Data.Sequence

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://www.soi.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.

Documentation

data Seq a

General-purpose finite sequences.

Instances

Monad Seq

Functor Seq

Typeable1 Seq

MonadPlus Seq

Applicative Seq

Foldable Seq

Traversable Seq

Alternative Seq

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)

Monoid (Seq a)

NFData a => NFData (Seq a)

Construction

empty :: Seq a

O(1). The empty sequence.

singleton :: a -> Seq a

O(1). A singleton sequence.

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

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

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

O(log(min(n1,n2))). Concatenate two sequences.

fromList :: [a] -> Seq a

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.

Repetition

replicate :: Int -> a -> Seq a

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

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)

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.

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

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

Deconstruction

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

Queries

null :: Seq a -> Bool

O(1). Is this the empty sequence?

length :: Seq a -> Int

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

Views

data ViewL a

View of the left end of a sequence.

Constructors

EmptyL

empty sequence

a :< (Seq a)

leftmost element and the rest of the sequence

Instances

Functor ViewL

Typeable1 ViewL

Foldable ViewL

Traversable ViewL

Eq a => Eq (ViewL a)

Data a => Data (ViewL a)

Ord a => Ord (ViewL a)

Read a => Read (ViewL a)

Show a => Show (ViewL a)

viewl :: Seq a -> ViewL a

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

data ViewR a

View of the right end of a sequence.

Constructors

EmptyR

empty sequence

(Seq a) :> a

the sequence minus the rightmost element, and the rightmost element

Instances

Functor ViewR

Typeable1 ViewR

Foldable ViewR

Traversable ViewR

Eq a => Eq (ViewR a)

Data a => Data (ViewR a)

Ord a => Ord (ViewR a)

Read a => Read (ViewR a)

Show a => Show (ViewR a)

viewr :: Seq a -> ViewR a

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

Scans

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

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

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

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

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

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

Sublists

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

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 ith 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)

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

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.

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

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

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

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)

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.

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

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

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

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

index :: Seq a -> Int -> a

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.

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.

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.

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

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

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.

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

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

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

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

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

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

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

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

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

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

Folds

General folds are available via the Foldable instance of Seq.

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

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.

reverse :: Seq a -> Seq a

O(n). The reverse of a sequence.

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.

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

O(min(n1,n2)). 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)

O(min(n1,n2,n3)). 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

O(min(n1,n2,n3)). 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)

O(min(n1,n2,n3,n4)). 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

O(min(n1,n2,n3,n4)). 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.