package sek

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This library offers efficient implementations of ephemeral sequences and persistent sequences, together with efficient conversions between these two data structures.

Type Abbreviations

The following type abbreviations help give readable types to some operations on sequences.

type index = int

An index into a sequence is an integer. It is comprised between 0 (included) and the length of the sequence (excluded or included, depending on the circumstances).

type length = int

The length of a sequence is a nonnegative integer.

type 'a segment = 'a array * index * length

An array segment is described by a triple of an array a, a start index i, and a length k.

type 'a segments = ('a segment -> unit) -> unit

A sequence of array segments is represented by a higher-order iterator.

type ('a1, 'a2) segments2 = ('a1 segment -> 'a2 segment -> unit) -> unit

A sequence of pairs of array segments of matching lengths is represented by a higher-order iterator.

type capacity = int

The capacity of a chunk is a nonnegative integer.

type depth = int

The depth of a chunk is a nonnegative integer.

type comparison = int

The result of a comparison is encoded as an integer value. A negative value means "less than"; zero means "equal"; a positive value means "greater than".

Library API

module type ITER = sig ... end

The signature ITER is the core iterator API. It is common to ephemeral and persistent sequences. Please follow the link for details.

module type ITER_EPHEMERAL = sig ... end

The signature ITER_EPHEMERAL extends the iterator API with concepts and operations that exist only in the case of iterators over ephemeral sequences. Please follow the link for details.

module type SEK = sig ... end

The signature SEK is the public API of the library. If you are a new user, you do not need to follow this link: the library's API appears below anyway. Just read on!

Ephemeral and Persistent Sequences

The submodules Ephemeral and Persistent offer implementations of ephemeral (mutable) and persistent (immutable) sequences.

type side
val front : side
val back : side

A side appears as a parameter to several operations, such as push and pop, which can operate at either end of a sequence.

val other : side -> side

other front is back. other back is front.

type direction
val forward : direction
val backward : direction

A direction appears as a parameter to several operations, such as iter, which can traverse the sequence either forward (from front to back) or backward (from back to front).

val opposite : direction -> direction

opposite forward is backward. opposite backward is forward.

val sign : direction -> int

sign forward is +1. sign backward is -1.

exception Empty

The exception Empty is raised by pop and peek operations when applied to an empty sequence.

exception End

The exception End is raised by the iterator operations get, get_segment, get_and_move, and get_segment_and_jump when the iterator has moved past the end of the sequence.

module Ephemeral : sig ... end

The submodule Ephemeral, also available under the name E, offers an implementation of ephemeral (mutable) sequences. Please follow the link for details.

module Persistent : sig ... end

The submodule Persistent, also available under the name P, offers an implementation of persistent (immutable) sequences. Please follow the link for details.

module E = Ephemeral

E is a short name for the submodule Ephemeral.

module P = Persistent

P is a short name for the submodule Persistent.

Conversion Functions

The following functions offer fast conversions between ephemeral and persistent sequences.

val snapshot : 'a Ephemeral.t -> 'a Persistent.t

snapshot s constructs and returns a persistent sequence whose elements are the elements of s. It is less efficient than snapshot_and_clear, whose use should be preferred, when possible.

Time complexity: O(K). Note that this operation introduces sharing: therefore, it may increase the cost of subsequent operations.

val snapshot_and_clear : 'a Ephemeral.t -> 'a Persistent.t

snapshot_and_clear s constructs and returns a persistent sequence whose elements are the elements of an ephemeral sequence s. As a side effect, it clears s.

Time complexity: O(min(K,n)). In other words, the cost is always O(K) and, in the specific case of short sequences, it is only O(n).

In the particular case where the ephemeral sequence s has been constructed by applying a series of push operations, the cost of snapshot_and_clear may be charged to these push operations, allowing one to consider that snapshot_and_clear costs O(1).

val edit : 'a Persistent.t -> 'a Ephemeral.t

edit s constructs and returns a new ephemeral sequence whose elements are the elements of s.

Time complexity: O(K).

Emulation Layers
module Emulated : sig ... end

The submodule Emulated contains Sek-based replacements for several modules in OCaml's standard library: Array, List, Queue, Stack.

val released : unit -> unit

The function call released() does nothing if the library was compiled in release mode, and fails (with an assertion failure) if the library was compiled with assertions enabled.

module Segment : sig ... end

The module Segment offers facilities for working with array segments. An array segment is a triple of an array, a start index, and a length.


The following settings can be controlled when the functor Make is applied.

Chunk Capacity

A sequence is represented in memory by a complex data structure that involves chunks, that is, arrays of elements. The data structure is organized in several layers. In the outermost layer, at depth 0, chunks of elements are used. In the next layer, at depth 1, chunks of chunks of elements are used, and so on: at depth k+1, chunks whose elements are depth-k chunks are used.

The function capacity, whose type is given by the signature CAPACITY, determines the desired capacity of these chunks. This capacity may depend on the depth.

module type CAPACITY = sig ... end
Overwriting Empty Slots

The Boolean flag overwrite_empty_slots, described by the signature OVERWRITE_EMPTY_SLOTS, determines whether the content of a just-emptied slot in an ephemeral sequence should be overwritten with the default value that was supplied when the sequence was created.

Setting this flag to true is safe. This is its default value.

Setting this flag to false can save time but can also cause a memory leak, because the obsolete value stored in the slot remains reachable and cannot be collected.

module type OVERWRITE_EMPTY_SLOTS = sig ... end
Compact Persistent Sequence Threshold

A persistent sequence whose length is less than or equal to a certain threshold can be represented in a simple and compact way (for instance, using an immutable array).

The integer value threshold, described by the signature is THRESHOLD, determines this threshold.

module type THRESHOLD = sig ... end
Dynamic Checking of Iterator Validity

An iterator on an ephemeral sequence is invalidated by most of the operations that affect the sequence. An invalidated iterator can no longer be used.

The flag check_iterator_validity determines whether a use of an invalidated iterator should be detected at runtime. When this flag is true, appropriate runtime checks are enabled, so that an attempt to use an invalidated iterator is detected and gives rise to an Invalid_argument exception. This is the default value. When this flag is false, the runtime checks are disabled, so a violation is not detected, and may cause arbitrary behavior. Setting this flag to false allows a small gain in performance, but is more dangerous.

module type CHECK_ITERATOR_VALIDITY = sig ... end

The Functor Make

module Make (Settings : sig ... end) : SEK

The functor Make constructs an implementation of the signature SEK, and allows the user to choose the value of the settings described above. Be warned, however, that the number and the nature of the settings expected by this functor may change in the future. A relatively forward-compatible way of using this functor is to always pass it a structure that is built by including the structure DefaultSettings and overriding the definition of one or more settings.

module DefaultSettings : sig ... end

The module DefaultSettings provides a set of recommended settings for the functor Make.

The Functor SupplyDefault

Every operation that constructs a sequence out of something other than a sequence requires the caller to supply a default value, which serves no visible purpose, but is used internally to initialize empty slots and (optionally) to overwrite a slot that becomes empty.

Because the presence of this argument can be bothersome, we provide a way of removing it by supplying a default value once and for all. To do so, apply the functor SupplyDefault to Sek (or to the result of Make), like so:

module S =
  SupplyDefault(Sek)(struct type element = int let default = 0 end)
module SupplyDefault (S : sig ... end) (D : sig ... end) : sig ... end

Complexity Guide

This section offers a high-level overview of the representation of sequences. We aim to give programmers a mental model of the time complexity of each operation and of the space usage of sequences in the heap.

The outermost layer

The elements of a sequence are stored internally in chunks, that is, arrays of a fixed capacity K. This is why this data structure is known as a chunk sequence. A larger value of K speeds up certain operations and reduces memory usage, but slows down other operations. The default value of K is 128.

More precisely, a sequence is represented using a front chunk, which stores the first K elements of the sequence, a back chunk, which stores the last K elements of the sequence, and a middle sequence, a sequence of chunks, which stores the remaining elements of the sequence.

As long as no concatenation operations are performed, all of the chunks in the middle sequence are full, that is, they contain exactly K elements. If concatenation operations are used, then some chunks in the middle sequence may not be full, that is, may contain fewer than K elements. Still, these chunks cannot be too sparsely populated: it is guaranteed that two consecutive chunks in the middle sequence always contain more than K elements. Thus, it is never possible to fit the contents of two consecutive chunks into just one. This guarantees that these chunks are on average at least half full.

As long as no concatenation operations are performed, the worst-case space usage of a sequence of length n is (1+10/K) * n + O(K) words. If concatenation operations are used, this bound is doubled and becomes 2 * (1+10/K) * n + O(K) words. Yet, this bound is unlikely to be reached in practice: one would need to repeatedly meet worst-case scenarios during a series of successive concatenation operations.

The inner layers

As explained above, the middle sequence is a sequence of chunks. How can one represent it efficiently? The answer, naturally, is to use the same idea as in the outermost layer, and to group these chunks together in chunks of chunks. Therefore, the middle sequence itself begins with a front chunk (a chunk of chunks), ends with a back chunk (a chunk of chunks), and in the middle, contains a middle sequence (of chunks of chunks). The construction continues down in the same way: this is a "polymorphic recursion" pattern.

Whereas the chunks of elements used in the outermost layer have capacity K, the chunks of chunks, chunks of chunks of chunks, and so on, used in the inner layers have capacity K'. To optimize the cost of concatenation and splitting, it is desirable for K' to be relatively small. The default value of K' is 16.

The number of layers in the data structure is logarithmic, and in practice, is typically very small. Because chunks are at least half-full on average, a data structure of depth d stores at least (K/2) (K'/2)^d elements. This means that the depth does not exceed D = 1 + log_(K'/2) (n/(K/2)). The depth D is thus O(log n). For the default values K = 128 and K' = 16, the depth of a sequence of up to 1 billion elements is at most 8, and the depth of a sequence of up to 18 million billion elements is at most 16.

In the presentation of the complexity bounds, we treat K and n as parameters, which means that O(K) and O(n) are not O(1). However, because K' is small, we treat it as a constant: we view O(K') as O(1).

Many operations that traverse the data structure have cost O(D), which is also O(log n). In several places, to simplify the statement of the complexity bounds, we make the hypothesis D <= K. This means that our analysis is restricted to a domain where the parameters n and K are correlated, and n is not allowed to grow unreasonably large relative to K. The numbers presented above show that, with the default values of K and K' and for all practical values of n, we are indeed within this domain.

Short persistent sequences

Below a certain threshold T, persistent sequences are represented in a more compact form: a persistent sequence of length less than or equal to T is represented as an immutable array of length n. The default value of T is 64.

This representation is optimal in terms of space, but suffers from an important drawback in terms of time: it takes time O(T^2) to build a persistent sequence of length T through a series of persistent push operations. For this reason, this representation may change in the future.

In most situations, the programmer can work around this issue by first building an ephemeral sequence through a series of ephemeral push operations, then converting it to a persistent sequence, for a total cost of O(T+K).

In order to simplify our time complexity bounds, we assume that T is O(K). Besides, our space complexity bound for persistent sequences (given above) requires K to be O(T). Thus, we need T and K to be reasonably close to each other.

Constant factors

The implementation of ephemeral sequences is carefully designed so as to minimize the constant factors associated with the operations E.push and E.pop and with the operations that allow iteration, such as E.Iter.get and E.Iter.move. As far as the performance of these operations is concerned, ephemeral sequences aim to be competitive with vectors (that is, resizable arrays).

The implementation of persistent sequences is carefully designed so as to minimize the average cost of the operations P.push and P.pop and the constant factors involved in iterating over elements. As far as the performance of these operations is concerned, persistent sequences aim to be as close as possible to linked lists, although it is quite likely that linked lists will always be more efficient when dealing with short-lived, short sequences.


The ephemeral data structure and the persistent data structure have the same representation of their middle sequence, allowing the main two conversion operations, namely snapshot_and_clear and edit, to be extremely efficient: their time complexity is O(K), regardless of the number of elements n in the data structure.

The operation E.copy, which creates a copy of an ephemeral sequence, exploits the same mechanism when its mode parameter is `Share. In this mode, the chunks are shared between the original sequence and the copy. The time complexity is then O(K).

Naturally, this efficiency comes at a cost. When a chunk is shared between several ephemeral or persistent data structures, its content cannot be modified in arbitrary ways. If one were not careful, an operation on a sequence s could have unintended effects on other sequences that share some chunks with s.

Thus, internally, the data structure keeps track of which chunks are definitely uniquely owned and which chunks are possibly shared.

The chunks that participate in a persistent sequence are always regarded as shared. Chunks that participate in an ephemeral sequence s may be either uniquely owned by s or shared with other (ephemeral or persistent) sequences.

Operations on uniquely-owned chunks are performed in place, whereas operations on shared chunks may involve a copy-on-write operation.

Thus, a copy operation such as let s' = E.copy ~mode:`Share s has a latent cost. Indeed, after the operation, both the original sequence s and its copy s' lose the unique ownership of their chunks. This implies that subsequent operations on s and s' will be slower than they would have been if no copy had taken place or if ~mode:`Copy had been used.


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