Library

Module

Module type

Parameter

Class

Class type

sectionYPositions = computeSectionYPositions($el), 10)" x-init="setTimeout(() => sectionYPositions = computeSectionYPositions($el), 10)">

On This Page

Legend:

Library

Module

Module type

Parameter

Class

Class type

Library

Module

Module type

Parameter

Class

Class type

The submodule `Persistent`

, also available under the name `P`

, offers an implementation of persistent (immutable) sequences. Please follow the link for details.

A sequence `s`

of type `'a t`

is an immutable data structure which represents a mathematical sequence of elements of type `'a`

.

In the documentation of the time complexity, we say that a sequence is *short* if its length is at most `T`

; it is *long* otherwise.

`val create : 'a -> 'a t`

`create default`

constructs and returns a new empty sequence. The default value `default`

is used to fill empty array slots.

Time complexity: *O(1)*.

`make default n v`

constructs and returns a fresh sequence whose length is `n`

and which consists of `n`

copies of the value `v`

. It is equivalent to `of_array default (Array.make n v)`

.

Time complexity: for short sequences, *O(n)*; for long sequences, *O(n/K + K)*.

`init default n f`

constructs and returns a fresh sequence whose length is `n`

and whose elements are the values produced by the calls `f 0`

, `f 1`

, ... `f (n-1)`

, in this order. It is equivalent to `of_array default (Array.init n f)`

.

Time complexity: *O(n)*, not counting the cost of the function `f`

.

`val default : 'a t -> 'a`

`default s`

returns the value that is used to fill empty array slots in the sequence `s`

.

Time complexity: *O(1)*.

`val is_empty : 'a t -> bool`

`is_empty s`

returns `true`

if the sequence `s`

is empty and `false`

otherwise. It is equivalent to `length s = 0`

.

Time complexity: *O(1)*.

`push side s x`

constructs and returns a new sequence obtained by pushing the element `x`

onto the front or back end of the sequence `s`

. The parameter `side`

determines which end of the sequence is acted upon.

Time complexity: for short sequences, *O(n)*; for long sequences, *O(K + log n)*. For long sequences, the total cost of *m* successive `push`

operations (performed in a single-threaded fashion) is *O(K + log n + m)*. This means that one can consider that the first `push`

operation costs *O(K + log n)* and that each of the successive calls has amortized cost *O(1)*.

If the sequence `s`

is nonempty, then `pop side s`

returns a pair of the element `x`

found at the front or back end of the sequence `s`

and of the sequence `s`

deprived of `x`

. The parameter `side`

determines which end of the sequence is acted upon. If the sequence `s`

is empty, the exception `Empty`

is raised.

Time complexity: for short sequences, *O(n)*; for long sequences, *O(log n)*. For long sequences, the total cost of *m* successive `pop`

operations is *O(log n + m)*. This means that one can consider that the first `pop`

operation costs *O(log n)* and that each of the successive calls has amortized cost *O(1)*.

If the sequence `s`

is nonempty, then `pop_opt side s`

returns a pair `(Some x, s')`

where `x`

is the element found at the front or back end of the sequence `s`

and `s'`

is the sequence `s`

deprived of `x`

. The parameter `side`

determines which end of the sequence is acted upon. If the sequence `s`

is empty, the pair `(None, s)`

is returned.

Time complexity: same as `pop`

.

If the sequence `s`

is nonempty, then `peek side s`

reads the element `x`

found at the front or back end of the sequence `s`

and returns `x`

. The parameter `side`

determines which end of the sequence is acted upon. If the sequence `s`

is empty, the exception `Empty`

is raised.

Time complexity: *O(1)*.

If the sequence `s`

is nonempty, then `peek_opt side s`

reads the element `x`

found at the front or back end of the sequence `s`

and returns `Some x`

. The parameter `side`

determines which end of the sequence is acted upon. If the sequence `s`

is empty, `None`

is returned.

Time complexity: *O(1)*.

`get s i`

returns the element `x`

located at index `i`

in the sequence `s`

. The index `i`

must lie in the semi-open interval `[0, length s)`

.

Time complexity: for short sequences, *O(1)*; for long sequences, *O(log n)*, or, more precisely, *O(log (min (i, n - i)))*.

`set s i x`

returns a new sequence obtained by replacing the element located at index `i`

in the sequence `s`

with the element `x`

. The index `i`

must lie in the semi-open interval `[0, length s)`

. The sequence `s`

is not affected.

Time complexity: for short sequences, *O(n)*; for long sequences, *O(K + log n)*, or, more precisely, *O(K + log (min (i, n - i)))*.

`concat s1 s2`

returns a new sequence obtained by concatenating the sequences `s1`

and `s2`

.

Time complexity: for short sequences, *O(n)*, where *n* is the length of the result of the concatenation. For long sequences, in pathological cases, `concat`

can cost as much as *O(K + log^2 n)*. In most cases, however, we expect `concat`

to cost *O(K + log n)*.

`split s i`

splits the sequence `s`

at index `i`

. It returns two sequences `s1`

and `s2`

such that the length of `s1`

is `i`

and the concatenation of `s1`

and `s2`

is `s`

. The index `i`

must lie in the closed interval `[0, length s]`

.

Time complexity: if `s1`

or `s2`

is short, *O(log n + min(|s1|, |s2|))*; otherwise *O(K + log^2 n)*, in the worst case, but in most cases, we expect `split`

to cost *O(K + log n)*, or, more precisely, *O(K + log (min (i, n - i)))*.

`take front s i`

splits the sequence `s`

at index `i`

and returns the first part. It is equivalent to `fst (split s i)`

. `take back s i`

also splits the sequence `s`

at index `i`

, and returns the second part. It is equivalent to `snd (split s i)`

. In either case, the index `i`

must lie in the closed interval `[0, length s]`

.

Time complexity: same as `split`

.

`drop side s i`

is equivalent to `take (other side) s i`

. The index `i`

must lie in the closed interval `[0, length s]`

.

Time complexity: same as `split`

.

`sub s head size`

extracts the sequence segment defined by the sequence `s`

, the start index `head`

, and the size `size`

.

Time complexity: if `size`

is at most *T*, then `sub`

has complexity *O(size + log n)*, or, more precisely *O(size + log (min (head, n - head)))*. Otherwise, `sub`

has complexity *O(log n)*, or, more precisely, *O(log size + log (min (head, n - head)))*.

`iter direction f s`

applies the function `f`

in turn to every element `x`

of the sequence `s`

. The parameter `direction`

determines in what order the elements are presented.

Time complexity: *O(n)*, not counting the cost of the function `f`

.

`iteri direction f s`

applies the function `f`

in turn to every index `i`

and matching element `x`

of the sequence `s`

. The parameter `direction`

determines in what order the elements are presented.

Time complexity: *O(n)*, not counting the cost of the function `f`

.

`iter_segments direction s f`

applies the function `f`

to a series of nonempty array segments whose concatenation represents the sequence `s`

. The function `f`

is allowed to *read* these array segments. **The function f is not allowed to write these array segments.** When iterating backward, each segment must be traversed in reverse order.

Time complexity: *O(n/K)*, not counting the cost of the function `f`

.

`val fold_left : ('a -> 'b -> 'a) -> 'a -> 'b t -> 'a`

`fold_left f a s`

applies the function `f`

in turn to each element of the sequence `s`

, in the forward direction. An accumulator is threaded through the calls to `f`

. `fold_left f a s`

is equivalent to `List.fold_left f a (to_list s)`

.

Time complexity: *O(n)*, not counting the cost of the function `f`

.

`val fold_right : ('a -> 'b -> 'b) -> 'a t -> 'b -> 'b`

`fold_right f a s`

applies the function `f`

in turn to each element of the sequence `s`

, in the backward direction. An accumulator is threaded through the calls to `f`

. `fold_right f s a`

is equivalent to `List.fold_right f (to_list s) a`

.

Time complexity: *O(n)*, not counting the cost of the function `f`

.

The submodule `Iter`

offers an implementation of iterators on persistent sequences.

`val to_list : 'a t -> 'a list`

`to_list s`

returns a list whose elements are the elements of the sequence `s`

.

Time complexity: *O(n)*.

`val to_array : 'a t -> 'a array`

`to_array s`

returns a fresh array whose elements are the elements of the sequence `s`

.

Time complexity: *O(n)*.

`to_seq direction s`

returns a fresh sequence whose elements are the elements of the sequence `s`

, enumerated according to `direction`

. The sequence `to_seq direction s`

is ephemeral: it can be consumed only once. This sequence occupies O(log n) space in memory: it is an iterator in disguise.

Time complexity: the creation of a sequence costs *O(1)*; then, demanding each element of the sequence has the same cost as a call to `Iter.get_and_move`

. If *k* elements of the resulting sequence are demanded by the user, then the total cost of producing these elements is *O(k)*.

`of_list_segment default n xs`

creates a new sequence out of the `n`

first elements of the list `xs`

. The list `xs`

must have at least `n`

elements.

Time complexity: *O(n)*. Remark: if *n > T* then the cost is *O(n + K)*, but this bound is equivalent to *O(n)* under our assumption that *K* is *O(T)*.

`val of_list : 'a -> 'a list -> 'a t`

`of_list default xs`

creates a new sequence out of the list `xs`

. If the length of the list `xs`

is known, then the use of `of_list_segment`

should be preferred.

Time complexity: *O(n)*.

`of_array_segment default a head size`

creates a new sequence out of the array segment defined by the array `a`

, the start index `head`

, and the size `size`

. The data is copied, so the array `a`

can still be used afterwards.

Time complexity: *O(n)*, where *n*, the length of the result sequence, is equal to `size`

.

`val of_array : 'a -> 'a array -> 'a t`

`of_array default a`

creates a new sequence out of the array `a`

. The data is copied, so the array `a`

can still be used afterwards. `of_array`

is *O(n)*.

`of_seq_segment default n xs`

creates a new sequence out of the `n`

first elements of the sequence `xs`

. The sequence `xs`

must have at least `n`

elements. It is consumed once.

Time complexity: *O(n)*, not counting the cost of demanding elements from the sequence `xs`

.

`val of_seq : 'a -> 'a Stdlib.Seq.t -> 'a t`

`of_seq d xs`

creates a new sequence out of the sequence `xs`

. The sequence `xs`

must be finite. It is consumed once. If the length of the sequence `xs`

is known, then the use of `of_seq_segment`

should be preferred.

Time complexity: *O(n)*, not counting the cost of demanding elements from the sequence `xs`

.

`find direction p s`

finds and returns the first element of the sequence `s`

, along the direction `direction`

, that satisfies the predicate `p`

. If no element of the sequence satisfies `p`

, the exception `Not_found`

is raised.

Time complexity: *O(i)*, where `i`

is the index of the first element that satisfies `p`

, or *n* if no element satisfies `p`

. This does not count the cost of the function `p`

.

`find_opt direction p s`

finds and returns the first element of the sequence `s`

, along the direction `direction`

, that satisfies the predicate `p`

. If no element of the sequence satisfies `p`

, `None`

is returned.

Time complexity: same as `find`

.

`find_map direction f s`

applies `f`

to each element of the sequence `s`

, along the direction `direction`

, and returns the first result other than `None`

. If there is no such result, it returns `None`

. If that `f`

is pure, it is equivalent to ```
find direction (fun o -> o <>
None) (map f s)
```

.

Time complexity: same as `find`

, not counting the cost of the function `f`

.

`val for_all : ('a -> bool) -> 'a t -> bool`

`for_all p s`

tests whether all elements of the sequence `s`

satisfy the predicate `p`

.

Time complexity: *O(i)*, where `i`

is the index of the first element that does not satisfy `p`

, or *n* if every element satisfies `p`

. This does not count the cost of the function `p`

.

`val exists : ('a -> bool) -> 'a t -> bool`

`exists p s`

tests whether some element of the sequence `s`

satisfies the predicate `p`

.

Time complexity: *O(i)*, where `i`

is the index of the first element that satisfies `p`

, or *n* if no element satisfies `p`

. This does not count the cost of the function `p`

.

`val mem : 'a -> 'a t -> bool`

`mem x s`

is equivalent to `exists (fun y -> x = y) s`

.

`val memq : 'a -> 'a t -> bool`

`memq x s`

is equivalent to `exists (fun y -> x == y) s`

.

`map default f s`

applies the function `f`

in turn to each element of the sequence `s`

, in the forward direction, and returns the sequence of the results.

Time complexity: *O(n)*.

`mapi default f s`

applies the function `f`

in turn to each index-and-element pair in the sequence `s`

, in the forward direction, and returns the sequence of the results.

Time complexity: *O(n)*.

`rev s`

returns a sequence whose elements are the elements of the sequence `s`

, in reverse order.

Time complexity: *O(n)*.

`zip s1 s2`

is the sequence of the pairs `(x1, x2)`

, where `x1`

and `x2`

are drawn *synchronously* from the sequences `s1`

and `s2`

. The lengths of the sequences `s1`

and `s2`

need not be equal: the length of the result is the minimum of the lengths of `s1`

and `s2`

.

Time complexity: *O(n)*, where *n* denotes the length of the result sequence.

`unzip s`

is equivalent to `(map _ fst s, map _ snd s)`

.

Time complexity: *O(n)*.

`filter p s`

returns the subsequence of the elements of `s`

that satisfy the predicate `p`

.

Time complexity: *O(n)*, not counting the cost of the function `p`

.

`filter_map default f s`

returns the subsequence of the defined images of the elements of `s`

through the partial function `f`

.

Time complexity: *O(n)*, not counting the cost of the function `f`

.

`partition p s`

returns a pair of the subsequence of the elements of `s`

that satisfy the predicate `p`

and those that do not satisfy `p`

.

Time complexity: *O(n)*, not counting the cost of the function `p`

.

`flatten ss`

is the iterated concatenation of the sequences in the sequence `ss`

.

Time complexity: same as a series of calls to `append`

.

`flatten_map d f s`

returns the concatenation of the images of the elements of `s`

through the function `f`

.

Time complexity: the current implementation is *O(n + K)*, where *n* denotes the length of the output sequence, not counting the cost of the function `f`

.

The following functions perform synchronous iteration on two sequences. Unlike the functions in OCaml's `List`

library, they do not require the two sequences to have the same length. If one of the sequences is strictly longer than the other, then its excess elements are ignored. If this behavior is deemed undesirable, then it is up to the user to check that the sequences have the same length. This can be done in constant time.

`iter2 direction f s1 s2`

repeatedly invokes `f x1 x2`

as long as a pair of elements `(x1, x2)`

can be drawn *synchronously* from the sequences `s1`

and `s2`

. The parameter `direction`

determines on what side iteration must begin and in which direction it must progress. The lengths of the sequences `s1`

and `s2`

need not be equal: iteration stops as soon as the shortest sequence is exhausted.

Time complexity: *O(min(n1,n2))*, where *n1* and *n2* denote the lengths of the arguments `s1`

and `s2`

, not counting the cost of the function `f`

.

`iter2_segments direction s1 s2 f`

repeatedly invokes `f seg1 seg2`

as long as a pair of nonempty array segments `seg1`

and `seg2`

of matching lengths can be drawn *synchronously* from the sequences `s1`

and `s2`

. The function `f`

is allowed to *read* these array segments. The parameter `direction`

determines on what side iteration must begin and in which direction it must progress. The lengths of the sequences `s1`

and `s2`

need not be equal: iteration stops as soon as the shortest sequence is exhausted.

Time complexity: *O(min(n1,n2)/K)*, where *n1* and *n2* denote the lengths of the arguments `s1`

and `s2`

, not counting the cost of the function `f`

.

`fold_left2`

is analogous to `iter2 forward`

, with the added feature that an accumulator is threaded through the calls to `f`

.

Time complexity: same as `iter2`

.

`fold_right2`

is analogous to `iter2 backward`

, with the added feature that an accumulator is threaded through the calls to `f`

.

Time complexity: same as `iter2`

.

`map2 d f s1 s2`

repeatedly invokes `f x1 x2`

as long as a pair of elements `(x1, x2)`

can be drawn *synchronously* from the sequences `s1`

and `s2`

, and returns the sequence of the results. Iteration is carried out in the forward direction. The lengths of the sequences `s1`

and `s2`

need not be equal: the length of the result is the minimum of the lengths of `s1`

and `s2`

.

Time complexity: *O(n)*, where *n* denotes the length of the result, not counting the cost of the function `f`

.

`for_all2 p s1 s2`

tests whether all pairs `(x1, x2)`

drawn synchronously from `s1`

and `s2`

satisfy the predicate `p`

. The sequences `s1`

and `s2`

need not have the same length: any excess elements are ignored.

Time complexity: *O(min(n1,n2))*, where *n1* and *n2* denote the lengths of the arguments `s1`

and `s2`

, not counting the cost of the function `p`

.

`exists2 p s`

tests whether some pair `(x1, x2)`

drawn synchronously from `s1`

and `s2`

satisfies the predicate `p`

. The sequences `s1`

and `s2`

need not have the same length: any excess elements are ignored.

Time complexity: *O(min(n1,n2))*, where *n1* and *n2* denote the lengths of the arguments `s1`

and `s2`

, not counting the cost of the function `p`

.

`equal p s1 s2`

tests whether the sequences `s1`

and `s2`

have the same length and all pairs `(x1, x2)`

drawn synchronously from `s1`

and `s2`

satisfy the predicate `p`

. If `p x1 x2`

compares the elements `x1`

and `x2`

for equality, then `equal p s1 s2`

compares the sequences `s1`

and `s2`

for equality.

Time complexity: *O(1)* if the sequences have distinct lengths; otherwise *O(i)*, where *i* is the index of the first pair that does not satisfy the predicate `p`

, or *n* if all pairs satisfy `p`

. This does not count the cost of the function `p`

.

`val compare : ('a -> 'b -> comparison) -> 'a t -> 'b t -> comparison`

If `cmp`

implements a preorder on elements, then `compare cmp`

implements the lexicographic preorder on sequences. (A preorder is an antisymmetric and transitive relation. For more details on comparison functions in OCaml, see the documentation of `Array.sort`

.)

Time complexity: same as `equal`

.

`val sort : ('a -> 'a -> comparison) -> 'a t -> 'a t`

`sort cmp s`

returns a copy of the sequence `s`

that is sorted according to the preorder `cmp`

. (For more details, see the documentation of `Array.sort`

.)

Time complexity: *O(n log n + K)*.

The current implementation converts the data to an array and back. A future release may provide a more efficient implementation.

`val stable_sort : ('a -> 'a -> comparison) -> 'a t -> 'a t`

`stable_sort cmp s`

returns a copy of the sequence `s`

that is sorted according to the preorder `cmp`

. (For more details, see the documentation of `Array.sort`

.) The sorting algorithm is stable: two elements that are equal according to `cmp`

are never permuted.

Time complexity: *O(n log n + K)*.

The current implementation converts the data to an array and back. A future release may provide a more efficient implementation.

`val uniq : ('a -> 'a -> comparison) -> 'a t -> 'a t`

`uniq cmp s`

filters the sequence `s`

by removing adjacent duplicate elements. That is, an element is dropped if it is equal (according to the preorder `cmp`

) to its left neighbor. If the sequence `s`

is sorted with respect to `cmp`

, then the sequence `uniq cmp s`

has no duplicate elements.

Time complexity: *O(n)*.

`val merge : ('a -> 'a -> comparison) -> 'a t -> 'a t -> 'a t`

`merge cmp s1 s2`

requires the sequences `s1`

and `s2`

to be sorted with respect to the preorder `cmp`

. It returns the sorted sequence `sort cmp (concat s1 s2)`

. `merge`

has complexity *O(n + K)*, where `n`

denotes the length of the result.

Time complexity: *O(n + K)*, where `n`

denotes the sum of the lengths of `s1`

and `s2`

, that is, the length of the result.

`val format : Stdlib.Format.formatter -> int t -> unit`

`format`

is a printer for sequences of integers. It can be installed in the OCaml toplevel loop by `#install_printer format`

. It is intended to be used only while debugging the library.

`val check : 'a t -> unit`

In a release build, `check s`

does nothing. In a development build, it checks that the data structure's internal invariant is satisfied.

On This Page