Copyright | (c) Ross Paterson 2005 (c) Louis Wasserman 2009 (c) Bertram Felgenhauer, David Feuer, Ross Paterson, and Milan Straka 2014 |
---|---|

License | BSD-style |

Maintainer | libraries@haskell.org |

Stability | experimental |

Portability | portable |

Safe Haskell | Safe-Inferred |

Language | Haskell98 |

General purpose finite sequences. Apart from being finite and having strict operations, sequences also differ from lists in supporting a wider variety of operations efficiently.

An amortized running time is given for each operation, with *n* referring
to the length of the sequence and *i* being the integral index used by
some operations. These bounds hold even in a persistent (shared) setting.

The implementation uses 2-3 finger trees annotated with sizes, as described in section 4.2 of

- Ralf Hinze and Ross Paterson,
"Finger trees: a simple general-purpose data structure",
*Journal of Functional Programming*16:2 (2006) pp 197-217. http://staff.city.ac.uk/~ross/papers/FingerTree.html

*Note*: Many of these operations have the same names as similar
operations on lists in the Prelude. The ambiguity may be resolved
using either qualification or the `hiding`

clause.

*Warning*: The size of a `Seq`

must not exceed `maxBound::Int`

. Violation
of this condition is not detected and if the size limit is exceeded, the
behaviour of the sequence is undefined. This is unlikely to occur in most
applications, but some care may be required when using `><`

, `<*>`

, `*>`

, or
`>>`

, particularly repeatedly and particularly in combination with
`replicate`

or `fromFunction`

.

- data Seq a
- empty :: Seq a
- singleton :: a -> Seq a
- (<|) :: a -> Seq a -> Seq a
- (|>) :: Seq a -> a -> Seq a
- (><) :: Seq a -> Seq a -> Seq a
- fromList :: [a] -> Seq a
- fromFunction :: Int -> (Int -> a) -> Seq a
- fromArray :: Ix i => Array i a -> Seq a
- replicate :: Int -> a -> Seq a
- replicateA :: Applicative f => Int -> f a -> f (Seq a)
- replicateM :: Monad m => Int -> m a -> m (Seq a)
- cycleTaking :: Int -> Seq a -> Seq a
- iterateN :: Int -> (a -> a) -> a -> Seq a
- unfoldr :: (b -> Maybe (a, b)) -> b -> Seq a
- unfoldl :: (b -> Maybe (b, a)) -> b -> Seq a
- null :: Seq a -> Bool
- length :: Seq a -> Int
- data ViewL a
- viewl :: Seq a -> ViewL a
- data ViewR a
- viewr :: Seq a -> ViewR a
- scanl :: (a -> b -> a) -> a -> Seq b -> Seq a
- scanl1 :: (a -> a -> a) -> Seq a -> Seq a
- scanr :: (a -> b -> b) -> b -> Seq a -> Seq b
- scanr1 :: (a -> a -> a) -> Seq a -> Seq a
- tails :: Seq a -> Seq (Seq a)
- inits :: Seq a -> Seq (Seq a)
- chunksOf :: Int -> Seq a -> Seq (Seq a)
- takeWhileL :: (a -> Bool) -> Seq a -> Seq a
- takeWhileR :: (a -> Bool) -> Seq a -> Seq a
- dropWhileL :: (a -> Bool) -> Seq a -> Seq a
- dropWhileR :: (a -> Bool) -> Seq a -> Seq a
- spanl :: (a -> Bool) -> Seq a -> (Seq a, Seq a)
- spanr :: (a -> Bool) -> Seq a -> (Seq a, Seq a)
- breakl :: (a -> Bool) -> Seq a -> (Seq a, Seq a)
- breakr :: (a -> Bool) -> Seq a -> (Seq a, Seq a)
- partition :: (a -> Bool) -> Seq a -> (Seq a, Seq a)
- filter :: (a -> Bool) -> Seq a -> Seq a
- sort :: Ord a => Seq a -> Seq a
- sortBy :: (a -> a -> Ordering) -> Seq a -> Seq a
- unstableSort :: Ord a => Seq a -> Seq a
- unstableSortBy :: (a -> a -> Ordering) -> Seq a -> Seq a
- lookup :: Int -> Seq a -> Maybe a
- (!?) :: Seq a -> Int -> Maybe a
- index :: Seq a -> Int -> a
- adjust :: (a -> a) -> Int -> Seq a -> Seq a
- adjust' :: forall a. (a -> a) -> Int -> Seq a -> Seq a
- update :: Int -> a -> Seq a -> Seq a
- take :: Int -> Seq a -> Seq a
- drop :: Int -> Seq a -> Seq a
- insertAt :: Int -> a -> Seq a -> Seq a
- deleteAt :: Int -> Seq a -> Seq a
- splitAt :: Int -> Seq a -> (Seq a, Seq a)
- elemIndexL :: Eq a => a -> Seq a -> Maybe Int
- elemIndicesL :: Eq a => a -> Seq a -> [Int]
- elemIndexR :: Eq a => a -> Seq a -> Maybe Int
- elemIndicesR :: Eq a => a -> Seq a -> [Int]
- findIndexL :: (a -> Bool) -> Seq a -> Maybe Int
- findIndicesL :: (a -> Bool) -> Seq a -> [Int]
- findIndexR :: (a -> Bool) -> Seq a -> Maybe Int
- findIndicesR :: (a -> Bool) -> Seq a -> [Int]
- foldMapWithIndex :: Monoid m => (Int -> a -> m) -> Seq a -> m
- foldlWithIndex :: (b -> Int -> a -> b) -> b -> Seq a -> b
- foldrWithIndex :: (Int -> a -> b -> b) -> b -> Seq a -> b
- mapWithIndex :: (Int -> a -> b) -> Seq a -> Seq b
- traverseWithIndex :: Applicative f => (Int -> a -> f b) -> Seq a -> f (Seq b)
- reverse :: Seq a -> Seq a
- intersperse :: a -> Seq a -> Seq a
- zip :: Seq a -> Seq b -> Seq (a, b)
- zipWith :: (a -> b -> c) -> Seq a -> Seq b -> Seq c
- zip3 :: Seq a -> Seq b -> Seq c -> Seq (a, b, c)
- zipWith3 :: (a -> b -> c -> d) -> Seq a -> Seq b -> Seq c -> Seq d
- zip4 :: Seq a -> Seq b -> Seq c -> Seq d -> Seq (a, b, c, d)
- zipWith4 :: (a -> b -> c -> d -> e) -> Seq a -> Seq b -> Seq c -> Seq d -> Seq e

# Documentation

data Seq a

General-purpose finite sequences.

Alternative Seq | |

Monad Seq | |

Functor Seq | |

MonadPlus Seq | |

Applicative Seq | |

Foldable Seq | |

Traversable Seq | |

IsList (Seq a) | |

Eq a => Eq (Seq a) | |

Data a => Data (Seq a) | |

Ord a => Ord (Seq a) | |

Read a => Read (Seq a) | |

Show a => Show (Seq a) | |

IsString (Seq Char) | |

Monoid (Seq a) | |

NFData a => NFData (Seq a) | |

Typeable (* -> *) Seq | |

type Item (Seq a) = a |

# Construction

(<|) :: a -> Seq a -> Seq a infixr 5

*O(1)*. Add an element to the left end of a sequence.
Mnemonic: a triangle with the single element at the pointy end.

(|>) :: Seq a -> a -> Seq a infixl 5

*O(1)*. Add an element to the right end of a sequence.
Mnemonic: a triangle with the single element at the pointy end.

fromFunction :: Int -> (Int -> a) -> Seq a

*O(n)*. Convert a given sequence length and a function representing that
sequence into a sequence.

fromArray :: Ix i => Array i a -> Seq a

*O(n)*. Create a sequence consisting of the elements of an `Array`

.
Note that the resulting sequence elements may be evaluated lazily (as on GHC),
so you must force the entire structure to be sure that the original array
can be garbage-collected.

## Repetition

replicateA :: Applicative f => Int -> f a -> f (Seq a)

`replicateA`

is an `Applicative`

version of `replicate`

, and makes
*O(log n)* calls to `<*>`

and `pure`

.

replicateA n x = sequenceA (replicate n x)

replicateM :: Monad m => Int -> m a -> m (Seq a)

`replicateM`

is a sequence counterpart of `replicateM`

.

replicateM n x = sequence (replicate n x)

cycleTaking :: Int -> Seq a -> Seq a

*O(log(k))*.

forms a sequence of length `cycleTaking`

k xs`k`

by
repeatedly concatenating `xs`

with itself. `xs`

may only be empty if
`k`

is 0.

cycleTaking k = fromList . take k . cycle . toList

## Iterative construction

iterateN :: Int -> (a -> a) -> a -> Seq a

*O(n)*. Constructs a sequence by repeated application of a function
to a seed value.

iterateN n f x = fromList (Prelude.take n (Prelude.iterate f x))

unfoldr :: (b -> Maybe (a, b)) -> b -> Seq a

Builds a sequence from a seed value. Takes time linear in the
number of generated elements. *WARNING:* If the number of generated
elements is infinite, this method will not terminate.

# Deconstruction

Additional functions for deconstructing sequences are available
via the `Foldable`

instance of `Seq`

.

## Queries

## Views

data ViewL a

View of the left end of a sequence.

data ViewR a

View of the right end of a sequence.

# Scans

# Sublists

*O(n)*. Returns a sequence of all suffixes of this sequence,
longest first. For example,

tails (fromList "abc") = fromList [fromList "abc", fromList "bc", fromList "c", fromList ""]

Evaluating the *i*th suffix takes *O(log(min(i, n-i)))*, but evaluating
every suffix in the sequence takes *O(n)* due to sharing.

*O(n)*. Returns a sequence of all prefixes of this sequence,
shortest first. For example,

inits (fromList "abc") = fromList [fromList "", fromList "a", fromList "ab", fromList "abc"]

Evaluating the *i*th prefix takes *O(log(min(i, n-i)))*, but evaluating
every prefix in the sequence takes *O(n)* due to sharing.

chunksOf :: Int -> Seq a -> Seq (Seq a)

*O(n)*. `chunksOf n xs`

splits `xs`

into chunks of size `n>0`

.
If `n`

does not divide the length of `xs`

evenly, then the last element
of the result will be short.

## Sequential searches

takeWhileL :: (a -> Bool) -> Seq a -> Seq a

*O(i)* where *i* is the prefix length. `takeWhileL`

, applied
to a predicate `p`

and a sequence `xs`

, returns the longest prefix
(possibly empty) of `xs`

of elements that satisfy `p`

.

takeWhileR :: (a -> Bool) -> Seq a -> Seq a

*O(i)* where *i* is the suffix length. `takeWhileR`

, applied
to a predicate `p`

and a sequence `xs`

, returns the longest suffix
(possibly empty) of `xs`

of elements that satisfy `p`

.

is equivalent to `takeWhileR`

p xs

.`reverse`

(`takeWhileL`

p (`reverse`

xs))

dropWhileL :: (a -> Bool) -> Seq a -> Seq a

*O(i)* where *i* is the prefix length.

returns
the suffix remaining after `dropWhileL`

p xs

.`takeWhileL`

p xs

dropWhileR :: (a -> Bool) -> Seq a -> Seq a

*O(i)* where *i* is the suffix length.

returns
the prefix remaining after `dropWhileR`

p xs

.`takeWhileR`

p xs

is equivalent to `dropWhileR`

p xs

.`reverse`

(`dropWhileL`

p (`reverse`

xs))

spanl :: (a -> Bool) -> Seq a -> (Seq a, Seq a)

*O(i)* where *i* is the prefix length. `spanl`

, applied to
a predicate `p`

and a sequence `xs`

, returns a pair whose first
element is the longest prefix (possibly empty) of `xs`

of elements that
satisfy `p`

and the second element is the remainder of the sequence.

spanr :: (a -> Bool) -> Seq a -> (Seq a, Seq a)

*O(i)* where *i* is the suffix length. `spanr`

, applied to a
predicate `p`

and a sequence `xs`

, returns a pair whose *first* element
is the longest *suffix* (possibly empty) of `xs`

of elements that
satisfy `p`

and the second element is the remainder of the sequence.

breakl :: (a -> Bool) -> Seq a -> (Seq a, Seq a)

*O(i)* where *i* is the breakpoint index. `breakl`

, applied to a
predicate `p`

and a sequence `xs`

, returns a pair whose first element
is the longest prefix (possibly empty) of `xs`

of elements that
*do not satisfy* `p`

and the second element is the remainder of
the sequence.

partition :: (a -> Bool) -> Seq a -> (Seq a, Seq a)

*O(n)*. The `partition`

function takes a predicate `p`

and a
sequence `xs`

and returns sequences of those elements which do and
do not satisfy the predicate.

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

*O(n)*. The `filter`

function takes a predicate `p`

and a sequence
`xs`

and returns a sequence of those elements which satisfy the
predicate.

# Sorting

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

*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 considerably
faster, and in particular uses less memory.

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

*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 considerably
faster, and in particular uses less memory.

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

*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`

,
and performs extremely well -- frequently twice as fast as `sort`

--
when the sequence is already nearly sorted.

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

*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`

, and performs extremely well --
frequently twice as fast as `sortBy`

-- when the sequence is already
nearly sorted.

# Indexing

lookup :: Int -> Seq a -> Maybe a

*O(log(min(i,n-i)))*. The element at the specified position,
counting from 0. If the specified position is negative or at
least the length of the sequence, `lookup`

returns `Nothing`

.

0 <= i < length xs ==> lookup i xs == Just (toList xs !! i)

i < 0 || i >= length xs ==> lookup i xs = Nothing

Unlike `index`

, this can be used to retrieve an element without
forcing it. For example, to insert the fifth element of a sequence
`xs`

into a `Map`

`m`

at key `k`

, you could use

```
case lookup 5 xs of
Nothing -> m
Just x ->
````insert`

k x m

@since 0.5.8

*O(log(min(i,n-i)))*. The element at the specified position,
counting from 0. The argument should thus be a non-negative
integer less than the size of the sequence.
If the position is out of range, `index`

fails with an error.

xs `index` i = toList xs !! i

Caution: `index`

necessarily delays retrieving the requested
element until the result is forced. It can therefore lead to a space
leak if the result is stored, unforced, in another structure. To retrieve
an element immediately without forcing it, use `lookup`

or '(!?)'.

adjust :: (a -> a) -> Int -> Seq a -> Seq a

*O(log(min(i,n-i)))*. Update the element at the specified position. If
the position is out of range, the original sequence is returned. `adjust`

can lead to poor performance and even memory leaks, because it does not
force the new value before installing it in the sequence. `adjust'`

should
usually be preferred.

adjust' :: forall a. (a -> a) -> Int -> Seq a -> Seq a

*O(log(min(i,n-i)))*. Update the element at the specified position.
If the position is out of range, the original sequence is returned.
The new value is forced before it is installed in the sequence.

adjust' f i xs = case xs !? i of Nothing -> xs Just x -> let !x' = f x in update i x' xs

@since 0.5.8

update :: Int -> a -> Seq a -> Seq a

*O(log(min(i,n-i)))*. Replace the element at the specified position.
If the position is out of range, the original sequence is returned.

*O(log(min(i,n-i)))*. The first `i`

elements of a sequence.
If `i`

is negative,

yields the empty sequence.
If the sequence contains fewer than `take`

i s`i`

elements, the whole sequence
is returned.

*O(log(min(i,n-i)))*. Elements of a sequence after the first `i`

.
If `i`

is negative,

yields the whole sequence.
If the sequence contains fewer than `drop`

i s`i`

elements, the empty sequence
is returned.

insertAt :: Int -> a -> Seq a -> Seq a

*O(log(min(i,n-i)))*.

inserts `insertAt`

i x xs`x`

into `xs`

at the index `i`

, shifting the rest of the sequence over.

insertAt 2 x (fromList [a,b,c,d]) = fromList [a,b,x,c,d] insertAt 4 x (fromList [a,b,c,d]) = insertAt 10 x (fromList [a,b,c,d]) = fromList [a,b,c,d,x]

insertAt i x xs = take i xs >< singleton x >< drop i xs

@since 0.5.8

deleteAt :: Int -> Seq a -> Seq a

*O(log(min(i,n-i)))*. Delete the element of a sequence at a given
index. Return the original sequence if the index is out of range.

deleteAt 2 [a,b,c,d] = [a,b,d] deleteAt 4 [a,b,c,d] = deleteAt (-1) [a,b,c,d] = [a,b,c,d]

@since 0.5.8

## Indexing with predicates

These functions perform sequential searches from the left or right ends of the sequence, returning indices of matching elements.

elemIndexL :: Eq a => a -> Seq a -> Maybe Int

`elemIndexL`

finds the leftmost index of the specified element,
if it is present, and otherwise `Nothing`

.

elemIndicesL :: Eq a => a -> Seq a -> [Int]

`elemIndicesL`

finds the indices of the specified element, from
left to right (i.e. in ascending order).

elemIndexR :: Eq a => a -> Seq a -> Maybe Int

`elemIndexR`

finds the rightmost index of the specified element,
if it is present, and otherwise `Nothing`

.

elemIndicesR :: Eq a => a -> Seq a -> [Int]

`elemIndicesR`

finds the indices of the specified element, from
right to left (i.e. in descending order).

findIndexL :: (a -> Bool) -> Seq a -> Maybe Int

finds the index of the leftmost element that
satisfies `findIndexL`

p xs`p`

, if any exist.

findIndicesL :: (a -> Bool) -> Seq a -> [Int]

finds all indices of elements that satisfy `findIndicesL`

p`p`

,
in ascending order.

findIndexR :: (a -> Bool) -> Seq a -> Maybe Int

finds the index of the rightmost element that
satisfies `findIndexR`

p xs`p`

, if any exist.

findIndicesR :: (a -> Bool) -> Seq a -> [Int]

finds all indices of elements that satisfy `findIndicesR`

p`p`

,
in descending order.

# Folds

General folds are available via the `Foldable`

instance of `Seq`

.

foldMapWithIndex :: Monoid m => (Int -> a -> m) -> Seq a -> m

*O(n)*. A generalization of `foldMap`

, `foldMapWithIndex`

takes a folding
function that also depends on the element's index, and applies it to every
element in the sequence.

@since 0.5.8

foldlWithIndex :: (b -> Int -> a -> b) -> b -> Seq a -> b

`foldlWithIndex`

is a version of `foldl`

that also provides access
to the index of each element.

foldrWithIndex :: (Int -> a -> b -> b) -> b -> Seq a -> b

`foldrWithIndex`

is a version of `foldr`

that also provides access
to the index of each element.

# Transformations

mapWithIndex :: (Int -> a -> b) -> Seq a -> Seq b

*O(n)*. A generalization of `fmap`

, `mapWithIndex`

takes a mapping
function that also depends on the element's index, and applies it to every
element in the sequence.

traverseWithIndex :: Applicative f => (Int -> a -> f b) -> Seq a -> f (Seq b)

`traverseWithIndex`

is a version of `traverse`

that also offers
access to the index of each element.

@since 0.5.8

intersperse :: a -> Seq a -> Seq a

Intersperse an element between the elements of a sequence.

intersperse a empty = empty intersperse a (singleton x) = singleton x intersperse a (fromList [x,y]) = fromList [x,a,y] intersperse a (fromList [x,y,z]) = fromList [x,a,y,a,z]

@since 0.5.8

## Zips

zip :: Seq a -> Seq b -> Seq (a, b)

*O(min(n1,n2))*. `zip`

takes two sequences and returns a sequence
of corresponding pairs. If one input is short, excess elements are
discarded from the right end of the longer sequence.