Safe Haskell | Safe |
---|---|

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

This module provides the various sorting implementations for Data.Sequence. Further notes are available in the file sorting.md (in this directory).

## Synopsis

- sort :: Ord a => Seq a -> Seq a
- sortBy :: (a -> a -> Ordering) -> Seq a -> Seq a
- sortOn :: Ord b => (a -> b) -> Seq a -> Seq a
- unstableSort :: Ord a => Seq a -> Seq a
- unstableSortBy :: (a -> a -> Ordering) -> Seq a -> Seq a
- unstableSortOn :: Ord b => (a -> b) -> Seq a -> Seq a
- data Queue e = Q !e (QList e)
- data QList e
- data IndexedQueue e = IQ !Int !e (IQList e)
- data IQList e
- = IQNil
- | IQCons !(IndexedQueue e) (IQList e)

- data TaggedQueue a b = TQ !a b (TQList a b)
- data TQList a b
- = TQNil
- | TQCons !(TaggedQueue a b) (TQList a b)

- data IndexedTaggedQueue e a = ITQ !Int !e a (ITQList e a)
- data ITQList e a
- = ITQNil
- | ITQCons !(IndexedTaggedQueue e a) (ITQList e a)

- mergeQ :: (a -> a -> Ordering) -> Queue a -> Queue a -> Queue a
- mergeIQ :: (a -> a -> Ordering) -> IndexedQueue a -> IndexedQueue a -> IndexedQueue a
- mergeTQ :: (a -> a -> Ordering) -> TaggedQueue a b -> TaggedQueue a b -> TaggedQueue a b
- mergeITQ :: (a -> a -> Ordering) -> IndexedTaggedQueue a b -> IndexedTaggedQueue a b -> IndexedTaggedQueue a b
- popMinQ :: (e -> e -> Ordering) -> Queue e -> (Queue e, e)
- popMinIQ :: (e -> e -> Ordering) -> IndexedQueue e -> (IndexedQueue e, e)
- popMinTQ :: (a -> a -> Ordering) -> TaggedQueue a b -> (TaggedQueue a b, b)
- popMinITQ :: (e -> e -> Ordering) -> IndexedTaggedQueue e b -> (IndexedTaggedQueue e b, b)
- buildQ :: (b -> b -> Ordering) -> (a -> Queue b) -> FingerTree a -> Maybe (Queue b)
- buildIQ :: (b -> b -> Ordering) -> (Int -> Elem y -> IndexedQueue b) -> Int -> FingerTree (Elem y) -> Maybe (IndexedQueue b)
- buildTQ :: (b -> b -> Ordering) -> (a -> TaggedQueue b c) -> FingerTree a -> Maybe (TaggedQueue b c)
- buildITQ :: (b -> b -> Ordering) -> (Int -> Elem y -> IndexedTaggedQueue b c) -> Int -> FingerTree (Elem y) -> Maybe (IndexedTaggedQueue b c)
- foldToMaybeTree :: (b -> b -> b) -> (a -> b) -> FingerTree a -> Maybe b
- foldToMaybeWithIndexTree :: (b -> b -> b) -> (Int -> Elem y -> b) -> Int -> FingerTree (Elem y) -> Maybe b

# Sort Functions

sort :: Ord a => Seq a -> Seq a Source #

\( O(n \log n) \). `sort`

sorts the specified `Seq`

by the natural
ordering of its elements. The sort is stable. If stability is not
required, `unstableSort`

can be slightly faster.

sortBy :: (a -> a -> Ordering) -> Seq a -> Seq a Source #

\( O(n \log n) \). `sortBy`

sorts the specified `Seq`

according to the
specified comparator. The sort is stable. If stability is not required,
`unstableSortBy`

can be slightly faster.

sortOn :: Ord b => (a -> b) -> Seq a -> Seq a Source #

\( O(n \log n) \). `sortOn`

sorts the specified `Seq`

by comparing
the results of a key function applied to each element.

is
equivalent to `sortOn`

f

, but has the
performance advantage of only evaluating `sortBy`

(`compare`

``on`

` f)`f`

once for each element in the
input list. This is called the decorate-sort-undecorate paradigm, or
Schwartzian transform.

An example of using `sortOn`

might be to sort a `Seq`

of strings
according to their length:

sortOn length (fromList ["alligator", "monkey", "zebra"]) == fromList ["zebra", "monkey", "alligator"]

If, instead, `sortBy`

had been used, `length`

would be evaluated on
every comparison, giving \( O(n \log n) \) evaluations, rather than
\( O(n) \).

If `f`

is very cheap (for example a record selector, or `fst`

),

will be faster than
`sortBy`

(`compare`

``on`

` f)

.`sortOn`

f

unstableSort :: Ord a => Seq a -> Seq a Source #

\( O(n \log n) \). `unstableSort`

sorts the specified `Seq`

by
the natural ordering of its elements, but the sort is not stable.
This algorithm is frequently faster and uses less memory than `sort`

.

unstableSortBy :: (a -> a -> Ordering) -> Seq a -> Seq a Source #

\( O(n \log n) \). A generalization of `unstableSort`

, `unstableSortBy`

takes an arbitrary comparator and sorts the specified sequence.
The sort is not stable. This algorithm is frequently faster and
uses less memory than `sortBy`

.

unstableSortOn :: Ord b => (a -> b) -> Seq a -> Seq a Source #

\( O(n \log n) \). `unstableSortOn`

sorts the specified `Seq`

by
comparing the results of a key function applied to each element.

is equivalent to `unstableSortOn`

f

,
but has the performance advantage of only evaluating `unstableSortBy`

(`compare`

``on`

` f)`f`

once for each
element in the input list. This is called the
decorate-sort-undecorate paradigm, or Schwartzian transform.

An example of using `unstableSortOn`

might be to sort a `Seq`

of strings
according to their length:

unstableSortOn length (fromList ["alligator", "monkey", "zebra"]) == fromList ["zebra", "monkey", "alligator"]

If, instead, `unstableSortBy`

had been used, `length`

would be evaluated on
every comparison, giving \( O(n \log n) \) evaluations, rather than
\( O(n) \).

If `f`

is very cheap (for example a record selector, or `fst`

),

will be faster than
`unstableSortBy`

(`compare`

``on`

` f)

.`unstableSortOn`

f

# Heaps

The following are definitions for various specialized pairing heaps.

All of the heaps are defined to be non-empty, which speeds up the merge functions.

data IndexedQueue e Source #

A pairing heap tagged with the original position of elements, to allow for stable sorting.

data TaggedQueue a b Source #

A pairing heap tagged with some key for sorting elements, for use
in `unstableSortOn`

.

data IndexedTaggedQueue e a Source #

A pairing heap tagged with both a key and the original position
of its elements, for use in `sortOn`

.

# Merges

The following are definitions for "merge" for each of the heaps above. Each takes a comparison function which is used to order the elements.

mergeIQ :: (a -> a -> Ordering) -> IndexedQueue a -> IndexedQueue a -> IndexedQueue a Source #

`mergeIQ`

merges two `IndexedQueue`

s, taking into account the
original position of the elements.

mergeTQ :: (a -> a -> Ordering) -> TaggedQueue a b -> TaggedQueue a b -> TaggedQueue a b Source #

`mergeTQ`

merges two `TaggedQueue`

s, based on the tag value.

mergeITQ :: (a -> a -> Ordering) -> IndexedTaggedQueue a b -> IndexedTaggedQueue a b -> IndexedTaggedQueue a b Source #

`mergeITQ`

merges two `IndexedTaggedQueue`

s, based on the tag
value, taking into account the original position of the elements.

# popMin

The following are definitions for `popMin`

, a function which
constructs a stateful action which pops the smallest element from the
queue, where "smallest" is according to the supplied comparison
function.

All of the functions fail on an empty queue.

Each of these functions is structured something like this:

popMinQ cmp (Q x ts) = (mergeQs ts, x)

The reason the call to `mergeQs`

is lazy is that it will be bottom
for the last element in the queue, preventing us from evaluating the
fully sorted sequence.

popMinQ :: (e -> e -> Ordering) -> Queue e -> (Queue e, e) Source #

Pop the smallest element from the queue, using the supplied comparator.

popMinIQ :: (e -> e -> Ordering) -> IndexedQueue e -> (IndexedQueue e, e) Source #

Pop the smallest element from the queue, using the supplied
comparator, deferring to the item's original position when the
comparator returns `EQ`

.

popMinTQ :: (a -> a -> Ordering) -> TaggedQueue a b -> (TaggedQueue a b, b) Source #

Pop the smallest element from the queue, using the supplied comparator on the tag.

popMinITQ :: (e -> e -> Ordering) -> IndexedTaggedQueue e b -> (IndexedTaggedQueue e b, b) Source #

Pop the smallest element from the queue, using the supplied
comparator on the tag, deferring to the item's original position
when the comparator returns `EQ`

.

# Building

The following are definitions for functions to build queues, given a comparison function.

buildIQ :: (b -> b -> Ordering) -> (Int -> Elem y -> IndexedQueue b) -> Int -> FingerTree (Elem y) -> Maybe (IndexedQueue b) Source #

buildTQ :: (b -> b -> Ordering) -> (a -> TaggedQueue b c) -> FingerTree a -> Maybe (TaggedQueue b c) Source #

buildITQ :: (b -> b -> Ordering) -> (Int -> Elem y -> IndexedTaggedQueue b c) -> Int -> FingerTree (Elem y) -> Maybe (IndexedTaggedQueue b c) Source #

# Special folds

A big part of what makes the heaps fast is that they're non empty,
so the merge function can avoid an extra case match. To take
advantage of this, though, we need specialized versions of `foldMap`

and `foldMapWithIndex`

, which can alternate between
calling the faster semigroup-like merge when folding over non empty
structures (like `Node`

and `Digit`

), and the
`Option`

-like mappend, when folding over structures
which can be empty (like `FingerTree`

).

foldToMaybeTree :: (b -> b -> b) -> (a -> b) -> FingerTree a -> Maybe b Source #