Data.Sequence

data Seq a where Source

General-purpose finite sequences.

 mzipWith = zipWith

 munzip = unzip

Pattern synonyms

Much like lists can be constructed and matched using the : and [] constructors, sequences can be constructed and matched using the Empty, :<|, and :|> pattern synonyms.

Note

These patterns are only available with GHC version 8.0 or later, and version 8.2 works better with them. When writing for such recent versions of GHC, the patterns can be used in place of empty, <|, |>, viewl, and viewr.

Pattern synonym examples

import Data.Sequence (Seq (..))

getFirst3 :: Seq a -> Maybe (a,a,a)
getFirst3 (x1 :<| x2 :<| x3 :<| _xs) = Just (x1,x2,x3)
getFirst3 _ = Nothing

> getFirst3 (fromList [1,2,3,4]) = Just (1,2,3)
> getFirst3 (fromList [1,2]) = Nothing

shift2Right :: Seq a -> Seq a -> (Seq a, Seq a)
shift2Right Empty ys = (Empty, ys)
shift2Right (Empty :|> x) ys = (Empty, x :<| ys)
shift2Right (xs :|> x1 :|> x2) = (xs, x1 :<| x2 :<| ys)

> shift2Right (fromList []) (fromList [10]) = (fromList [], fromList [10])
> shift2Right (fromList [9]) (fromList [10]) = (fromList [], fromList [9,10])
> shift2Right (fromList [8,9]) (fromList [10]) = (fromList [], fromList [8,9,10])
> shift2Right (fromList [7,8,9]) (fromList [10]) = (fromList [7], fromList [8,9,10])

empty :: Seq a Source

\( O(1) \). The empty sequence.

singleton :: a -> Seq a Source

\( O(1) \). A singleton sequence.

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

\( O(1) \). 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 Source

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

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

\( O(\log(\min(n_1,n_2))) \). Concatenate two sequences.

fromList :: [a] -> Seq a Source

\( O(n) \). Create a sequence from a finite list of elements. There is a function toList in the opposite direction for all instances of the Foldable class, including Seq.

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

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

Since: containers-0.5.6.2

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

\( O(n) \). 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.

Since: containers-0.5.6.2

replicate :: Int -> a -> Seq a Source

\( O(\log n) \). replicate n x is a sequence consisting of n copies of x.

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

replicateA is an Applicative version of replicate, and makes \( O(\log n) \) calls to liftA2 and pure.

replicateA n x = sequenceA (replicate n x)

replicateM :: Applicative m => Int -> m a -> m (Seq a) Source

replicateM is a sequence counterpart of replicateM.

replicateM n x = sequence (replicate n x)

For base >= 4.8.0 and containers >= 0.5.11, replicateM is a synonym for replicateA.

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

O(log k). cycleTaking k xs forms a sequence of length k by repeatedly concatenating xs with itself. xs may only be empty if k is 0.

cycleTaking k = fromList . take k . cycle . toList

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

\( O(n) \). 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 Source

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.

unfoldl :: (b -> Maybe (b, a)) -> b -> Seq a Source

unfoldl f x is equivalent to reverse (unfoldr (fmap swap . f) x).

Additional functions for deconstructing sequences are available via the Foldable instance of Seq.

null :: Seq a -> Bool Source

\( O(1) \). Is this the empty sequence?

length :: Seq a -> Int Source

\( O(1) \). The number of elements in the sequence.

data ViewL a Source

View of the left end of a sequence.

viewl :: Seq a -> ViewL a Source

\( O(1) \). Analyse the left end of a sequence.

data ViewR a Source

View of the right end of a sequence.

viewr :: Seq a -> ViewR a Source

\( O(1) \). Analyse the right end of a sequence.

scanl :: (a -> b -> a) -> a -> Seq b -> Seq a Source

scanl is similar to foldl, but returns a sequence of reduced values from the left:

scanl f z (fromList [x1, x2, ...]) = fromList [z, z `f` x1, (z `f` x1) `f` x2, ...]

scanl1 :: (a -> a -> a) -> Seq a -> Seq a Source

scanl1 is a variant of scanl that has no starting value argument:

scanl1 f (fromList [x1, x2, ...]) = fromList [x1, x1 `f` x2, ...]

scanr :: (a -> b -> b) -> b -> Seq a -> Seq b Source

scanr is the right-to-left dual of scanl.

scanr1 :: (a -> a -> a) -> Seq a -> Seq a Source

scanr1 is a variant of scanr that has no starting value argument.

tails :: Seq a -> Seq (Seq a) Source

\( O(n) \). 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.

inits :: Seq a -> Seq (Seq a) Source

\( O(n) \). 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) Source

\(O \Bigl(\bigl(\frac{n}{c}\bigr) \log c\Bigr)\). chunksOf c xs splits xs into chunks of size c>0. If c does not divide the length of xs evenly, then the last element of the result will be short.

Side note: the given performance bound is missing some messy terms that only really affect edge cases. Performance degrades smoothly from \( O(1) \) (for \( c = n \)) to \( O(n) \) (for \( c = 1 \)). The true bound is more like \( O \Bigl( \bigl(\frac{n}{c} - 1\bigr) (\log (c + 1)) + 1 \Bigr) \)

Since: containers-0.5.8

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

\( O(i) \) 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 Source

\( O(i) \) 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.

takeWhileR p xs is equivalent to reverse (takeWhileL p (reverse xs)).

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

\( O(i) \) where \( i \) is the prefix length. dropWhileL p xs returns the suffix remaining after takeWhileL p xs.

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

\( O(i) \) where \( i \) is the suffix length. dropWhileR p xs returns the prefix remaining after takeWhileR p xs.

dropWhileR p xs is equivalent to reverse (dropWhileL p (reverse xs)).

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

\( O(i) \) 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) Source

\( O(i) \) 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) Source

\( O(i) \) 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.

breakl p is equivalent to spanl (not . p).

breakr :: (a -> Bool) -> Seq a -> (Seq a, Seq a) Source

breakr p is equivalent to spanr (not . p).

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

\( O(n) \). 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 Source

\( O(n) \). The filter function takes a predicate p and a sequence xs and returns a sequence of those elements which satisfy the predicate.

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.

Since: containers-0.3.0

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.

Since: containers-0.3.0

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. sortOn f is equivalent to sortBy (compare `on` f), but has the performance advantage of only evaluating 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), sortBy (compare `on` f) will be faster than sortOn f.

Since: containers-0.5.11

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.

Since: containers-0.3.0

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. unstableSortOn f is equivalent to unstableSortBy (compare `on` f), but has the performance advantage of only evaluating 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), unstableSortBy (compare `on` f) will be faster than unstableSortOn f.

Since: containers-0.5.11

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

\( O(\log(\min(i,n-i))) \). 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: containers-0.5.8

(!?) :: Seq a -> Int -> Maybe a Source

\( O(\log(\min(i,n-i))) \). A flipped, infix version of lookup.

Since: containers-0.5.8

index :: Seq a -> Int -> a Source

\( O(\log(\min(i,n-i))) \). 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 Source

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

Since: containers-0.5.8

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

\( O(\log(\min(i,n-i))) \). 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: containers-0.5.8

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

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

take :: Int -> Seq a -> Seq a Source

\( O(\log(\min(i,n-i))) \). The first i elements of a sequence. If i is negative, take i s yields the empty sequence. If the sequence contains fewer than i elements, the whole sequence is returned.

drop :: Int -> Seq a -> Seq a Source

\( O(\log(\min(i,n-i))) \). Elements of a sequence after the first i. If i is negative, drop i s yields the whole sequence. If the sequence contains fewer than i elements, the empty sequence is returned.

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

\( O(\log(\min(i,n-i))) \). insertAt i x xs inserts 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: containers-0.5.8

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

\( O(\log(\min(i,n-i))) \). 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: containers-0.5.8

splitAt :: Int -> Seq a -> (Seq a, Seq a) Source

\( O(\log(\min(i,n-i))) \). Split a sequence at a given position. splitAt i s = (take i s, drop i s).

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 Source

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

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

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 Source

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

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

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

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

findIndexL p xs finds the index of the leftmost element that satisfies p, if any exist.

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

findIndicesL p finds all indices of elements that satisfy p, in ascending order.

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

findIndexR p xs finds the index of the rightmost element that satisfies p, if any exist.

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

findIndicesR p finds all indices of elements that satisfy p, in descending order.

General folds are available via the Foldable instance of Seq.

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

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

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 Source

foldrWithIndex is a version of foldr that also provides access to the index of each element.

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

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) Source

traverseWithIndex is a version of traverse that also offers access to the index of each element.

Since: containers-0.5.8

reverse :: Seq a -> Seq a Source

\( O(n) \). The reverse of a sequence.

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

\( O(n) \). 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: containers-0.5.8

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

\( O(\min(n_1,n_2)) \). 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.

zipWith :: (a -> b -> c) -> Seq a -> Seq b -> Seq c Source

\( O(\min(n_1,n_2)) \). zipWith generalizes zip by zipping with the function given as the first argument, instead of a tupling function. For example, zipWith (+) is applied to two sequences to take the sequence of corresponding sums.

zip3 :: Seq a -> Seq b -> Seq c -> Seq (a, b, c) Source

\( O(\min(n_1,n_2,n_3)) \). zip3 takes three sequences and returns a sequence of triples, analogous to zip.

zipWith3 :: (a -> b -> c -> d) -> Seq a -> Seq b -> Seq c -> Seq d Source

\( O(\min(n_1,n_2,n_3)) \). zipWith3 takes a function which combines three elements, as well as three sequences and returns a sequence of their point-wise combinations, analogous to zipWith.

zip4 :: Seq a -> Seq b -> Seq c -> Seq d -> Seq (a, b, c, d) Source

\( O(\min(n_1,n_2,n_3,n_4)) \). zip4 takes four sequences and returns a sequence of quadruples, analogous to zip.

zipWith4 :: (a -> b -> c -> d -> e) -> Seq a -> Seq b -> Seq c -> Seq d -> Seq e Source

\( O(\min(n_1,n_2,n_3,n_4)) \). zipWith4 takes a function which combines four elements, as well as four sequences and returns a sequence of their point-wise combinations, analogous to zipWith.

unzip :: Seq (a, b) -> (Seq a, Seq b) Source

Unzip a sequence of pairs.

unzip ps = ps `seq` (fmap fst ps) (fmap snd ps)

Example:

unzip $ fromList [(1,"a"), (2,"b"), (3,"c")] =
  (fromList [1,2,3], fromList ["a", "b", "c"])

See the note about efficiency at unzipWith.

Since: containers-0.5.11

unzipWith :: (a -> (b, c)) -> Seq a -> (Seq b, Seq c) Source

\( O(n) \). Unzip a sequence using a function to divide elements.

 unzipWith f xs == unzip (fmap f xs)

Efficiency note:

unzipWith produces its two results in lockstep. If you calculate unzipWith f xs and fully force either of the results, then the entire structure of the other one will be built as well. This behavior allows the garbage collector to collect each calculated pair component as soon as it dies, without having to wait for its mate to die. If you do not need this behavior, you may be better off simply calculating the sequence of pairs and using fmap to extract each component sequence.

Since: containers-0.5.11