package sek

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.

Library API

Ephemeral and Persistent Sequences

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

Conversion Functions

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

Emulation Layers

Miscellaneous

Settings

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.

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.

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.

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.

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)

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.

Conversions

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.