Data.Sequence.NonEmpty

A general-purpose non-empty (by construction) finite sequence type.

Non-emptiness means that:

• Functions that take an NESeq can safely operate on it with the assumption that it has at least value.

• Functions that return an NESeq provide an assurance that the result has at least one value.

Data.Sequence.NonEmpty re-exports the API of Data.Sequence, faithfully reproducing asymptotics, typeclass constraints, and semantics. Functions that ensure that input and output maps are both non-empty (like <|) return NESeq, but functions that might potentially return an empty map (like tail) return a Seq instead.

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

1. The nonEmptySeq smart constructor will convert a Seq a into a Maybe (NESeq a), returning Nothing if the original Seq was empty.

2. You can use :<||, :||>, and insertSeqAt to insert a value into a Seq to create a guaranteed NESeq.

3. You can use the IsNonEmpty and IsEmpty patterns to "pattern match" on a Seq to reveal it as either containing a NESeq or an empty sequence.

4. withNonEmpty offers a continuation-based interface for deconstructing a Seq and treating it as if it were an NESeq.

You can convert an NESeq into a Seq with toSeq or IsNonEmpty, essentially "obscuring" the non-empty property from the type.

O(1). An abstract constructor for an NESeq that consists of a "head" a and a "tail" Seq a. Similar to :| for NonEmpty.

Can be used to match on the head and tail of an NESeq, and also used to construct an NESeq by consing an item to the beginnong of a Seq, ensuring that the result is non-empty.

O(1). An abstract constructor for an NESeq that consists of a "init" Seq a and a "last" a. Similar to :| for NonEmpty, but at the end of the list instead of at the beginning.

Can be used to match on the init and last of an NESeq, and also used to construct an NESeq by snocing an item to the end of a Seq, ensuring that the result is non-empty.

O(1) . A singleton sequence.

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

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

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

O(\log(\min(n_1,n_2))) . Concatenate a non-empty sequence with a potentially empty sequence (Seq), to produce a guaranteed non-empty sequence. Mnemonic: like ><, but a pipe for the guarunteed non-empty side.

O(\log(\min(n_1,n_2))) . Concatenate a non-empty sequence with a potentially empty sequence (Seq), to produce a guaranteed non-empty sequence. Mnemonic: like ><, but a pipe for the guarunteed non-empty side.

O(n) . Create a sequence from a finite list of elements. There is a function toNonEmpty in the opposite direction for all instances of the Foldable1 class, including NESeq.

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

O(1). Retrieve the left-most item in a non-empty sequence. Note that this function is total.

O(1). Delete the left-most item in a non-empty sequence. Returns a potentially empty sequence (Seq) in the case that the original NESeq contained only a single element. Note that this function is total.

O(1). Retrieve the right-most item in a non-empty sequence. Note that this function is total.

O(1). Delete the right-most item in a non-empty sequence. Returns a potentially empty sequence (Seq) in the case that the original NESeq contained only a single element. Note that this function is total.

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

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

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

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

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

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

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 non-empty prefixes of this sequence, shortest first. For example,

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

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.

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. Is only defined if c is a positive number.

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)

O(n \log n) . sort sorts the specified NESeq by the natural ordering of its elements. The sort is stable. If stability is not required, unstableSort can be slightly faster.

O(n \log n) . sortBy sorts the specified NESeq according to the specified comparator. The sort is stable. If stability is not required, unstableSortBy can be slightly faster.

O(n \log n) . sortOn sorts the specified NESeq by comparing the results of a key function applied to each element. sortOn f is equivalent to sortBy (compare `Data.Function.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 NESeq 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 `Data.Function.on` f) will be faster than sortOn f.

O(n \log n) . unstableSort sorts the specified NESeq by the natural ordering of its elements, but the sort is not stable. This algorithm is frequently faster and uses less memory than sort.

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.

O(n \log n) . unstableSortOn sorts the specified NESeq by comparing the results of a key function applied to each element. unstableSortOn f is equivalent to unstableSortBy (compare `Data.Function.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 NESeq 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 `Data.Function.on` f) will be faster than unstableSortOn f.

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.

Unlike index, this can be used to retrieve an element without forcing it.

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

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.

prop> 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 (!?).

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.

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

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, take i s yields the empty sequence. If the sequence contains fewer than 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, drop i s yields the whole sequence. If the sequence contains fewer than i elements, the empty sequence is returned.

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])

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

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]

O(\log(\min(i,n-i))) . Split a sequence at a given position.

• This ys means that the given position was longer than the length of the list, and ys is the entire original system.

• That zs means that the given position was zero or smaller, and so zs is the entire original sequence.

• These ys zs gives ys (the sequence of elements before the given position, take n xs) and zs (the sequence of elements after the given position, drop n xs).

O(n). A generalization of foldMap1, foldMapWithIndex takes a folding function that also depends on the element's index, and applies it to every element in the sequence.

foldlWithIndex is a version of foldl that also provides access to the index of each element.

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

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

Is a more restrictive version of traverseWithIndex1; traverseWithIndex1 should be used whenever possible.

O(n). traverseWithIndex1 is a version of traverse1 that also offers access to the index of each element.

O(n) . The reverse of a sequence.

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]