package sek

A sequence s of type 'a t is a mutable data structure which represents a mathematical sequence of elements of type 'a.

create default constructs and returns a new empty sequence. The default value default is used to fill empty array slots.

Time complexity: O(K).

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: O(n + 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 + K), not counting the cost of the function f.

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

Time complexity: O(1).

length s returns the length of the sequence s.

Time complexity: O(1).

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).

clear s empties the sequence s.

Time complexity: O(K), unless the global parameter overwrite_empty_slots is false, in which case the complexity is O(1).

copy ~mode s constructs and returns a copy s' of the sequence s. The sequences s and s' initially have the same elements, and can thereafter be modified independently of one another. Several copying modes are available, which have the same observable behavior, but offer distinct performance characteristics:

• copy ~mode:`Copy s creates a sequence that is physically disjoint from s. All of the elements are copied one by one. It takes linear time, which is slow, but on the upside, it has no latent cost. The sequence s is unaffected, and the sequence s' has unique ownership of its chunks.

• copy ~mode:`Share s creates a sequence whose chunks are physically shared with those of s. The copying of individual chunks is delayed until s or s' is actually modified. This operation has complexity O(K), which is fast, but on the downside, there is a latent cost: subsequent update operations on s and s' are more costly.

The default mode is `Copy. That is, copy s is a short-hand for copy ~mode:`Copy s.

Time complexity: O(n + K) in `Share mode; O(K) in `Copy mode.

If s1 and s2 are distinct sequences, then assign s1 s2 moves s2's elements into s1, overwriting s1's previous content, and clears s2. If s1 and s2 are the same sequence, then assign s1 s2 has no effect.

Time complexity: O(1) in the special case where s2 is never used afterwards; otherwise O(K).

push side s x pushes 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: O(log n) in the worst case. That said, in practice, most push operations execute in O(1).

If the sequence s is nonempty, then pop side s pops an element x off 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: same as push.

If the sequence s is nonempty, then pop_opt side s pops an element x off 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: 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: O(log n), or, more precisely, O(log (min (i, n - i))).

set s i x replaces 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 updated in place.

Time complexity: if the set operation hits a chunk that is marked as potentially shared with other sequences, then its complexity is O(K + log n), and this chunk is replaced in the process with one that is not shared with any other sequence. Otherwise, the complexity is O(log n), or, more precisely, O(log (min (i, n - i))).

concat s1 s2 creates and returns a new sequence whose content is the concatenation of the sequences s1 and s2. The sequences s1 and s2 are cleared. The sequences s1 and s2 must be distinct. concat is slightly less efficient than append, whose use should be preferred.

Time complexity: in pathological cases, concat can cost as much as O(K + log^2 n), where n is the length of the result of the concatenation. In most cases, however, we expect concat to cost O(K + log n). In the particular case of a concatenation that involves sequences whose chunks have never been shared, a more precise bound is O(K + log (min(n1, n2))), where n1 and n2 denote the lengths of the two sequences.

append back s1 s2 is equivalent to assign s1 (concat s1 s2). Thus, s1 is assigned the concatenation of the sequences s1 and s2, while s2 is cleared. In other words, append back s1 s2 appends the sequence s2 at the back end of the sequence s1.

append front s1 s2 is equivalent to assign s1 (concat s2 s1). Thus, s1 is assigned the concatenation of the sequences s2 and s1, while s2 is cleared. In other words, append front s1 s2 prepends the sequence s2 at the front end of the sequence s1.

In either case, the sequences s1 and s2 must be distinct.

Time complexity: same as concat.

split s i splits the sequence s at index i. It returns two new sequences s1 and s2 such that the length of s1 is i and the concatenation of s1 and s2 is s. The sequence s is cleared. The index i must lie in the closed interval [0, length s]. split is slightly less efficient than carve, whose use should be preferred.

Time complexity: in pathological cases, split can cost as much as O(K + log^2 n), where n is the length of the result of the concatenation. In most cases, however, we expect split to cost O(K + log n), or, more precisely, O(K + log (min (i, n - i))). The latter bound holds, in particular, when the operation involves a sequence whose chunks have never been shared.

carve back s i is equivalent to let s1, s2 = split s i in assign s s1; s2. Thus, it splits the sequence s at index i into two parts: the first part is written to s, while the second part is returned.

carve front s i is equivalent to let s1, s2 = split s i in assign s s2; s1. Thus, it splits the sequence s at index i into two parts: the second part is written to s, while the first part is returned.

In either case, the index i must lie in the closed interval [0, length s].

Time complexity: same as split.

take side s i is equivalent to (and faster than) ignore (carve side s i). In other words, take front s i truncates the sequence s at index i, and keeps only the front part; take back s i truncates the sequence s at index i, and keeps only the back part. 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. The sequence s is unaffected.

Time complexity: O(size + K).

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. The function f is not allowed to modify the sequence s while iteration is ongoing. If the flag check_iterator_validity is enabled (which it is, by default), then an illegal modification is detected at runtime.

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. The function f is not allowed to modify the sequence s while iteration is ongoing. If the flag check_iterator_validity is enabled (which it is, by default), then an illegal modification is detected at runtime.

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. When iterating backward, each segment must be traversed in reverse order. The function f is not allowed to write these array segments. The function f is not allowed to modify the sequence s while iteration is ongoing. If the flag check_iterator_validity is enabled (which it is, by default), then an illegal modification is detected at runtime.

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

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. The function f is not allowed to modify the sequence s while iteration is ongoing. Subject to this condition, fold_left f a s is equivalent to List.fold_left f a (to_list s). If the flag check_iterator_validity is enabled (which it is, by default), then an illegal modification is detected at runtime.

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

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. The function f is not allowed to modify the sequence s while iteration is ongoing. Subject to this condition, fold_right f s a is equivalent to List.fold_right f (to_list s) a. If the flag check_iterator_validity is enabled (which it is, by default), then an illegal modification is detected at runtime.

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

The submodule Iter offers an implementation of iterators on ephemeral sequences.

to_list s returns a list whose elements are the elements of the sequence s.

Time complexity: O(n).

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 + K).

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 + K).

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 + K), where n, the length of the result, is equal to size.

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.

Time complexity: O(n + K).

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 + K), not counting the cost of demanding elements from the sequence xs.

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 + K), 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 f is pure, then 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.

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.

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.

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

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 a sequence of the results. The sequence s is unaffected.

Time complexity: O(n + K), not counting the cost of the function f.

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 a sequence of the results. The sequence s is unaffected.

Time complexity: O(n + K), not counting the cost of the function f.

rev s returns a sequence whose elements are the elements of the sequence s, in reverse order. The sequence s is unaffected.

Time complexity: O(n + K).

zip s1 s2 is a 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 + K), where n denotes the length of the result sequence.

unzip s is equivalent to (map _ fst s, map _ snd s).

Time complexity: O(n + K).

filter p s returns a subsequence of the elements of s that satisfy the predicate p. The sequence s is unaffected.

Time complexity: O(n + K), 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 + K), 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. The sequence s is unaffected.

Time complexity: O(n + K), not counting the cost of the function p.

flatten ss returns the iterated concatenation of the sequences in the sequence ss. As a side effect, every sequence in the sequence ss is cleared, and the sequence ss itself is cleared.

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

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 + K), 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.

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.

sort cmp s sorts the sequence s 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.

stable_sort cmp s sorts the sequence s 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.

uniq cmp s returns a copy of the sequence s where adjacent duplicate elements have been filtered out. 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. The sequence s is unaffected.

Time complexity: O(n + K).

merge cmp s1 s2 requires the sequences s1 and s2 to be sorted with respect to the preorder cmp. It constructs a new sequence that is a sorted copy of the sequence concat s1 s2. The sequences s1 and s2 are unaffected.

Time complexity: O(n + K), where n denotes the sum of the lengths of s1 and s2, that is, the length of the result.

fill s i k x modifies the sequence s by overwriting the elements in the range [i, i+k) with the element x.

Time complexity: if the sequence involves chunks that may be shared with other sequences, the complexity is O(k + k/K * log n + K in the worst case. If none of the chunks that correspond to the range [i, i+k) were ever shared, then the cost is only O(k + log n).

blit s1 i1 s2 i2 k modifies the sequence s2 by overwriting the elements of s2 in the range [i2, i2+k) with the elements found in the sequence s1 in the range [i1, i1+k). It is permitted for s1 and s2 to be the same sequence; in that case, all reads appear to take place before any writes.

Time complexity: same as fill.

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.

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