :: Int -> ReadS (NESet a) Source # readList :: ReadS [NESet a] Source # readPrec :: ReadPrec (NESet a) Source # readListPrec :: ReadPrec [NESet a] Source #
Show a => Show (NESet a) Source # |
Instance details Defined in Data.Set.NonEmpty.Internal Methods showsPrec :: Int -> NESet a -> ShowS Source # show :: NESet a -> String Source # showList :: [NESet a] -> ShowS Source #
Eq a => Eq (NESet a) Source # |
Instance details Defined in Data.Set.NonEmpty.Internal Methods (==) :: NESet a -> NESet a -> Bool Source # (/=) :: NESet a -> NESet a -> Bool Source #
Ord a => Ord (NESet a) Source # |
Instance details Defined in Data.Set.NonEmpty.Internal Methods compare :: NESet a -> NESet a -> Ordering Source # (<) :: NESet a -> NESet a -> Bool Source # (<=) :: NESet a -> NESet a -> Bool Source # (>) :: NESet a -> NESet a -> Bool Source # (>=) :: NESet a -> NESet a -> Bool Source # max :: NESet a -> NESet a -> NESet a Source # min :: NESet a -> NESet a -> NESet a Source #

## Conversions between empty and non-empty sets

pattern IsNonEmpty :: NESet a -> Set a Source #

O(1) match, O(log n) usage of contents. The `IsNonEmpty` and `IsEmpty` patterns allow you to treat a `Set` as if it were either a ``IsNonEmpty` n` (where `n` is a `NESet`) or an `IsEmpty`.

For example, you can pattern match on a `Set`:

    myFunc :: Set X -> Y
    myFunc (IsNonEmpty n) =  -- here, the user provided a non-empty set, and n is the NESet
    myFunc IsEmpty        =  -- here, the user provided an empty set

Matching on ``IsNonEmpty` n` means that the original `Set` was not empty, and you have a verified-non-empty `NESet` `n` to use.

Note that patching on this pattern is O(1). However, using the contents requires a O(log n) cost that is deferred until after the pattern is matched on (and is not incurred at all if the contents are never used).

A case statement handling both `IsNonEmpty` and `IsEmpty` provides complete coverage.

This is a bidirectional pattern, so you can use `IsNonEmpty` to convert a `NESet` back into a `Set`, obscuring its non-emptiness (see `toSet`).

pattern IsEmpty :: Set a Source #

O(1). The `IsNonEmpty` and `IsEmpty` patterns allow you to treat a `Set` as if it were either a ``IsNonEmpty` n` (where `n` is a `NESet`) or an `IsEmpty`.

Matching on `IsEmpty` means that the original `Set` was empty.

A case statement handling both `IsNonEmpty` and `IsEmpty` provides complete coverage.

This is a bidirectional pattern, so you can use `IsEmpty` as an expression, and it will be interpreted as `empty`.

See `IsNonEmpty` for more information.

nonEmptySet :: Set a -> Maybe (NESet a) Source #

O(log n). Smart constructor for an `NESet` from a `Set`. Returns `Nothing` if the `Set` was originally actually empty, and ``Just` n` with an `NESet`, if the `Set` was not empty.

`nonEmptySet` and ``maybe` `empty` `toSet`` form an isomorphism: they are perfect structure-preserving inverses of eachother.

See `IsNonEmpty` for a pattern synonym that lets you "match on" the possiblity of a `Set` being an `NESet`.

    nonEmptySet (Data.Set.fromList [3,5]) == Just (fromList (3:|[5]))

toSet :: NESet a -> Set a Source #

O(log n). Convert a non-empty set back into a normal possibly-empty map, for usage with functions that expect `Set`.

Can be thought of as "obscuring" the non-emptiness of the set in its type. See the `IsNotEmpty` pattern.

`nonEmptySet` and ``maybe` `empty` `toSet`` form an isomorphism: they are perfect structure-preserving inverses of eachother.

    toSet (fromList ((3,"a") :| [(5,"b")])) == Data.Set.fromList [(3,"a"), (5,"b")]

withNonEmpty Source #

Arguments

:: r | value to return if set is empty
-> (NESet a -> r) | function to apply if set is not empty
-> Set a |
-> r |

O(log n). A general continuation-based way to consume a `Set` as if it were an `NESet`. ``withNonEmpty` def f` will take a `Set`. If set is empty, it will evaluate to `def`. Otherwise, a non-empty set `NESet` will be fed to the function `f` instead.

    nonEmptySet == withNonEmpty Nothing Just

insertSet :: Ord a => a -> Set a -> NESet a Source #

O(log n). Convert a `Set` into an `NESet` by adding a value. Because of this, we know that the set must have at least one element, and so therefore cannot be empty.

See `insertSetMin` for a version that is constant-time if the new value is strictly smaller than all values in the original set

    insertSet 4 (Data.Set.fromList [5, 3]) == fromList (3 :| [4, 5])
    insertSet 4 Data.Set.empty == singleton 4 "c"

insertSetMin :: a -> Set a -> NESet a Source #

O(1) Convert a `Set` into an `NESet` by adding a value where the value is strictly less than all values in the input set The values in the original map must all be strictly greater than the new value. The precondition is not checked.

    insertSetMin 2 (Data.Set.fromList [5, 3]) == fromList (2 :| [3, 5])
    valid (insertSetMin 2 (Data.Set.fromList [5, 3])) == True
    valid (insertSetMin 7 (Data.Set.fromList [5, 3])) == False
    valid (insertSetMin 3 (Data.Set.fromList [5, 3])) == False

insertSetMax :: a -> Set a -> NESet a Source #

O(log n) Convert a `Set` into an `NESet` by adding a value where the value is strictly less than all values in the input set The values in the original map must all be strictly greater than the new value. The precondition is not checked.

While this has the same asymptotics as `insertSet`, it saves a constant factor for key comparison (so may be helpful if comparison is expensive) and also does not require an `Ord` instance for the key type.

    insertSetMin 7 (Data.Set.fromList [5, 3]) == fromList (3 :| [5, 7])
    valid (insertSetMin 7 (Data.Set.fromList [5, 3])) == True
    valid (insertSetMin 2 (Data.Set.fromList [5, 3])) == False
    valid (insertSetMin 5 (Data.Set.fromList [5, 3])) == False

unsafeFromSet :: Set a -> NESet a Source #

O(log n). Unsafe version of `nonEmptySet`. Coerces a `Set` into an `NESet`, but is undefined (throws a runtime exception when evaluation is attempted) for an empty `Set`.

# Construction

singleton :: a -> NESet a Source #

O(1). Create a singleton set.

fromList :: Ord a => NonEmpty a -> NESet a Source #

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

fromAscList :: Eq a => NonEmpty a -> NESet a Source #

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

fromDescList :: Eq a => NonEmpty a -> NESet a Source #

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

fromDistinctAscList :: NonEmpty a -> NESet a Source #

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.

fromDistinctDescList :: NonEmpty a -> NESet a Source #

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.

powerSet :: NESet a -> NESet (NESet a) Source #

Calculate the power set of a non-empty: the set of all its (non-empty) subsets.

    t `member` powerSet s == t `isSubsetOf` s

Example:

    powerSet (fromList (1 :| [2,3])) =
      fromList (singleton 1 :| [ singleton 2
                               , singleton 3
                               , fromList (1 :| [2])
                               , fromList (1 :| [3])
                               , fromList (2 :| [3])
                               , fromList (1 :| [2,3])
                               ]
               )

We know that the result is non-empty because the result will always at least contain the original set.

# Insertion

insert :: Ord a => a -> NESet a -> NESet a Source #

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.

# Deletion

delete :: Ord a => a -> NESet a -> Set a Source #

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

# Query

member :: Ord a => a -> NESet a -> Bool Source #

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

notMember :: Ord a => a -> NESet a -> Bool Source #

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

lookupLT :: Ord a => a -> NESet a -> Maybe a Source #

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 -> NESet a -> Maybe a Source #

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

    lookupLT 4 (fromList (3 :| [5])) == Just 5
    lookupLT 5 (fromList (3 :| [5])) == Nothing

lookupLE :: Ord a => a -> NESet a -> Maybe a Source #

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

    lookupLT 2 (fromList (3 :| [5])) == Nothing
    lookupLT 4 (fromList (3 :| [5])) == Just 3
    lookupLT 5 (fromList (3 :| [5])) == Just 5

lookupGE :: Ord a => a -> NESet a -> Maybe a Source #

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

    lookupLT 3 (fromList (3 :| [5])) == Just 3
    lookupLT 4 (fromList (3 :| [5])) == Just 5
    lookupLT 6 (fromList (3 :| [5])) == Nothing

size :: NESet a -> Int Source #

O(1). The number of elements in the set. Guaranteed to be greater than zero.

isSubsetOf :: Ord a => NESet a -> NESet a -> Bool Source #

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

isProperSubsetOf :: Ord a => NESet a -> NESet a -> Bool Source #

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

disjoint :: Ord a => NESet a -> NESet a -> Bool Source #

O(n+m). Check whether two sets are disjoint (i.e. their intersection is empty).

    disjoint (fromList (2:|[4,6]))   (fromList (1:|[3]))     == True
    disjoint (fromList (2:|[4,6,8])) (fromList (2:|[3,5,7])) == False
    disjoint (fromList (1:|[2]))     (fromList (1:|[2,3,4])) == False

# Combine

union :: Ord a => NESet a -> NESet a -> NESet a Source #

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

unions :: (Foldable1 f, Ord a) => f (NESet a) -> NESet a Source #

The union of a non-empty list of sets

difference :: Ord a => NESet a -> NESet a -> Set a Source #

O(m*log(n/m + 1)), m <= n. Difference of two sets.

Returns a potentially empty set (`Set`) because the first set might be a subset of the second set, and therefore have all of its elements removed.

(\\) :: Ord a => NESet a -> NESet a -> Set a Source #

Same as `difference`.

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

O(m*log(n/m + 1)), m <= n. The intersection of two sets.

Returns a potentially empty set (`Set`), because the two sets might have an empty intersection.

Elements of the result come from the first set, so for example

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

prints `(fromList (A:|[]),fromList (B:|[]))`.

cartesianProduct :: NESet a -> NESet b -> NESet (a, b) Source #

Calculate the Cartesian product of two sets.

    cartesianProduct xs ys = fromList $ liftA2 (,) (toList xs) (toList ys)

Example:

    cartesianProduct (fromList (1:|[2])) (fromList ('a':|['b'])) =
      fromList ((1,'a') :| [(1,'b'), (2,'a'), (2,'b')])

disjointUnion :: NESet a -> NESet b -> NESet (Either a b) Source #

Calculate the disjoint union of two sets.

     disjointUnion xs ys = map Left xs `union` map Right ys

Example:

    disjointUnion (fromList (1:|[2])) (fromList ("hi":|["bye"])) =
      fromList (Left 1 :| [Left 2, Right "hi", Right "bye"])

# Filter

filter :: (a -> Bool) -> NESet a -> Set a Source #

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

Returns a potentially empty set (`Set`) because the predicate might filter out all items in the original non-empty set.

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

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`.

Returns a potentially empty set (`Set`) because the predicate might fail on the first input.

    takeWhileAntitone p = Data.Set.fromDistinctAscList . Data.List.NonEmpty.takeWhile p . toList
    takeWhileAntitone p = filter p

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

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`.

Returns a potentially empty set (`Set`) because the predicate might be true for all items.

    dropWhileAntitone p = Data.Set.fromDistinctAscList . Data.List.NonEmpty.dropWhile p . toList
    dropWhileAntitone p = filter (not . p)

spanAntitone :: (a -> Bool) -> NESet a -> These (NESet a) (NESet a) Source #

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`.

Returns a `These` with potentially two non-empty sets:

- ``This` n1` means that the predicate never failed for any item, returning the original set
- ``That` n2` means that the predicate failed for the first item, returning the original set
- ``These` n1 n2` gives `n1` (the set up to the point where the predicate stops holding) and `n2` (the set starting from the point where the predicate stops holding)

    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) -> NESet a -> These (NESet a) (NESet a) Source #

O(n). Partition the map according to a predicate.

Returns a `These` with potentially two non-empty sets:

- ``This` n1` means that the predicate was true for all items.
- ``That` n2` means that the predicate was false for all items.
- ``These` n1 n2` gives `n1` (all of the items that were true for the predicate) and `n2` (all of the items that were false for the predicate).

See also `split`.

    partition (> 3) (fromList (5 :| [3])) == These (singleton 5) (singleton 3)
    partition (< 7) (fromList (5 :| [3])) == This  (fromList (3 :| [5]))
    partition (> 7) (fromList (5 :| [3])) == That  (fromList (3 :| [5]))

split :: Ord a => a -> NESet a -> Maybe (These (NESet a) (NESet a)) Source #

O(log n). The expression (``split` x set`) is potentially a `These` containing up to two `NESet`s based on splitting the set into sets containing items before and after the value `x`. It will never return a set that contains `x` itself.

- `Nothing` means that `x` was the only value in the the original set, and so there are no items before or after it.
- ``Just` (`This` n1)` means `x` was larger than or equal to all items in the set, and `n1` is the entire original set (minus `x`, if it was present)
- ``Just` (`That` n2)` means `x` was smaller than or equal to all items in the set, and `n2` is the entire original set (minus `x`, if it was present)
- ``Just` (`These` n1 n2)` gives `n1` (the set of all values from the original set less than `x`) and `n2` (the set of all values from the original set greater than `x`).

    split 2 (fromList (5 :| [3])) == Just (That  (fromList (3 :| [5]))      )
    split 3 (fromList (5 :| [3])) == Just (That  (singleton 5)              )
    split 4 (fromList (5 :| [3])) == Just (These (singleton 3) (singleton 5))
    split 5 (fromList (5 :| [3])) == Just (This  (singleton 3)              )
    split 6 (fromList (5 :| [3])) == Just (This  (fromList (3 :| [5]))      )
    split 5 (singleton 5)         == Nothing

splitMember :: Ord a => a -> NESet a -> (Bool, Maybe (These (NESet a) (NESet a))) Source #

O(log n). The expression (``splitMember` x set`) splits a set just like `split` but also returns ``member` x set` (whether or not `x` was in `set`)

    splitMember 2 (fromList (5 :| [3])) == (False, Just (That  (fromList (3 :| [5)]))))
    splitMember 3 (fromList (5 :| [3])) == (True , Just (That  (singleton 5)))
    splitMember 4 (fromList (5 :| [3])) == (False, Just (These (singleton 3) (singleton 5)))
    splitMember 5 (fromList (5 :| [3])) == (True , Just (This  (singleton 3))
    splitMember 6 (fromList (5 :| [3])) == (False, Just (This  (fromList (3 :| [5])))
    splitMember 5 (singleton 5)         == (True , Nothing)

splitRoot :: NESet a -> NonEmpty (NESet a) Source #

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).

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

# Indexed

lookupIndex :: Ord a => a -> NESet a -> Maybe Int Source #

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 -> NESet a -> Int Source #

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

elemAt :: Int -> NESet a -> a Source #

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 -> NESet a -> Set a Source #

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.

Returns a potentially empty set (`Set`), because this could potentailly delete the final element in a singleton set.

    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 -> NESet a -> Set a Source #

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

Returns a potentailly empty set (`Set`), which can only happen when calling `take 0`.

    take n = Data.Set.fromDistinctAscList . Data.List.NonEmpty.take n . toAscList

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

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

Returns a potentailly empty set (`Set`), in the case that `drop` is called with a number equal to or greater the number of items in the set, and we drop every item.

    drop n = Data.Set.fromDistinctAscList . Data.List.NonEmpty.drop n . toAscList

splitAt :: Int -> NESet a -> These (NESet a) (NESet a) Source #

O(log n). Split a set at a particular index `i`.

- ``This` n1` means that there are less than `i` items in the set, and `n1` is the original set.
- ``That` n2` means `i` was 0; we dropped 0 items, so `n2` is the original set.
- ``These` n1 n2` gives `n1` (taking `i` items from the original set) and `n2` (dropping `i` items from the original set))

# Map

map :: Ord b => (a -> b) -> NESet a -> NESet b Source #

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) -> NESet a -> NESet b Source #

O(n). ``mapMonotonic` f s == `map` f s`, but works only when `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 = Data.Foldable.toList s

# Folds

foldr :: (a -> b -> b) -> b -> NESet a -> b Source #

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,

    elemsList set = foldr (:) [] set

foldl :: (a -> b -> a) -> a -> NESet b -> a Source #

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,

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

foldr1 :: Foldable t => (a -> a -> a) -> t a -> a Source #

A variant of `foldr` that has no base case, and thus may only be applied to non-empty structures.

This function is non-total and will raise a runtime exception if the structure happens to be empty.

#### Examples

Expand

Basic usage:

    >>> foldr1 (+) [1..4]
    10

    >>> foldr1 (+) []
    Exception: Prelude.foldr1: empty list

    >>> foldr1 (+) Nothing
    *** Exception: foldr1: empty structure

    >>> foldr1 (-) [1..4]
    -2

    >>> foldr1 (&&) [True, False, True, True]
    False

    >>> foldr1 (||) [False, False, True, True]
    True

    >>> foldr1 (+) [1..]
    * Hangs forever *

foldl1 :: Foldable t => (a -> a -> a) -> t a -> a Source #

A variant of `foldl` that has no base case, and thus may only be applied to non-empty structures.

This function is non-total and will raise a runtime exception if the structure happens to be empty.

    foldl1 f = foldl1 f . toList

#### Examples

Expand

Basic usage:

    >>> foldl1 (+) [1..4]
    10

    >>> foldl1 (+) []
    *** Exception: Prelude.foldl1: empty list

    >>> foldl1 (+) Nothing
    *** Exception: foldl1: empty structure

    >>> foldl1 (-) [1..4]
    -8

    >>> foldl1 (&&) [True, False, True, True]
    False

    >>> foldl1 (||) [False, False, True, True]
    True

    >>> foldl1 (+) [1..]
    * Hangs forever *

## Strict folds

foldr' :: (a -> b -> b) -> b -> NESet a -> b Source #

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 -> NESet b -> a Source #

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.

foldr1' :: (a -> a -> a) -> NESet a -> a Source #

O(n). A strict version of `foldr1`. Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value.

foldl1' :: (a -> a -> a) -> NESet a -> a Source #

O(n). A strict version of `foldl1`. Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value.

# Min/Max

findMin :: NESet a -> a Source #

O(1). The minimal element of a set. Note that this is total, making `lookupMin` obsolete. It is constant-time, so has better asymptotics than `Data.Set.lookupMin` and `Data.Map.findMin` as well.

    findMin (fromList (5 :| [3])) == 3

findMax :: NESet a -> a Source #

O(log n). The maximal key of a set Note that this is total, making `lookupMin` obsolete.

    findMax (fromList (5 :| [3])) == 5

deleteMin :: NESet a -> Set a Source #

O(1). Delete the minimal element. Returns a potentially empty set (`Set`), because we might delete the final item in a singleton set. It is constant-time, so has better asymptotics than `Data.Set.deleteMin`.

    deleteMin (fromList (5 :| [3, 7])) == Data.Set.fromList [5, 7]
    deleteMin (singleton 5) == Data.Set.empty

deleteMax :: NESet a -> Set a Source #

O(log n). Delete the maximal element. Returns a potentially empty set (`Set`), because we might delete the final item in a singleton set.

    deleteMax (fromList (5 :| [3, 7])) == Data.Set.fromList [3, 5]
    deleteMax (singleton 5) == Data.Set.empty

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

O(1). Delete and find the minimal element. It is constant-time, so has better asymptotics that `Data.Set.minView` for `Set`.

Note that unlike `Data.Set.deleteFindMin` for `Set`, this cannot ever fail, and so is a total function. However, the result `Set` is potentially empty, since the original set might have contained just a single item.

    deleteFindMin (fromList (5 :| [3, 10])) == (3, Data.Set.fromList [5, 10])

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

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

Note that unlike `Data.Set.deleteFindMax` for `Set`, this cannot ever fail, and so is a total function. However, the result `Set` is potentially empty, since the original set might have contained just a single item.

    deleteFindMax (fromList (5 :| [3, 10])) == (10, Data.Set.fromList [3, 5])

# Conversion

## List

elems :: NESet a -> NonEmpty a Source #

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

toList :: NESet a -> NonEmpty a Source #

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

toAscList :: NESet a -> NonEmpty a Source #

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

toDescList :: NESet a -> NonEmpty a Source #

O(n). Convert the set to a descending non-empty list of elements.

# Debugging

valid :: Ord a => NESet a -> Bool Source #

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