mdarray: An Owning Multidimensional Array Analog of mdspan

Document #: P1684R2
Date: 2022-03-21
Project: Programming Language C++
Library Evolution
Reply-to: Christian Trott
<>
Daisy Hollman
<>
Mark Hoemmen
<>
Daniel Sunderland
<>
Damien Lebrun-Grandie
<>

1 Revision History

1.1 P1684r2: To submit to 2022-04 Mailing

1.1.0.1 Changes from R1

1.2 P1684r1: 2022-03 Mailing

1.2.0.1 Changes from R0

2 Motivation

[P0009R16?] proposes mdspan, a nonowning multidimensional array abstraction. It is currently under LWG review and targets C++23. This proposal builds on mdspan by introducing mdarray, an owning multidimensional array that interoperates with mdspan. The mdarray class is to vector as mdspan is to span. Owning semantics can make it easier for users to express common cases, like returning an array from a function.

P0009 Only:

void make_random_rotation(mdspan<float, 3, 3> output);
void apply_rotation(mdspan<float, 3, 3>, mdspan<float, 3>);
void random_rotate(mdspan<float, dynamic_extent, 3> points) {
  float buffer[9] = { };
  auto rotation = mdspan<float, 3, 3>(buffer);
  make_random_rotation(rotation);
  for(int i = 0; i < points.extent(0); ++i) {
    apply_rotation(rotation, subspan(points, i, std::all));
  }
}

This Work:

mdarray<float, 3, 3> make_random_rotation();
void apply_rotation(mdarray<float, 3, 3>&, const mdspan<float, 3>&);
void random_rotate(mdspan<float, dynamic_extent, 3> points) {
  auto rotation = make_random_rotation();
  for(int i = 0; i < points.extent(0); ++i) {
    apply_rotation(rotation, subspan(points, i, std::all));
  }
}

In addition, especially for arrays with small, compile-time extents, mdarray’s owning semantics make fewer demands on interprocedural analysis for optimization. In particular, the lack of indirection due to mdarray owning its data makes it easier for compilers to deduce that the data could be stored in registers.

3 Design

One major goal of the design for mdarray is to parallel the design of mdspan as much as possible, with the goals of reducing cognitive load for users already familiar with mdspan and of incorporating the lessons learned from over a decade of experience with [P0009R16?] and libraries of similar design. This paper assumes the reader has read and is already familiar with [P0009R16?].

3.1 Design overview

The analogy to mdspan can be seen in the declaration of the proposed design for mdarray.

template<class ElementType,
         class Extents,
         class LayoutPolicy = layout_right,
         class Container = see-below>
  class mdarray;

This intentionally parallels the design of mdspan in [P0009R16?], which has the following signature.

template<class ElementType,
         class Extents,
         class LayoutPolicy = layout_right,
         class AccessorPolicy = default_accessor<ElementType>>
  class mdspan;

Our original mdarray proposal [P1684R0] had a ContainerPolicy instead of a Container template parameter. The ContainerPolicy provided functionality like that of mdspan’s AccessorPolicy, and would also select the actual container type used to store the mdarray’s data. For the current revision of this proposal, we have decided that the complexity of a ContainerPolicy is not actually required for mdarray. The vast majority of cases where customization of the AccessorPolicy is required to modify the access behavior, are local contextual requirements that are better served by mdspan. For example, one might have code that creates and uses an mdarray in several different ways. One loop over the array might have write conflicts that an atomic accessor would resolve. Changing how one views data in a way limited to a particular context is the job of a view. That is, in this case, it would make the most sense to create a temporary mdspan with an atomic accessor that views the original mdarray, for use in the particular loop that needs atomic access.

3.2 Differences between mdarray and mdspan

By design, mdarray is as similar as possible to mdspan, except with container semantics instead of reference semantics. However, the use of container semantics calls for a few differences.

3.2.1 Deep constness

The most notable difference from mdspan is deep constness. Like all reference semantic types in the standard, mdspan has shallow constness, but container types in the standard library propagate const through their access functions. Thus, mdarray needs const and non-const versions of every analogous operation in mdspan that interacts with the underlying data.

template<class ElementType, class Extents, class LayoutPolicy, class Container>
class mdarray {
  /* ... */

  // also in mdspan:
  using pointer = /* ... */;
  // only in mdarray:
  using const_pointer = /* ... */;

  // also in mdspan:
  using reference = /* ... */;
  // only in mdarray:
  using const_reference = /* ... */;

  // analogous to mdspan, except with const_reference return type:
  template<class... IndexType>
    constexpr const_reference operator[](IndexType...) const;
  template<class IndexType, size_t N>
    constexpr const_reference operator[](const array<IndexType, N>&) const;
  // non-const overloads only in mdarray:
  template<class... IndexType>
    constexpr reference operator[](IndexType...);
  template<class IndexType, size_t N>
    constexpr reference operator[](const array<IndexType, N>&);

  // also in mdspan, except with const_pointer return type:
  constexpr const_pointer data() const noexcept;
  // non-const overload only in mdarray:
  constexpr pointer data() noexcept;

  /* ... */
};

3.2.2 Interoperability between mdarray and mdspan

The mdarray class needs a means of interoperating with mdspan in roughly the same way as contiguous containers interact with span, or as string interacts with string_view. One way we could do this would be by adding a constructor to mdspan. This which would be more consistent with the analogous features in span and string_view. However, in the interest of avoiding modifications to an in-flight proposal, we instead propose using a member function of mdarray for this functionality for now. This member function is tentatively named view(), but we welcome suggestions for other names. We would be happy to change this based on design direction from LEWG.

template<class ElementType, class Extents, class LayoutPolicy, class Container>
class mdarray {
  /* ... */
  
  // only in mdarray:
  using view_type = /* ... */;
  using const_view_type = /* ... */;
  view_type view() noexcept;
  const_view_type view() const noexcept;

  /* ... */
};

3.2.3 Container instead of AccessorPolicy

As we discuss elsewhere, the mdspan class has an AccessorPolicy template parameter, while mdarray has a Container template parameter.

template<class ElementType, class Extents, class LayoutPolicy, class Container>
class mdarray {
  /* ... */
  
  // only in mdarray
 using container_type = Container;

  /* ... */
};

3.2.4 Constructors and assignment operators

The constructors and assignment operators of mdspan and mdarray have a few differences. The mdspan class provides a compatible mdspan copy-like constructor and copy-like assignment operator, with proper constraints and expectations to enforce compatibility of shape, layout, and size. Since mdarray has owning semantics, we also need move-like versions of these:

template<class ElementType, class Extents, class LayoutPolicy, class ContainerPolicy>
class mdarray {
  /* ... */
  
  // analogous to mdspan:
  template<class ET, class Exts, class LP, class CP>
    constexpr mdarray(const mdarray<ET, Exts, LP, CP>&);
  template<class ET, class Exts, class LP, class CP>
    constexpr mdarray& operator=(const mdarray<ET, Exts, LP, CP>&);
  // only in mdarray:
  template<class ET, class Exts, class LP, class CP>
    constexpr mdarray(mdarray<ET, Exts, LP, CP>&&) noexcept(see-below);
  template<class ET, class Exts, class LP, class CP>
    constexpr mdarray& operator=(mdarray<ET, Exts, LP, CP>&&) noexcept;

  /* ... */
};

(The noexcept clauses on these constructors and operators should probably actually derive from noexcept clauses on the analogous functionality for the element type and policy types.)

Additionally, the analog of the mdspan(pointer, IndexType...) constructor for mdarray does not need a pointer argument, since the mdarray owns the data and thus should be able to construct it from sizes:

template<class ElementType, class Extents, class LayoutPolicy, class ContainerPolicy>
class mdarray {
  /* ... */

  // only in mdarray
  template <class... IndexType>
  explicit constexpr mdarray(IndexType...);

  /* ... */
};

Note that in the completely static extents case, this is ambiguous with the default constructor. For consistency in generic code, the semantics of this constructor should be preferred over those of the default constructor in that case.

By this same logic, we arrive at the mapping_type constructor analogs:

template<class ElementType, class Extents, class LayoutPolicy, class Container>
class mdarray {
  /* ... */

  // only in mdarray
  explicit constexpr mdarray(const mapping_type&);

  /* ... */
};

There is some question as to whether we should also have constructors that take container_type instances in addition to indices. Consistency with standard container adapters like std::priority_queue would dictate that we should. Note that these constructors would deep-copy the input container.

template<class ElementType, class Extents, class LayoutPolicy, class Container>
class mdarray {
  /* ... */

  // only in mdarray
  explicit constexpr mdarray(const mapping_type&);
  constexpr mdarray(const container_type&, const mapping_type&);

  /* ... */
};

Finally, mdarray does not have the analog of mdspan’s constructor that takes an array<IndexType, N> of dynamic extents. This avoids any ambiguity or confusion with a constructor that takes a container instance, in the case where the container happens to be an array<IndexType, N>.

3.3 Extents design reused

As with mdspan, the Extents template parameter to mdarray shall be a template instantiation of std::extents, as described in [P0009R16?]. The concerns addressed by this aspect of the design are exactly the same in mdarray and mdspan, so using the same form and mechanism seems like the right thing to do here.

3.4 LayoutPolicy design reused

While not quite as straightforward, the decision to use the same design for LayoutPolicy from mdspan in mdarray is still quite obviously the best choice. The only pieces are perhaps a less perfect fit are the is_contiguous() and is_always_contiguous() requirements. While noncontiguous use cases for mdspan are quite common (e.g., subspan()), noncontiguous use cases for mdarray are expected to be a bit more arcane. Nonetheless, reasonable use cases do exist (for instance, padding of the fast-running dimension in anticipation of a resize operation), and the reduction in cognitive load due to concept reuse certainly justifies reusing LayoutPolicy for mdarray.

3.5 AccessorPolicy replaced by Container

By far the most complicated aspect of the design for mdarray is the analog of the AccessorPolicy in mdspan. The AccessorPolicy for mdspan is designed for nonowning semantics. It provides a pointer type, a reference type, and a means of converting from a pointer and an offset to a reference. Beyond the lack of an allocation mechanism (that would be needed by mdarray), the AccessorPolicy requirements address concerns normally addressed by the allocation mechanism itself. For instance, the C++ named requirements for Allocator allow for the provision of the pointer type to std::vector and other containers. Arguably, consistency between mdarray and standard library containers is far more important than with mdspan in this respect. Several approaches to addressing this incongruity are discussed below.

3.5.1 Expected behavior of motivating use cases

Regardless of the form of the solution, there are several use cases where we have a clear understanding of how we want them to work. As alluded to above, perhaps the most important motivating use case for mdarray is that of small, fixed-size extents. Consider a fictitious (not proposed) function, get-underlying-container, that somehow retrieves the underlying storage of an mdarray. For an mdarray of entirely fixed sizes, we would expect the default implementation to return something that is (at the very least) convertible to array of the correct size:

auto a = mdarray<int, 3, 3>();
std::array<int, 9> data = get-underlying-container(a);

(Whether or not a reference to the underlying container should be obtainable is slightly less clear, though we see no reason why this should not be allowed.) The default for an mdarray with variable extents is only slightly less clear, though it should almost certainly meet the requirements of contiguous container ([container.requirements.general]/13). The default model for contiguous container of variable size in the standard library is vector, so an entirely reasonable outcome would be to have:

auto a = mdarray<int, 3, dynamic_extent>();
std::vector<int> data = get-underlying-container(a);

Moreover, taking a view of a mdarray should yield an analogous mdspan with consistent semantics (except, of course, that the latter is nonowning). We provisionally call the method for taking a view of an mdarray “view()”:

template <class T, class Extents, class LayoutPolicy, class ContainerPolicy>
void frobnicate(mdarray<T, Extents, LayoutPolicy, ContainerPolicy> data)
{
  auto data_view = data.view();
  /* ... */
}

In order for this to work, Container::data() should be required to return T*. That way, interoperability with mdspan is trivial, since it can simply be created as:

template <class T, class Extents, class LayoutPolicy, class Container>
void frobnicate(mdarray<T, Extents, LayoutPolicy, Container> a)
{
  mdspan a_view(a.data(), a.mapping());
  /* ... */
}

3.5.2 Analogs in the standard library: Container adapters

Perhaps the best analogs for what mdarray is doing with respect to allocation and ownership are the container adaptors ([container.adaptors]), since they imbue additional semantics to what is otherwise an ordinary container. These all take a Container template parameter, which defaults to deque for stack and queue, and to vector for priority_queue. The allocation concern is thus delegated to the container concept, reducing the cognitive load associated with the design. While this design approach overconstrains the template parameter slightly (that is, not all of the requirements of the Container concept are needed by the container adaptors), the simplicity arising from concept reuse more than justifies the cost of the extra constraints.

It is difficult to say whether the use of Container directly, as with the container adaptors, is also the correct approach for mdarray. There are pieces of information that may need to be customized in some very reasonable use cases that are not provided by the standard container concept. The most important of these is the ability to produce a semantically consistent AccessorPolicy when creating a mdspan that refers to a mdarray. (Interoperability between mdspan and mdarray is considered a critical design requirement because of the nearly complete overlap in the set of algorithms that operate on them.) For instance, given a Container instance c and an AccessorPolicy instance a, the behavior of a.access(p, n) should be consistent with the behavior of c[n] for a mdspan wrapping a that is a view of a mdarray wrapping c (if p is c.begin()). But because c[n] is part of the container requirements and thus may encapsulate any arbitrary mapping from an offset of c.begin() to a reference, the only reasonable means of preserving these semantics for arbitrary container types is to reference the original container directly in the corresponding AccessorPolicy. In other words, the signature for the view() method of mdarray would need to look something like (ignoring, for the moment, whether the name for the type of the accessor is specified or implementation-defined):

template<class ElementType,
         class Extents,
         class LayoutPolicy,
         class Container>
struct mdarray {
 /* ... */
 mdspan<
   ElementType, Extents, LayoutPolicy,
   container_reference_accessor<Container>>
 view() const noexcept;
 /* ... */
};

template <class Container>
struct __container_reference_accessor { // not proposed
  using pointer = Container*;
  /* ... */
  template <class Integer>
  reference access(pointer p, Integer offset) {
    return (*p)[offset];
  }
  /* ... */
};

However, this approach comes at the cost of an additional indirection: one for the pointer to the container, and one for the container dereference itself. This is likely unacceptable cost in a facility designed to target performance-sensitive use cases. The situation for the offset requirement (which is used by submdspan) is potentially even worse for arbitrary non-contiguous containers, adding up to one indirection per invocation of submdspan. This is likely unacceptable in many contexts.

Nonetheless, using refinements of the existing Container concept directly with mdarray is an incredibly attractive option. This is because it avoids the introduction of an extra concept, and thus significantly decreases the cognitive cost of the abstraction. Thus, direct use of the existing Container concept hierarchy should be preferred to other options unless the shortcomings of the existing concept are so irreconcilable (or so complicated to reconcile) as to create more cognitive load than is needed for an entirely new concept.

One straightforward way to resolve the above concerns with arbitrary container types, is to simply restrict what type of containers can be used. Specifically, we would restrict it such that creating an mdspan with default_accessor is straightforward. Thus we would require decltype(Container::data()) to denote ElementType*, and &c[i] to equal c.data() + i for all i in the range of [0, c.size()).

In what follows, we discuss two design alternatives.

3.5.3 (Not proposed) alternative: A dedicated ContainerPolicy concept

Despite the additional cognitive load, there are a few arguments in favor of using a dedicated concept for the container description of mdarray. As is often the case with concept-driven design, the implementation of mdarray only needs a relatively small subset of the interface elements in the Container concept hierarchy. This alone is not enough to justify an additional concept external to the existing hierarchy; however, there are also quite a few features missing from the existing container concept hierarchy, without which an efficient mdarray implementation may be difficult or impossible. As alluded to above, conversion to an AccessorPolicy for the creation of a mdspan is one missing piece. (Another, interestingly, is sized construction of the container mixed with allocator awareness, which is surprisingly lacking in the current hierarchy somehow.) For these reasons, it is worth exploring a design based on analogy to the AccessorPolicy concept rather than on analogy to Container. If we make that abstraction owning, we might call it something like _ContainerLikeThing (not proposed here; included for discussion). In that case, a model of the _ContainerLikeThing concept that meets the needs of mdarray might look something like:

template <class ElementType, class Allocator=std::allocator<ElementType>>
struct vector_container_like_thing // models _ContainerLikeThing
{
public:
  using element_type = ElementType;
  using container_type = std::vector<ElementType, Allocator>;
  using allocator_type = typename container_type::allocator_type;
  using pointer = typename container_type::pointer;
  using const_pointer = typename container_type::const_pointer;
  using reference = typename container_type::reference;
  using const_reference = typename container_type::const_reference;
  using accessor_policy = std::accessor_basic<element_type>;
  using const_accessor_policy = std::accessor_basic<const element_type>;

  // analogous to `access` method in `AccessorPolicy`
  reference access(ptrdiff_t offset) { return __c[size_t(offset)]; }
  const_reference access(ptrdiff_t offset) const { return __c[size_t(offset)]; }

  // Interface for mdspan creation
  accessor_policy make_accessor_policy() { return { }; }
  const_accessor_policy make_accessor_policy() const { return { }; }
  typename pointer data() { return  __c.data(); }
  typename const_pointer data() const { return  __c.data(); }

  // Interface for sized construction
  static vector_container_policy create(size_t n) {
    return vector_container_like_thing{container_type(n, element_type{})};
  }
  static vector_container_policy create(size_t n, allocator_type const& alloc) {
    return vector_container_like_thing{container_type(n, element_type{}, alloc)};
  }

  container_type __c;
};

This approach solves many of the problems associated with using the Container concept directly. It is the most flexible and provides the best compatibility with mdspan, since the conversion to analogous AccessorPolicy is fully customizable. This comes at the cost of additional cognitive load, but this can be justified based on the observation that almost half of the functionality in the above sketch is absent from the container hierarchy: the make_accessor_policy() requirement and the sized, allocator-aware container creation (create(n, alloc)) have no analogs in the container concept hierarchy. Non-allocator-aware creation (create(n)) is analogous to sized construction from the sequence container concept, the data() method is analogous to begin() on the contiguous container concept, and access(n) is analogous to operator[] or at(n) from the optional sequence container requirements. Even for these latter pieces of functionality, though, we are required to combine several different concepts from the Container hierarchy. Based on this analysis, we have decided it is reasonable to pursue designs for this customization point that diverge from Container, including ones that use AccessorPolicy as a starting point. Given a better design, we would definitely consider reversing direction on this decision, but despite significant effort, we were unable to find a design that was more than an awkward and forced fit for the Container concept hierarchy.

3.5.4 (Not proposed) alternative: ContainerPolicy subsumes AccessorPolicy

The above approach has the significant drawback that the _ContainerLikeThing is an owning abstraction fairly similar to a container that diverges from the Container hierarchy. We initially explored this direction because it avoids having to provide a mdarray constructor that takes both a Container and a ContainerPolicy, which we felt was a “design smell.” Another alternative along these lines is to make the mdarray itself own the container instance and have the ContainerPolicy (name subject to bikeshedding; maybe ContainerFactory or ContainerAccessor is more appropriate?) be a nonowning abstraction that describes the container creation and access. While this approach leads to an ugly mdarray(container_type, mapping_type, ContainerPolicy) constructor, the analog that constructor affords to the mdspan(pointer, mapping_type, AccessorPolicy) constructor is a reasonable argument in favor of this design despite its quirkiness. Furthermore, this approach affords the opportunity to explore a ContainerPolicy design that subsumes AccessorPolicy, thus providing the needed conversion to AccessorPolicy for the analogous mdspan by simple subsumption. More importantly, this subsumption would significantly decrease the cognitive load for users already familiar with mdspan. A model of ContainerPolicy for this sort of approach might look something like:

template <class ElementType, class Allocator=std::allocator<ElementType>>
struct vector_container_policy // models ContainerPolicy (and thus AccessorPolicy)
{
public:
  using element_type = ElementType;
  using container_type = std::vector<ElementType, Allocator>;
  using allocator_type = typename container_type::allocator_type;
  using pointer = typename container_type::pointer;
  using const_pointer = typename container_type::const_pointer;
  using reference = typename container_type::reference;
  using const_reference = typename container_type::const_reference;
  using offset_policy = vector_container_policy<ElementType, Allocator>

  // ContainerPolicy requirements:
  reference access(container_type& c, ptrdiff_t i) { return c[size_t(i)]; }
  const_reference access(container_type const& ptrdiff_t i) const { return c[size_t(i)]; }

  // ContainerPolicy requirements (interface for sized construction):
  container_type create(size_t n) {
    return container_type(n, element_type{});
  }
  container_type create(size_t n, allocator_type const& alloc) {
    return container_type(n, element_type{}, alloc);
  }

  // AccessorPolicy requirement:
  reference access(pointer p, ptrdiff_t i) { return p[i]; }
  // For the const analog of AccessorPolicy:
  const_reference access(const_pointer p, ptrdiff_t i) const { return p[i]; }

  // AccessorPolicy requirement:
  pointer offset(pointer p, ptrdiff_t i) { return p + i; }
  // For the const analog of AccessorPolicy:
  const_pointer offset(const_pointer p, ptrdiff_t i) const { return p + i; }

  // AccessorPolicy requirement:
  element_type* decay(pointer p) { return p; }
  // For the const analog of AccessorPolicy:
  const element_type* decay(pointer p) const { return p; }

};

The above sketch makes clear the biggest challenge with this approach: the mismatch in shallow versus deep constness in for an abstractions designed to support mdspan and mdarray, respectively. The ContainerPolicy concept thus requires additional const-qualified overloads of the basis operations. Moreover, while the ContainerPolicy itself can be obtained directly from the corresponding AccessorPolicy in the case of the non-const method for creating the corresponding mdspan (provisionally called view()), the const-qualified version needs to adapt the policy, since the nested types have the wrong names (e.g., const_pointer should be named pointer from the perspective of the mdspan that the const-qualified view() needs to return). This could be fixed without too much mess using an adapter (that does not need to be part of the specification):

template <ContainerPolicy P>
class __const_accessor_policy_adapter { // models AccessorPolicy
public:
  using element_type = add_const_t<typename P::element_type>;
  using pointer = typename P::const_pointer;
  using reference = typename P::const_reference;
  using offset_policy = __const_accessor_policy_adapter<typename P::offset_policy>;

  reference access(pointer p, ptrdiff_t i) { return acc_.access(p, i); }
  pointer offset(pointer p, ptrdiff_t i) { return acc_.offset(p, i); }
  element_type* decay(pointer p) { return acc_.decay(p); }

private:
  [[no_unique_address]] add_const_t<P> acc_;
};

Compared to simply using a container as the argument, this approach has the benefit of enabling mdarray to use containers for which data()[i] is not giving access to the same element as container[i]. However, after more consideration we believe that the need for supporting such containers as underlying storage for mdarray is likely fairly niche. Furthermore, we believe one could later extent the design of mdarray to allow for such ContainerPolicies, even if the initial design only allows for a restricted set of containers.

4 Wording

24.6.� Class template mdarray [mdarray]


24.6.�.1 mdarray overview [mdarray.overview]

1 mdarray maps a multidimensional index in its domain to a reference to an element in its codomain.

2 The domain of an mdarray object is a multidimensional index space defined by an extents.

3 The codomain of an mdarray object is a set of elements accessible from a contiguous range of integer indices.

namespace std {

template<class ElementType, class Extents, class LayoutPolicy, class Container = 
         vector<ElementType>>
class mdarray {
public:
  using extents_type = Extents;
  using layout_type = LayoutPolicy;
  using container_type = Container;
  using mapping_type = typename layout_type::template mapping<extents_type>;
  using element_type = ElementType;
  using value_type = element_type;
  using size_type = typename Extents::size_type;
  using pointer = typename container_type::pointer;
  using reference = typename container_type::reference;
  using const_pointer = typename container_type::const_pointer;
  using const_reference = typename container_type::const_reference;

  // [mdarray.ctors], mdarray constructors
  constexpr mdarray() requires(rank_dynamic() != 0) = default;
  constexpr mdarray(const mdarray& rhs) = default;
  constexpr mdarray(mdarray&& rhs) = default;

  template<class... SizeTypes>
    requires(rank()>0 || rank_dynamic()==0)
    explicit constexpr mdarray(SizeTypes... exts);
  constexpr mdarray(const Extents& ext);
  constexpr mdarray(const mapping_type& m);
  

  template<class... SizeTypes>
    explicit constexpr mdarray(const container_type& c, SizeTypes... exts);
  constexpr mdarray(const container_type& c, const extents_type& ext);
  constexpr mdarray(const container_type& c, const mapping_type& m);

  template<class... SizeTypes>
    explicit constexpr mdarray(container_type&& c, SizeTypes... exts);
  constexpr mdarray(container_type&& c, const extents_type& ext);
  constexpr mdarray(container_type&& c, const mapping_type& m);

  template<class... SizeTypes>
    explicit constexpr mdarray(initializer_list<element_type> init, SizeTypes... exts);
  constexpr mdarray(initializer_list<element_type> init, const extents_type& ext);
  constexpr mdarray(initializer_list<element_type> init, const mapping_type& m);

  template<container-compatible-range<element_type> R, class... SizeTypes>
    explicit constexpr mdarray(from_range_t, R&& rg, SizeTypes... exts);
  template<container-compatible-range<element_type> R>
    constexpr mdarray(from_range_t, R&& rg, const extents_type& ext);
  template<container-compatible-range<element_type> R>
    constexpr mdarray(from_range_t, R&& rg, const mapping_type& m);

  template<class OtherElementType, class OtherExtents, 
           class OtherLayoutPolicy, class OtherContainer>
    explicit(see below)
    constexpr mdarray(
      const mdarray<OtherElementType, OtherExtents, 
                   OtherLayoutPolicy, OtherContainer>& other);

  template<class OtherElementType, class OtherExtents,
           class OtherLayoutPolicy, class Accessor>
    explicit(see below)
    constexpr mdarray(mdspan<OtherElementType, OtherExtents,
                             OtherLayoutPolicy, Accessor> other);

  // [mdarray.ctors.alloc], mdarray constructors with allocators
  template<class Alloc>
    constexpr mdarray(const Extents& ext, const Alloc& a);
  template<class Alloc>
    constexpr mdarray(const mapping_type& m, const Alloc& a);

  template<class Alloc>
    constexpr mdarray(const container_type& c, const extents_type& ext, const Alloc& a);
  template<class Alloc>
    constexpr mdarray(const container_type& c, const mapping_type& m, const Alloc& a);

  template<class Alloc>
    constexpr mdarray(container_type&& c, const extents_type& ext, const Alloc& a);
  template<class Alloc>
    constexpr mdarray(container_type&& c, const mapping_type& m, const Alloc& a);

  template<class Alloc>
    constexpr mdarray(initializer_list<element_type> init, const extents_type& ext, const Alloc& a);
  template<class Alloc>
    constexpr mdarray(initializer_list<element_type> init, const mapping_type& m, const Alloc& a);
  
  template<container-compatible-range<element_type> R, class Alloc>
    constexpr mdarray(from_range_t, R&& rg, const extents_type& ext, const Alloc& a);
  template<container-compatible-range<element_type> R, class Alloc>
    constexpr mdarray(from_range_t, R&& rg, const mapping_type&, const Alloc& a);

  template<class OtherElementType, class OtherExtents, 
           class OtherLayoutPolicy, class OtherContainer, 
           class Alloc>
    explicit(see below)
    constexpr mdarray(
      const mdarray<OtherElementType, OtherExtents, 
                   OtherLayoutPolicy, OtherContainer>& other, const Alloc& a);

  template<class OtherElementType, class OtherExtents,
           class OtherLayoutPolicy, class Accessor,
           class Alloc>
    explicit(see below)
    constexpr mdarray(mdspan<OtherElementType, OtherExtents,
                             OtherLayoutPolicy, Accessor> other,
                      const Alloc& a);

  constexpr mdarray& operator=(const mdarray& rhs) = default;
  constexpr mdarray& operator=(mdarray&& rhs) = default;

  // [mdarray.members], mdarray members
  template<class... SizeTypes>
    constexpr reference operator[](SizeTypes... indices);
  template<class... SizeTypes>
    constexpr const_reference operator[](SizeTypes... indices) const;
  template<class SizeType, size_t N>
    constexpr reference operator[](const array<SizeType, N>& indices);
  template<class SizeType, size_t N>
    constexpr const_reference operator[](const array<SizeType, N>& indices) const;

  static constexpr size_t rank() { return Extents::rank(); }
  static constexpr size_t rank_dynamic() { return Extents::rank_dynamic(); }
  static constexpr size_type static_extent(size_t r) { return Extents::static_extent(r); }

  constexpr const extents_type& extents() const { return map_.extents(); }
  constexpr size_type extent(size_t r) const { return extents().extent(r); }
  constexpr size_type size() const;

  constexpr pointer data() { return ctr\_.data(); }
  constexpr const_pointer data() const { return ctr\_.data(); }
  constexpr const mapping_type& mapping() const { return map_; }

  template<class OtherElementType, class OtherExtents,
           class OtherLayoutType, class OtherAccessorType>
  operator mdspan ();

  template<class OtherAccessorType>
    constexpr mdspan<element_type, extents_type, layout_type, OtherAccessorType>
      view(const OtherAccessorType& a = default_accessor<element_type>);
  template<class OtherAccessorType>
    constexpr mdspan<const element_type, extents_type, layout_type, OtherAccessorType>
      view(const OtherAccessorType& a = default_accessor<element_type>) const;

  static constexpr bool is_always_unique() {
    return mapping_type::is_always_unique();
  }
  static constexpr bool is_always_contiguous() {
    return mapping_type::is_always_contiguous();
  }
  static constexpr bool is_always_strided() {
    return mapping_type::is_always_strided();
  }

  constexpr bool is_unique() const {
    return map_.is_unique();
  }
  constexpr bool is_contiguous() const {
    return map_.is_contiguous();
  }
  constexpr bool is_strided() const {
    return map_.is_strided();
  }
  constexpr size_type stride(size_t r) const {
    return map_.stride(r);
  }

private:
  container_type ctr_;
  mapping_type map_; // exposition only
};

4 mdarray<ElementType, Extents, LayoutPolicy, Accessor> is a trivially copyable type if Container and LayoutPolicy::mapping_type<Extents> are trivially copyable types.

5 ElementType is required to be a non-const complete object type that is neither an abstract class type nor an array type. If Extents is not a specialization of extents, then the program is ill-formed. LayoutPolicy shall meet the layout mapping policy requirements. Container shall meet the requirements of ContiguousContainer. If is_same_v<typename Container::value_type, ElementType> equals false, then the program is ill-formed.

24.6.�.2 mdarray constructors [mdarray.ctors]

template<class... SizeTypes>
  requires(rank()>0 || rank_dynamic()==0)
  explicit constexpr mdarray(SizeTypes... exts);

1 Constraints:

3 Preconditions: If container_type is a specialization of array, mapping_type(Extents(static_cast<size_type>(std::move(exts))...)).required_span_size() == size(container_type()) is true.

2 Effects:

constexpr mdarray(const extents_type& ext);

3 Constraints:

3 Preconditions: If container_type is a specialization of array, mapping_type(Extents(std::move(ext))).required_span_size() == size(container_type()) is true.

4 Effects:

constexpr mdarray(const mapping_type& m);

5 Constraints: if container_type is not a specialization of array, is_constructible_v<container_type, size_t> is true.

3 Preconditions: If container_type is a specialization of array, mapping_type(Extents(m.required_span_size() == size(container_type()) is true.

6 Effects:

template<class... SizeTypes>
  explicit constexpr mdarray(const container_type& c, SizeTypes... exts);

7 Constraints:

8 Preconditions: c.size() == mapping_type(extents_type(static_cast<size_type>(std::move(exts))...)).required_span_size() is true.

9 Effects:

constexpr mdarray(const container_type& c, const extents_type& ext);

10 Constraints: is_constructible_v<mapping_type, const extents_type&> is true, and

11 Preconditions: c.size() == mapping_type(ext).required_span_size() is true.

12 Effects:

constexpr mdarray(const container_type& c, const mapping_type& m);

13 Preconditions: c.size() == m.required_span_size() is true.

14 Effects:

template<class... SizeTypes>
  explicit constexpr mdarray(container_type&& c, SizeTypes... exts);

15 Constraints:

16 Preconditions: c.size() == mapping_type(extents_type(static_cast<size_type>(std::move(exts))...)).required_span_size() is true.

17 Effects:

constexpr mdarray(container_type&& c, const extents_type& ext);

18 Constraints: is_constructible_v<mapping_type, const extents_type&> is true, and

18 Preconditions: c.size() == mapping_type(ext).required_span_size() is true.

19 Effects:

constexpr mdarray(container_type&& c, const mapping_type& m);

20 Preconditions: c.size() == m.required_span_size() is true.

21 Effects:

template<class... SizeTypes>
  explicit constexpr mdarray(std::initializer_list<element_type> init, SizeTypes... exts);

7 Constraints:

8 Preconditions: init.size() == mapping_type(extents_type(static_cast<size_type>(std::move(exts))...)).required_span_size() is true.

9 Effects:

constexpr mdarray(std::initializer_list<element_type> init, const extents_type& ext);

10 Constraints:

11 Preconditions: init.size() == mapping_type(ext).required_span_size() is true.

12 Effects:

constexpr mdarray(std::initializer_list<element_type> init, const mapping_type& m);

10 Constraints: is_constructible_v<container_type, std::initializer_list<element_type>> is true.

13 Preconditions: init.size() == m.required_span_size() is true.

14 Effects:

template<container-compatible-range<element_type> R, class... SizeTypes>
  explicit constexpr mdarray(from_range_t, R&& rg, SizeTypes... exts);

7 Constraints:

8 Preconditions: init.size() == mapping_type(extents_type(static_cast<size_type>(std::move(exts))...)).required_span_size() is true.

9 Effects:

template<container-compatible-range<element_type> R>
  constexpr mdarray(from_range_t, R&& rg, const extents_type& ext);

10 Constraints:

11 Preconditions: size(rg) == mapping_type(ext).required_span_size() is true.

12 Effects:

template<container-compatible-range<element_type> R>
  constexpr mdarray(from_range_t, R&& rg, const mapping_type& m);

10 Constraints: the expression ranges::to<container_type>(std::forward<R>(rg)) is well formed.

13 Preconditions: size(rg) == m.required_span_size() is true.

14 Effects:

14 Constraints:

template<class OtherElementType, class OtherExtents,
         class OtherLayoutPolicy, class OtherContainer>
  explicit(see below)
  constexpr mdarray(const mdarray<OtherElementType, OtherExtents, 
                                OtherLayoutPolicy, OtherContainer>& other);

23 Mandates:

24 Constraints:

25 Preconditions:

26 Effects:

27 Remarks: The expression inside explicit is:

  !is_convertible_v<const typename OtherLayoutPolicy::mapping_type&, mapping_type> ||
  !is_convertible_v<const OtherContainer&, Container>
  template<class OtherElementType, class OtherExtents,
           class OtherLayoutPolicy, class Accessor>
    explicit(see below)
    constexpr mdarray(mdspan<OtherElementType, OtherExtents,
                             OtherLayoutPolicy, Accessor> other);

28 Mandates:

29 Constraints:

30 Preconditions:

31 Effects:

[Note: Requiring default constructibility of value_type means that ctr_ may first be constructed with its required span size, and then filled by iterating over all unique multidimensional indices i... in the mdarray’s domain. Alternately, ctr_ may be constructed via ranges::to, if the elements of other can be viewed by a range. The intent is to permit ranges::to initialization of ctr_ if possible, without requiring a particular iteration order (as the best-performing order can depend sensitively on the two layouts) or even requiring all mdspan to be iterable by a range.— end note]

32 Remarks: The expression inside explicit is:

  !is_convertible_v<const typename OtherLayoutPolicy::mapping_type&, mapping_type> ||
  !is_convertible_v<Accessor::reference, value_type>

24.6.�.3 mdarray constructors with allocators [mdarray.ctors.alloc]

template<class Alloc>
  constexpr mdarray(const extents_type& ext, const Alloc& a);

28 Constraints:

29 Effects:

template<class Alloc>
  constexpr mdarray(const mapping_type& m, const Alloc& a);

30 Constraints: is_constructible_v<container_type, size_t, Alloc> is true.

31 Effects:

template<class Alloc>
  constexpr mdarray(const container_type& c, const extents_type& ext, const Alloc& a);

32 Constraints:

33 Preconditions: c.size() == mapping_type(ext).required_span_size() is true.

34 Effects:

template<class Alloc>
  constexpr mdarray(const container_type& c, const mapping_type& m, const Alloc& a);

35 Constraints: is_constructible_v<container_type, container_type, Alloc> is true.

36 Preconditions: c.size() == m.required_span_size() is true.

37 Effects:

template<class Alloc>
  constexpr mdarray(container_type&& c, const extents_type& ext, const Alloc& a);

38 Constraints:

39 Preconditions: c.size() == mapping_type(ext).required_span_size() is true.

40 Effects:

template<class Alloc>
  constexpr mdarray(container_type&& c, const mapping_type& m, const Alloc& a);

41 Constraints: is_constructible_v<container_type, container_type, Alloc> is true.

42 Preconditions: c.size() == m.required_span_size() is true.

43 Effects:

template<class Alloc>
constexpr mdarray(std::initializer_list<element_type> init, const extents_type& ext, const Alloc& a);

10 Constraints:

11 Preconditions: init.size() == mapping_type(ext).required_span_size() is true.

12 Effects:

template<class Alloc>
constexpr mdarray(std::initializer_list<element_type> init, const mapping_type& m, const Alloc& a);

10 Constraints: is_constructible_v<container_type, std::initializer_list<element_type>, Alloc> is true.

13 Preconditions: init.size() == m.required_span_size() is true.

14 Effects:

template<container-compatible-range<element_type> R, class Alloc>
  constexpr mdarray(from_range_t, R&& rg, const extents_type& ext, const Alloc& a);

10 Constraints:

11 Preconditions: size(rg) == mapping_type(ext).required_span_size() is true.

12 Effects:

template<container-compatible-range<element_type> R, class Alloc>
  constexpr mdarray(from_range_t, R&& rg, const mapping_type& m, const Alloc& a);

10 Constraints: the expression ranges::to<container_type>(std::forward<R>(rg), a) is well formed.

13 Preconditions: size(rg) == m.required_span_size() is true.

14 Effects:

template<class OtherElementType, class OtherExtents,
         class OtherLayoutPolicy, class OtherContainer, class Alloc>
  explicit(see below)
  constexpr mdarray(const mdarray<OtherElementType, OtherExtents, 
                                  OtherLayoutPolicy, OtherContainer>& other,
                    const Alloc& a);

44 Mandates:

45 Constraints:

46 Preconditions:

47 Effects:

48 Remarks: The expression inside explicit is:

  !is_convertible_v<const typename OtherLayoutPolicy::mapping_type&, mapping_type> ||
  !is_convertible_v<const OtherContainer&, Container>
  template<class OtherElementType, class OtherExtents,
           class OtherLayoutPolicy, class Accessor,
           class Alloc>
    explicit(see below)
    constexpr mdarray(mdspan<OtherElementType, OtherExtents,
                             OtherLayoutPolicy, Accessor> other,
                      const Alloc& a);

28 Mandates:

29 Constraints:

30 Preconditions:

31 Effects:

[Note: For intent, please see Note on the mdarray constructor taking an mdspan with no allocator.— end note]

32 Remarks: The expression inside explicit is:

  !is_convertible_v<const typename OtherLayoutPolicy::mapping_type&, mapping_type> ||
  !is_convertible_v<Accessor::reference, value_type> ||
  !is_convertible_v<Alloc, container_type::allocator_type>


24.6.�.4 mdarray members [mdarray.members]

template<class... SizeTypes>
  constexpr reference operator[](SizeTypes... indices);

1 Constraints:

2 Let I be static_cast<size_type>(std::move(indices)).

3 Preconditions: I is a multidimensional index into extents(). [Note: This implies that map_(I...) <map_.required_span_size() is true.— end note];

4 Effects: Equivalent to: return ctr_[map_(I...)];.

template<class... SizeTypes>
  constexpr const_reference operator[](SizeTypes... indices) const;

5 Constraints:

6 Let I be static_cast<size_type>(std::move(indices)).

7 Preconditions: I is a multidimensional index into extents(). [Note: This implies that map_(I...) <map_.required_span_size() is true.— end note];

8 Effects: Equivalent to: return ctr_[map_(I...)];.

template<class SizeType, size_t N>
  constexpr reference operator[](const array<SizeType, N>& indices);

9 Constraints:

10 Effects: Let P be the parameter pack such that is_same_v<make_index_sequence<rank()>, index_sequence<P...>> is true.
Equivalent to: return operator[](static_cast<size_type>(indices[P])...);

template<class SizeType, size_t N>
  constexpr const_reference operator[](const array<SizeType, N>& indices) const;

11 Constraints:

12 Effects: Let P be the parameter pack such that is_same_v<make_index_sequence<rank()>, index_sequence<P...>> is true.
Equivalent to: return operator[](static_cast<size_type>(indices[P])...);

constexpr size_type size() const;

13 Precondition: The size of extents() is a representable value of size_type ([basic.fundamental]).

14 Returns: extents().fwd-prod-of-extents(rank()).

  template<class OtherElementType, class OtherExtents,
           class OtherLayoutType, class OtherAccessorType>
  operator mdspan ();

15 Constraints: is_assignable_<mdspan<element_type, extents_type, layout_type>, mdspan<OtherElementType, OtherExtents, OtherLayoutType, OtherAccessorType>> is true.

16 Returns: mdspan(data(),map_)`

  template<class OtherAccessorType>
    constexpr mdspan<element_type, extents_type, layout_type, OtherAccessorType>
      view(const OtherAccessorType& a = default_accessor<element_type>());

17 Constraints: is_assignable_<mdspan<element_type, extents_type, layout_type>, mdspan<element_type, extents_type, layout_type, OtherAccessorType>> is true.

18 Returns: mdspan(data(),map_, a)

  template<class OtherAccessorType>
    constexpr mdspan<const element_type, extents_type, layout_type, OtherAccessorType>
      view(const OtherAccessorType& a = default_accessor<element_type>()) const;

19 Constraints: is_same_v<OtherAccessorType::element_type, element_type> is true.

20 Returns: mdspan(data(),map_, a)

5 References

[P1684R0]
David Hollman, Christian Trott, Mark Hoemmen, Daniel Sundernland. 2019. mdarray: An Owning Multidimensional Array Analog of mdspan.
https://wg21.link/p1684r0