1 Revision History

1.1 P0009r13: 2021-10 Mailing

LEWG reviewed P0009r12 together with P2299r3 on 2021-06-08.

LEWG Poll Approve the direction of P2299R3 and merge it into P0009.

Attendance: 25; Number of authors: 1 [presumably for P2299, as P0009 coauthors were also attending]; Author’s Position: SF.

• Incorporated changes proposed by P2299r3
  • Added dextents alias
  • Removed mdspan alias and renamed basic_mdspan to mdspan (which is now a class type, not an alias). This undoes a change introduced in P0009r6. P2299r3 explains the rationale. Existing code using the mdspan alias will need to change by replacing the list of extents template arguments with a single extents type.
  • Added mdspan deduction guides
  • As needed for deduction guides, added layout_type alias to layout mapping requirements and to layout_left, layout_right, and layout_stride
• Adapted new LWG wording guidelines by replacing “Expects” with “Preconditions”
• Minor formatting corrections
• Added design discussion as requested by LEWG
• Remove unconditional noexcept from mdspan
• Fix layout required_span_size for rank-0 mdspan
• Fix layout_stride::required_span_size for mdspans with at least one extent being zero
• Use operator[] in mdspan for multidimensional array access, and add explanation to Discussion section
• Remove reference to span in the mdspan wording, since mdspan does not necessarily require a backing span (because pointer need not be ElementType*)
• added conversion constructor for strided and unique layouts to layout_stride
• added constructor for mdspan from pointer and extents
• added requirement for layout policy mapping to be nothrow move constructible and assignable
• added requirement for accessor policy to be nothrow move constructible and assignable
• added requirement for accessor policy pointer to be nothrow move constructible and assignable
• remove throws nothing clauses from mdspan and submdspan.

1.2 P0009r12: post 2021-05 Mailing

• Fixed definition of static_extent
• Added converting constructor for default_accessor (when the pointers to elements are convertible) for things like default_accessor<double> to default_accessor<const double>
• Changed [mdspan.accessor.basic] to [mdspan.accessor.default], to correspond with the name change in P0009r11
• Minor formatting corrections

1.3 P0009r11: 2021-05 Mailing

• Ask LEWG to poll on targeting P0009 for C++23
• Change all the sizes from ptrdiff_t to size_t and index_type to size_type, for consistency with span and the rest of the standard library`
• Renamed IndexType to SizeType or SizeTypes (depending if it is a single type or a parameter pack)
• Changed comparisons to hidden friends
• Explicitly mention which types are trivially copyable or empty. This is important as a major intended use case for this is heterogeneous computing. A trivially copyable is heavily used as a proxy for types which can be copied between a host (such as a CPU) and a device (such as a GPU) or between two devices by just copying the bytes which make up the object representation of the type. If they are not trivially copyable, heterogeneous computing would not be able to use these types as vocabulary types.
• State the conditions that make basic_mdspan trivially default constructible
• In layout_* types, made operator() and stride() constexpr
• In layout_stride, made assignment operators and required_span_size() constexpr to match the other layout_* types
• Renamed subspan to submdspan, as this only applies to mdspan
• Made submdspan() constexpr
• Tweak the wording of is_strided
• Renamed all_type to full_extent_t and all to full_extent
• Renamed accessor_basic to default_accessor
• Removed accessor policy decay(p) member function as it was an artifact from an earlier version of this proposal when basic_mdspan had a span() member function that returned a std::span
• Removed span() from [mdspan.basic.members] description as .span() was removed from an earlier version of this proposal

1.4 P0009r10: Pre 2020-02-Prague Mailing

• Switched to mpark/wg21 pandoc format
• Add general description of span and mdspan
• Removed mdspan_subspan expo only type; use basic_mdspan<see below> instead
• Fixed typos in accessor table
• Made editorial changes to wording based on San Diego feedback
• Updated operational semantics subsection heading based on new style guidelines

1.5 P0009r9: Pre 2019-02-Kona Mailing

• Wording fixes based on guidance: LWG small group at 2018-11-SanDiego

1.6 P0009r8: Pre 2018-11-SanDiego Mailing

• Refinement based upon updated prototype / reference implementation

1.7 P0009r7: Post 2018-06-Rapperswil Mailing

• wording reworked based on guidance: LWG review at 2018-06-Rapperswil
• usage of span requires reference to C++20 working draft
• namespace for library TS std::experimental::fundamentals_v3

1.8 P0009r6 : Pre 2018-06-Rapperswil Mailing

P0009r5 was not taken up at 2018-03-Jacksonville meeting. Related LEWG review of P0900 at 2018-03-Jacksonville meeting

LEWG Poll We want the ability to customize the access to elements of span (ability to restrict, etc):

span<T, N, Accessor=...>

LEWG Poll We want the customization of basic_mdspan to be two concepts Mapper and Accessor (akin to Allocator design).

basic_mdspan<T, Extents, Mapper, Accessor>
mdspan<T, N...>

LEWG Poll: We want the customization of basic_mdspan to be an arbitrary (and potentially user-extensible) list of properties.

basic_mdspan<T, Extents, Properties...>

Changes from P0009r5 due to related LEWG reviews:

• Replaced variadic property list with extents, layout mapping, and accessor properties.
• Incorporated P0454r1.
  • Added accessor policy concept.
  • Renamed mdspan to basic_mdspan.
  • Added a mdspan alias to basic_mdspan.

1.9 P0009r5 : Pre 2018-03-Jacksonville Mailing

LEWG review of P0009r4 at 2017-11-Albuquerque meeting

LEWG Poll: We should be able to index with span<int type[N]> (in addition to array).

Against comment - there is not a proven needs for this feature.

LEWG Poll: We should be able to index with 1d mdspan.

LEWG Poll: We should put the requirement on “rank() <= N” back to “rank()==N”.

Unanimous consent

LEWG Poll: With the editorial changes from small group, plus the above polls, forward this to LWG for Fundamentals v3.

Unanimous consent

Changes from P0009r4:

• Removed nullptr constructor.
• Added constexpr to indexing operator.
• Indexing operator requires that rank()==sizeof...(indices).
• Fixed typos in examples and moved them to appendix.
• Converted note on how extentions to access properties may cause reference to be a proxy type to an “see below” to make it normative.

1.10 P0009r4 : Pre 2017-11-Albuquerque Mailing

LEWG review at 2017-03-Kona meeting

LEWG review of P0546r1 at 2017-03-Kona meeting

LEWG Poll: Should we have a single template that covers both single and multi-dimensional spans?

Changes from P0009r3:

• Align with P0122r5 span proposal.
• Rename to mdspan, multidimensional span, to align with span.
• Move preferred array extents mechanism to appendix.
• Expose codomain as a span.
• Add layout mapping concept.

1.11 P0009r3 : Post 2016-06-Oulu Mailing

LEWG review at 2016-06-Oulu

LEWG did not like the name array_ref, and suggested the following alternatives: - sci_span - numeric_span - multidimensional_span - multidim_span - mdspan - md_span - vla_span - multispan - multi_span

LEWG Poll: Are member begin()/end() still good?

LEWG Poll: Want this proposal to provide range-producing functions outside array_ref?

LEWG Poll: Want a separate proposal to explore iteration design space?

Changes from P0009r2:

• Removed iterator support; a future paper will be written on the subject.
• Noted difference between multidimensional array versus language’s array-of-array-of-array…
• Clearly describe requirements for the embedded type aliases (element_type, reference, etc).
• Expanded description of how the variadic properties list would work.
• Stopped allowing array_ref<T[N]> in addition to array_ref<extents<N>>.
• Clarified domain, codomain, and domain -> codomain mapping specifications.
• Consistently use extent and extents for the multidimensional index space.

1.12 P0009r2 : Pre 2016-06-Oulu Mailing

LEWG review at 2016-02-Jacksonville.

Changes from P0009r1:

• Adding details for extensibility of layout mapping.
• Move motivation, examples, and relaxed incomplete array type proposal to separate papers.
  • P0331: Motivation and Examples for Polymorphic Multidimensional Array.
  • P0332: Relaxed Incomplete Multidimensional Array Type Declaration.

1.13 P0009r1 : Pre 2016-02-Jacksonville Mailing

LEWG review at 2015-10-Kona.

LEWG Poll: What should this feature be called?

LEWG Poll: Do we want 0-length static extents?

LEWG POLL: Do we want the language to support syntaxes like X[3][][][5]?

LEWG POLL: Do we want the variadic property list in template args (either raw or in properties<>)? Note there is no precedence for this in the library.

LEWG POLL: Do we want the per-view bounds-checking knob?

Changes from P0009r0:

• Renamed view to array_ref.
• How are users allowed to add properties? Needs elaboration in paper.
• view<int[][][]>::layout should be named.
• Rename is_regular (possibly to is_affine) to avoid overloading the term with the Regular concept.
• Make static span(), operator(), constructor, etc variadic.
• Demonstrate the need for improper access in the paper.
• In operator(), take integral types by value.

1.14 P0009r0 : Pre 2015-10-Kona Mailing

Original non-owning multidimensional array reference (view) paper with motivation, specification, and examples.

1.15 Related Activity

Related LEWG review of P0546r1 at 2017-11-Albuquerque meeting

LEWG Poll: span should specify the dynamic extent as the element type of the first template parameter rather than the (current) second template parameter

LEWG Poll: span should support the addition of access properties variadic template parameters

Authors agreed to bring a separate paper ([[P0900r0]]) discussing how the variadic properties will work.

2 Description

2.1 What we propose to add

This paper proposes adding to the C++ Standard Library a multidimensional array view, mdspan, along with classes, class templates, and constants for describing and creating multidimensional array views. It also proposes adding the submdspan function that “slices” (returns an mdspan that views a subset of) an existing mdspan`.

The mdspan class template can represent arbitrary mixes of compile-time or run-time extents. Its element type can be any complete object type that is neither an abstract class type nor an array type. It has two customization opportunities for users: the layout mapping and the accessor. The layout mapping specifies the formula, and properties of the formula, for mapping a multidimensional index to an element of the array. The accessor governs how elements are read and written.

2.2 Definitions

A multidimensional array view views a multidimensional array, just as a span views a one-dimensional array or vector.

A multidimensional array of rank $R$ maps from a tuple of $R$ indices to a single offset index. Each of the $R$ indices in the tuple is in a bounded range whose inclusive lower bound is zero, and whose nonnegative exclusive upper bound is that index’s extent. The array thus has $R$ extents. The offset index ranges over a subset of a bounded contiguous index range whose lower bound is zero, and whose upper bound is the product of the $R$ extents.

More formally, a multidimensional array of rank $R$ maps from its domain, a multidimensional index space of rank $R$, to its codomain, elements of a contiguous span of objects. A multidimensional index space of rank $R$ is the Cartesian product $[0,\; N$₀) ⨯ [0, N₁) ⨯ … ⨯ [0, N_R-1) of half-open integer intervals, where the $N$_k for $k\; =\; 0$, …, $R-1$ are the array’s extents. A multidimensional index is a element of a multidimensional index space.

2.3 Why do we need multidimensional arrays?

Multidimensional arrays are fundamental concepts in many fields, including graphics, mathematics, statistics, engineering, and the sciences. Many programming languages thus come with multidimensional array data structures either as a core language feature, or as a tightly integrated standard library. Example languages include Ada, ANSI Common Lisp, APL, C#, Fortran, Julia, Matlab, Mathematica, Pascal, Python (via NumPy), and Visual Basic. The original version of the Fortran language for the IBM 704 featured arrays with one, two, or three extents (Backus 1956, pp. 10-11).

Multidimensional arrays have long been useful for representing large amounts of data, describing points in physical space, or expressing approximations of functions. They are a natural way to represent mathematical objects like matrices and tensors. This makes multidimensional arrays a critical data structure for many computations at the heart of modern machine learning. In fact, one of the predominant machine learning frameworks is called TensorFlow.

2.4 Why are existing C++ data structures not enough?

C++ currently has the following approaches that could be used to represent multidimensional arrays:

1. “native” arrays where all the extents are compile-time constants, like int[3][4][5];

2. pointer-of-pointers(-of-pointers…), like int***, set up as a data structure to view multidimensional data;

3. arrays-of-arrays(-of-arrays…) data structures, like vector<vector<array<int, N>>>; or

4. gslice, which selects a subset of indices of a valarray and can be used to impose a multidimensional array layout on the valarray, in a way analogous to layout_stride.

If a multidimensional array has any extents that are not known at compile time, Approach (1) does not work.

Approach (2) does not suffice as a stand-alone data structure, because a pointer-of-pointers does not carry along the array’s run-time extents. Users thus end up building some subset of mdspan’s functionality to represent a multidimensional array view. Every run-time extent other than the rightmost requires a separate memory allocation for an array of pointers. A pointer-of-pointers also loses information about any dimensions known at compile time. Users cannot arbitrarily mix compile-time and run-time extents.

Approach (3) can mix vector and array to represent extents known at run time resp. compile time. However, any use of vector at any position other than the outermost results in the data structure no longer having a contiguous memory allocation (or a subset thereof) for the elements. This makes the data structure incompatible with many libraries that expect a subset of a contiguous allocation. Also, every run-time extent other than the rightmost requires a separate memory allocation for an array of arrays. In addition, each element access requires reading multiple memory locations (“pointer chasing”). Finally, the inlining depth for an element access is proportional to the array’s rank.

Approach (4) is meant for addressing many elements of a valarray all at once. Even though valarray itself is a one-dimensional array, one can use gslice to make the valarray represent multidimensional data. Giving a gslice to valarray::operator[] returns something that references a subset of elements of the original valarray. However, the result (a gslice_array in the nonconst case, some type that might be an expression template in the const case) is not guaranteed to have an operator[]. Thus, it’s not a view, whereas our proposed submdspan function always takes and returns a view. In the const case, the result might even be a (deep) copy of the input. Finally, gslice offers no efficient way to address a single element. The gslice constructor takes strides and lengths as valarrays and is meant for array-based computation. Accessing a single element requires accessing the memory of three valarrays.

2.5 Mixing compile-time and run-time extents

The fundamental reason to allow expressing extents at compile time is performance. Knowing an extent at compile time enables many compiler optimizations, such as unrolling and precomputing of offsets. These can significantly improve the generated code. Not storing extents at run time may help conserve registers and stack space.

In many fields, some extents are naturally known at compile time. For many physics and engineering algorithms, some extents are dictated by fundamental properties of the physical world or the discretization scheme. For example, the position of a particle in space requires a rank-3 array, since physical space has three dimensions. At the same time, other extents are only known at run time, such as the number of particles in a simulation. A natural data structure for storing a list of particles would thus be a rank-2 array, where the one run-time extent is the number of particles and the one compile-time extent is three. In graphics, some of the most fundamental objects are square matrices with 2, 3, or 4 rows and columns. The number of matrices with which one would like to compute might only be known at run time. This would make a rank-3 array with two compile-time extents a natural data structure for the matrices.

2.6 Why custom memory layouts?

Our mdspan class template permits custom layouts. Our proposal comes with three memory layouts:

• layout_right: C or C++ style, row major, where the rightmost index gives stride-1 access to the underlying memory;

• layout_left: Fortran or Matlab style, column major, where the leftmost index gives stride-1 access to the underlying memory;

• layout_stride: a generalization of the two layouts above, which stores a separate stride (possibly not one) for each extent.

“Custom” layouts besides these could include space-filling curves or “tiled” layouts.

An important reason we allow different layouts is language interoperability. For example, C++ and Fortran have different “native” layouts. Python’s NumPy arrays have a configurable layout, to provide compatibility with both languages.

Control of the layout can also be used to write code that performs well on different computer architectures when only changing a template argument. Consider the following implementation of a parallel dense matrix-vector product.

using layout = /* see-below */;

std::mdspan<double, std::extents<N, M>, layout> A = ...;
std::mdspan<double, std::extents<N>> y = ...;
std::mdspan<double, std::extents<M>> x = ...;

std::ranges::iota_view range{0, N};

std::for_each(std::execution::par_unseq, 
  std::ranges::begin(range), std::ranges::end(range),
  [=](int i) {
     double sum = 0.0;
     for(int j = 0; j < M; ++j) {
       sum += A[i, j] * x[j];
     }
     y[i] = sum;
  });

On conventional CPU architectures, this code performs well with layout = layout_right, the native C++ row-major layout. However, when offloading the for_each to NVIDIA GPUs (which NVIDIA’s nvc++ compiler can do), layout = layout_left (Fortran’s column-major layout) performs much better, since it enables coalesced data access on the matrix A.

However, it is not enough to have just C++ and Fortran memory mappings. For instance, one way to compute tensor products is to decompose them into many matrix-matrix multiplications. The resulting decomposition may involve matrices with non-unit strides in both extents. This means that they have neither a row-major nor a column-major layout.

More complex layouts can improve performance significantly for some algorithms. For instance, tiling (a “matrix of small matrices” layout) can improve data locality for many computations relevant to linear algebra and the discretization of partial differential equations. Tiled layouts can also improve vectorization. For example, Intel’s Math Kernel Library introduced the Vectorized Compact Routines. These provide “batched” matrix operations that increase available parallelism by operating on many matrices at once. The Vectorized Compact Routines accept matrices in an “interleaved” layout that optimizes vectorized memory access.

Another design goal for our custom layouts is to permit nonunique layouts. A nonunique layout lets multiple index tuples refer to the same element. This can save memory for data structures that have natural symmetry. For example, if A is a symmetric matrix, then A[i,j] and A[j,i] refer to the same element, so the element can and should only be stored once.

2.7 Why custom accessors?

Custom accessors can provide information to the compiler, or permit the injection of special ways of doing data access. Most hardware today has more ways to access data than simple reads and writes. For example, some instructions affect caching behavior, by making loads and/or stores nontemporal (not cached at some level) or even noncoherent. Other instructions implement atomic access. This is why several of us proposed atomic_ref, as the heart of an “atomic accessor” for mdspan. C’s restrict qualifier conveys whether an array is assumed never to alias another array in some context. The volatile keyword is yet another qualifier which limits compiler optimizations around data access. Custom mdspan accessors can apply restrict (if the C++ implementation supports this extension) or volatile to array accesses.

Custom accessors also address concerns relating to heterogeneous memory. Standard C++ does not have the idea of “memory spaces that normal code cannot access,” but many extensions to C++ do have this idea. For example, a custom accessor could convey accessibility by CPU or GPU threads, so that the compiler would prevent users from accessing GPU memory while running on the CPU, or vice versa. Multiple memory spaces occur in programming models other than for GPUs. For example, “partitioned global address space” models have a “global shared memory” that requires special operations to access. C++ libraries like Kokkos expose access to such memory using an analog of a custom accessor. Other accessors could expose an array interface to a persistent storage device that is not directly byte addressable. We do not propose such accessors here, but this is a customization point third-party libraries could directly use, and is available for any future extensions of the C++ standard for supporting heterogeneous memory.

For a discussion of the idea of accessors and several examples, please see (Keryell and Falcou 2016).

2.8 Subspan Support

A critical feature of this proposal is submdspan, the subspan or “slicing” function that returns a view of a subset of an existing mdspan. The result may have any rank up to and including the rank of the input. All of the aforementioned languages with multidimensional array support provide subspan capabilities. Subspans are important because they enable code reuse. For example, the inner loop in the dense matrix-vector product described above actually represents a dot product – an inner product of two vectors. If one already has a function for such an inner product, then a natural implementation would simply reuse that function. The LAPACK linear algebra library depends on subspan reuse for the performance of its one-sided “blocked” matrix factorizations (Cholesky, LU, and QR). These factorizations reuse textbook non-blocked algorithms by calling them on groups of contiguous columns at a time. This lets LAPACK spend as much time in dense matrix-matrix multiply (or algorithms with analogous performance) as possible.

2.9 Why propose a multidimensional array view before a container?

Factoring views from containers generally makes sense. For example, one often sees functions that take vector by reference when they only need to access the vector’s elements or call .size() on it. This is one reason for span. Some of us have proposed a multidimensional array container, mdarray P1684, but we have focused on mdspan because we consider views more fundamental.

Many fields that compute with multidimensional arrays rely heavily on shared-memory parallel programming, where multiple processing units (threads, vector units, etc.) access different elements of the same array in parallel. Memory allocation and deallocation are “synchronization points” for parallel processing units, and thus hinder parallelization. This makes just viewing a multidimensional array, rather than managing its ownership, the most fundamental way for parallel computations to express how they access an array.

It is often necessary to view previously allocated memory as a multidimensional array. An important special case is when C++ code is calling or being called from another programming language, such as C, Fortran, or Python. This use case matters enough to Python that its C API defines a Buffer Protocol for viewing multidimensional arrays across languages. Language interoperability is key to the success of the various Python-based data analysis frameworks built up around NumPy.

2.10 Use multiple-parameter operator[] for array access

We welcome multiple-parameter operator[] as the preferred multidimensional array access operator. P1161R3, now part of C++20, prepared the way for this by deprecating comma expressions inside operator[] invocations. P2128R6, which proposed changing operator[] to accept multiple parameters, was approved at the October 2021 WG21 Plenary meeting. Please refer to P2128 for an extensive discussion.

Many existing libraries use the function call operator() for multidimensional array access, with operator[] available for rank-1 (single-dimensional) mdspan. P2128 gives examples. It’s straightforward to adapt these libraries to transition to mdspan. For example, a subclass or wrapper of mdspan can provide an operator() that simply forwards to mdspan::operator[]. The subclass or wrapper can then deprecate operator() to help developers find and change all the code that uses it.

2.11 Reference Implementation

A reference implementation of this proposal under BSD license is available at: mdspan. This implementation is also available on godbolt for experimentation: godbolt.

2.12 References

• J. W. Backus et al. “Programmer’s Reference Manual: Fortran Automatic Coding System for the IBM 704.” Applied Science Division and Programming Research Department, International Business Machines Corporation, Oct. 15, 1956. Available online (last accessed Oct. 10, 2021).

• D. Hollman, C. Trott, M. Hoemmen, and D. Sunderland. “mdarray: An Owning Multidimensional Array Analog of mdspan.” P1684r0, May 28, 2019. Available online (last accessed Oct. 10, 2021).

• R. Keryell and J. Falcou. “Accessors: A C++ standard library class to qualify data accesses.” P0367r0, May 29, 2016. Available online (last accessed Oct. 10, 2021).

3 Editing Notes

The proposed changes are relative to the working draft of the standard as of N4842.

The � character is used to denote a placeholder section number, table number, or paragraph number which the editor shall determine.

Add the header <mdspan> to the “C++ library headers” table in [headers] in a place that respects the table’s current alphabetic order.

Add the header <mdspan> to the “Containers library summary” table in [containers.general] below the listing for <span>.

4 Wording

The � character is used to denote a placeholder section number which the editor shall determine.

Make the following changes to 22.7.1 [views.general],

1 The header <span> defines the view span. The header <mdspan> defines the class template mdspan and other facilities for interacting with these nonowning multidimensional views.

Add the following subclauses to the end of the [views] subclause (after span):

22.7.� Header <mdspan> synopsis [mdspan.syn]

namespace std {
  // [mdspan.extents], class template extents
  template<size_t... Extents>
    class extents;

  template<size_t Rank>
    using dextents = decltype(
      [] <size_t... Pack> (index_sequence<Pack...>) constexpr {
        return extents<
          [] (auto) constexpr { return dynamic_extent; } (
            integral_constant<size_t, Pack>{})...>{};
      }(make_index_sequence<Rank>{}));

  template<class>
  constexpr size_t make_dynamic_extent() { return dynamic_extent; } // exposition only

  template <class... Integrals>
  extents(Integrals...)
    -> extents<make_dynamic_extent<Integrals>()...>;


  // [mdspan.layout], Layout mapping policies
  class layout_left;
  class layout_right;
  class layout_stride;

  // [mdspan.accessor.default]
  template<class ElementType>
    class default_accessor;

  // [mdspan.basic], class template mdspan
  template<class ElementType, class Extents, class LayoutPolicy = layout_right,
           class AccessorPolicy = default_accessor<ElementType>>
    class mdspan;

  template <class ElementType, class... Integrals>
  explicit mdspan(ElementType*, Integrals...)
    -> mdspan<ElementType, dextents<sizeof...(Integrals)>;

  template <class ElementType, class SizeType, size_t N>
  mdspan(ElementType*, const array<SizeType, N>&)
    -> mdspan<ElementType, dextents<N>>;

  template <class ElementType, size_t... ExtentsPack>
  mdspan(ElementType*, const extents<ExtentsPack...>&)
    -> mdspan<ElementType, extents<ExtentsPack...>>;

  template <class ElementType, class MappingType>
  mdspan(ElementType*, const MappingType&)
    -> mdspan<ElementType, typename MappingType::extents_type, typename MappingType::layout_type>;

  template <class ElementType, class MappingType, class AccessorType>
  mdspan(ElementType*, const MappingType&, const AccessorType&)
    -> mdspan<ElementType, typename MappingType::extents_type, typename MappingType::layout_type, AccessorType>;

  // [mdspan.submdspan]
  template<class ElementType, class Extents, class LayoutPolicy,
           class AccessorPolicy, class... SliceSpecifiers>
    constexpr mdspan<see below> submdspan(
      const mdspan<ElementType, Extents, LayoutPolicy, AccessorPolicy>&,
      SliceSpecifiers ...);

  // tag supporting submdspan
  struct full_extent_t { explicit full_extent_t() = default; };
  inline constexpr full_extent_t full_extent = full_extent_t{};
}

22.7.� Overview [mdspan.overview]

1 A multidimensional index space is a Cartesian product of integer intervals. Each interval can be represented by a half-open range [I_b, I_e), where I_b and I_e are the lower and upper bounds of the i^th dimension. The rank of a multidimensional index space is the number of intervals it represents.

2 A multidimensional index is an element within the a multidimensional index space and can be represented as a pack of integer types. The multidimensional index idx... refers to an element within the domain of a multidimensional index space if both the following are true:

• (2.1) sizeof...(idx) is equal to rank, and

• (2.2) For all i in the range [0,rank), the i^th value of idx is in the range [I_b, I_e).

3 For the following subsections, let r be a value in the range [0,rank).

4 full_extent_t is an empty class type.

22.7.� Class template extents [mdspan.extents]

22.7.�.1 Overview [mdspan.extents.syn]

namespace std {

template<size_t... Extents>
class extents {
public:
  using size_type = size_t;

  // [mdspan.extents.cons], Constructors and assignment
  constexpr extents() noexcept = default;
  constexpr extents(const extents&) noexcept = default;
  constexpr extents& operator=(const extents&) noexcept = default;
  constexpr extents(extents&&) noexcept = default;
  constexpr extents& operator=(extents&&) noexcept = default;

  template<size_t... OtherExtents>
    constexpr extents(const extents<OtherExtents...>&) noexcept;
  template<class... SizeTypes>
    constexpr extents(SizeTypes...) noexcept;
  template<class SizeType>
    constexpr extents(const array<SizeType, rank_dynamic()>&) noexcept;
  template<size_t... OtherExtents>
    constexpr extents& operator=(const extents<OtherExtents...>&) noexcept;

  // [mdspan.extents.obs], Observers of the domain multidimensional index space
  static constexpr size_t rank() noexcept { return sizeof...(Extents); }
  static constexpr size_t rank_dynamic() noexcept
    { return ((Extents == dynamic_extent) + ...); }
  static constexpr size_type static_extent(size_t) noexcept;
  constexpr size_type extent(size_t) const noexcept;

  // [mdspan.extents.compare], extents comparison operators
  template<size_t... OtherExtents>
    friend constexpr bool operator==(const extents&, const extents<OtherExtents...>&) noexcept;

private:
  static constexpr size_t dynamic_index(size_t) noexcept; // exposition only
  array<size_type, rank_dynamic()> dynamic_extents_{}; // exposition only
};

}

22.7.�.2 Overview [mdspan.extents.overview]

1 The class template extents represents a multidimensional index space of rank equal to sizeof...(Extents).

2 extents<Extents...> is a trivially copyable type.

3 E_r is a dynamic extent if it is equal to dynamic_extent, otherwise E_r is a static extent. For each E_r equivalent to dynamic_extent,
the upper bound of the interval is stored in the exposition-only array dynamic_extents_ at dynamic_extents_[dynamic_index(r)].

4 If E_r is a dynamic extent, let D_r be the value of dynamic_extents_[dynamic_index(r)]. The r^th interval of an extents is as follows:

• (4.1) [0, E_r) if E_r is a static extent,

• (4.2) otherwise [0, D_r).

constexpr size_t dynamic_index(size_t i) noexcept; // exposition only

• 5 Returns: If i <= sizeof...(Extents) is true, the number of arguments before the ith template parameter in the template parameter pack Extents equivalent to dynamic_extent. Otherwise, rank_dynamic().

22.7.�.3 Constructors and assignment [mdspan.extents.cons]

template<size_t... OtherExtents>
  constexpr extents(const extents<OtherExtents...>& other) noexcept;

• 1 Constraints:

  • (1.1) sizeof...(OtherExtents) equals rank().

  • (1.2) For all r where static_extent(r) != dynamic_extent and other.static_extent(r) != dynamic_extent are both true, static_extent(r) == other.static_extent(r) is true.

• 2 Preconditions: For all r, static_extent(r) == dynamic_extent || static_extent(r) == other.extent(r) is true.

• 3 Effects: For each r where static_extent(r) == dynamic_extent is true, assigns other.extent(r) to dynamic_extent[dynamic_index(r)].

template<class... SizeTypes>
  constexpr extents(SizeTypes... dynamic) noexcept;

• 4 Constraints:

  • (4.1) (is_convertible_v<SizeTypes, size_type> && ...) is true, and

  • (4.2) sizeof...(SizeTypes) == rank_dynamic() is true.

• 5 Effects: Initializes dynamic_extents_ with dynamic....

template<class SizeType>
constexpr extents(const array<SizeType, rank_dynamic()>& dynamic) noexcept;

• 6 Constraints: is_convertible_v<SizeType, size_type> is true.

• 7 Effects: Initializes dynamic_extents_ with dynamic.

template<size_t... OtherExtents>
constexpr extents& operator=(const extents<OtherExtents...>& other) noexcept;

• 8 Constraints:

  • (10.1) sizeof...(OtherExtents) equals rank().

  • (10.2) For all r where static_extent(r) != dynamic_extent and other.static_extent(r) != dynamic_extent are both true, static_extent(r) == other.static_extent(r) is true.

• 9 Preconditions: For all r, static_extent(r) == dynamic_extent || static_extent(r) == other.extent(r) is true.

• 10 Effects: For each r where static_extent(r) == dynamic_extent is true, assigns other.extent(r) to dynamic_extent[dynamic_index(r)].

• 11 Returns: *this.

22.7.�.3 Observers of the domain multidimensional index space [mdspan.extents.obs]

constexpr size_type static_extent(size_t i) const noexcept;

• 1 Preconditions: i < rank() is true.

• 2 Returns: the i^th value of Extents.

constexpr size_type extent(size_t i) const noexcept;

• 3 Preconditions: i < rank() is true.

• 4 Returns: dynamic_extents_[dynamic_index(i)] if static_extent(i) == dynamic_extent is true, otherwise static_extent(i).

22.7.�.4 extents comparison operators [mdspan.extents.compare]

template<size_t... OtherExtents>
  friend constexpr bool operator==(const extents& lhs, const extents<OtherExtents...>& rhs) noexcept;

• 1 Returns: true if lhs.rank() equals rhs.rank() and lhs.extents(r) equals rhs.extents(r) for all r, otherwise false.

22.7.� Layout mapping policy [mdspan.layout]

22.7.�.1 Layout mapping requirements [mdspan.layout.reqs]

1. A layout mapping policy is a class that contains a layout mapping, which is a nested class template.

2. A layout mapping policy and its layout mapping nested class template meet the requirements in Table �.

3. A layout mapping meets the requirements of Cpp17DefaultConstructible, Cpp17CopyAssignable, and Cpp17EqualityComparable.

4. In Table �:

   • MP denotes a layout mapping policy.
   • M denotes a specialization of the layout mapping policy’s nested layout mapping template class.
   • E denotes a specialization of extents.
   • e denotes an object of type E.
   • m denotes an object of type M.
   • i... and j... are multidimensional indices in the multidimensional index space defined by e.
   • r is an integral value in the range [0, e.rank()).
   • dr... is an integer pack where sizeof...(dr) == e.rank() is true and the rth element is equal to 1 and all other elements are 0.

22.7.�.2 Class template layout_left [mdspan.layout.left]

1 layout_left meets the requirements of layout mapping policy.

2 layout_left is a trivially copyable type. layout_left::mapping<Extents> is a trivially copyable type.

3 layout_left gives a layout mapping where the left-most extent is stride one and strides increase left-to-right as the product of extents.

4 If Extents is not a (possibly cv-qualified) specialization of extents, then the program is ill-formed.

namespace std {

struct layout_left {
  template<class Extents>
  class mapping {
  public:
    using size_type = typename Extents::size_type;
    using extents_type = Extents;
    using layout_type = layout_left;

    constexpr mapping() noexcept = default;
    constexpr mapping(const mapping&) noexcept = default;
    constexpr mapping(mapping&&) noexcept = default;
    constexpr mapping(const Extents&) noexcept;
    template<class OtherExtents>
      constexpr mapping(const mapping<OtherExtents>&) noexcept;

    constexpr mapping& operator=(const mapping&) noexcept = default;
    constexpr mapping& operator=(mapping&&) noexcept = default;
    template<class OtherExtents>
      constexpr mapping& operator=(const mapping<OtherExtents>&) noexcept;

    constexpr Extents extents() const noexcept { return extents_; }

    constexpr size_type required_span_size() const noexcept;

    template<class... Indices>
      constexpr size_type operator()(Indices...) const noexcept; 

    static constexpr bool is_always_unique() noexcept { return true; }
    static constexpr bool is_always_contiguous() noexcept { return true; }
    static constexpr bool is_always_strided() noexcept { return true; }

    constexpr bool is_unique() const noexcept { return true; }
    constexpr bool is_contiguous() const noexcept { return true; }
    constexpr bool is_strided() const noexcept { return true; }

    constexpr size_type stride(size_t) const noexcept;

    template<class OtherExtents>
      friend constexpr bool operator==(const mapping&, const mapping<OtherExtents>&) noexcept;

  private:
    Extents extents_{}; // exposition only
  };
};
}

22.7.�.2.1 layout_left::mapping members [mdspan.layout.layout_left]

constexpr mapping(const Extents& e) noexcept;

• 1 Effects: Initializes extents_ with e.

template<class OtherExtents>
  constexpr mapping(const mapping<OtherExtents>& other) noexcept;

• 2 Constraints: is_convertible_v<OtherExtents,Extents> is true.

• 3 Effects: Initializes extents_ with other.extents().

template<class OtherExtents>
  constexpr mapping& operator=(const mapping<OtherExtents>& other) noexcept;

• 4 Effects: Equivalent to:

  extents_ = other.extents();
  return *this;

constexpr size_type required_span_size() const noexcept;

• 5 Returns: If Extents::rank() > 0 is true, the product of extents().extent(r) for all r in the range [0, Extents::rank() ). Otherwise, 1.

template<class... Indices> 
  constexpr size_type operator()(Indices... i) const noexcept;

• 6 Constraints:

  • (6.1) sizeof...(Indices) == Extents::rank() is true, and

  • (6.1) (is_convertible_v<Indices, size_type> && ...) is true.

• 7 Preconditions: 0 ≤ array{i...}[r] < extents().extent(r) for all r in the range [0, Extents::rank() ).

• 8 Effects: Let P... be the parameter pack such that is_same_v<make_index_sequence<size_type, sizeof...(Indices)>, integer_sequence<size_type, P...>> is true.
  Equivalent to: return Extents::rank() > 0 ? (i*stride(P()) + ...) : 0;

constexpr size_type stride(size_t r) const;

• 9 Returns: 1 if r equals zero, otherwise, the product of extents().extent(k) for all k in the range [0, r ).

template<class OtherExtents>
  friend constexpr bool operator==(const mapping& x, const mapping<OtherExtents>& y) noexcept;

• 10 Effects: Equivalent to: return x.extents() == y.extents();.

22.7.�.3 Class template layout_right [mdspan.layout.right]

1 layout_right meets the requirements of layout mapping policy.

2 layout_right is a trivially copyable type. layout_right::mapping<Extents> is a trivially copyable type.

3 The layout mapping property layout_right gives a layout mapping where the right-most extent is stride one and strides increase right-to-left as the product of extents.

4 If Extents is not a (possibly cv-qualified) specialization of extents, then the program is ill-formed.

namespace std {

struct layout_right {
  template<class Extents>
  class mapping {
  public:
    using size_type = typename Extents::size_type;
    using extents_type = Extents;
    using layout_type = layout_right;

    constexpr mapping() noexcept = default;
    constexpr mapping(const mapping&) noexcept = default;
    constexpr mapping(const Extents&) noexcept;
    constexpr mapping(mapping&&) noexcept = default;
    template<class OtherExtents>
      constexpr mapping(const mapping<OtherExtents>&) noexcept;

    constexpr mapping& operator=(const mapping&) noexcept = default;
    constexpr mapping& operator=(mapping&&) noexcept = default;
    template<class OtherExtents>
      constexpr mapping& operator=(const mapping<OtherExtents>&) noexcept;

    constexpr Extents extents() const noexcept { return extents_; }

    constexpr size_type required_span_size() const noexcept;

    template<class... Indices>
      constexpr size_type operator()(Indices...) const noexcept;

    static constexpr bool is_always_unique() noexcept { return true; }
    static constexpr bool is_always_contiguous() noexcept { return true; }
    static constexpr bool is_always_strided() noexcept { return true; }

    constexpr bool is_unique() const noexcept { return true; }
    constexpr bool is_contiguous() const noexcept { return true; }
    constexpr bool is_strided() const noexcept { return true; }

    constexpr size_type stride(size_t) const noexcept;

    template<class OtherExtents>
      friend constexpr bool operator==(const mapping&, const mapping<OtherExtents>&) noexcept;

  private:
    Extents extents_{}; // exposition only
  };
};
}

22.7.�.3.1 layout_right::mapping members [mdspan.layout.layout_right]

constexpr mapping(const Extents& e) noexcept;

• 1 Effects: Initializes extents_ with e.

template<class OtherExtents>
  constexpr mapping(const mapping<OtherExtents>& other) noexcept;

• 2 Constraints: is_convertible_v<OtherExtents,Extents> is true.

• 3 Effects: Initializes extents_ with other.extents().

template<class OtherExtents>
  constexpr mapping& operator=(const mapping<OtherExtents>& other) noexcept;

• 4 Effects: Equivalent to:

  extents_ = other.extents();
  return *this;

size_type required_span_size() const noexcept;

• 5 Returns: If Extents::rank() > 0 is true, the product of extents().extent(r) for all r in the range [0, Extents::rank() ). Otherwise, 1.

template<class... Indices> 
  constexpr size_type operator()(Indices... i) const noexcept;

• 6 Constraints:

  • (6.1) sizeof...(Indices) == Extents::rank() is true, and

  • (6.1) (is_convertible_v<Indices, size_type> && ...) is true.

• 7 Preconditions: 0 ≤ array{i...}[r] < extents().extent(r) for all r in the range [0, Extents::rank() ).

• 8 Effects: Let P... be the parameter pack such that is_same_v<make_index_sequence<size_type, sizeof...(Indices)>, integer_sequence<size_type, P...>> is true.
  Equivalent to: return Extents::rank() > 0 ? (i*stride(P()) + ...) : 0;

constexpr size_type stride(size_t r) const noexcept;

• 9 Returns: 1 if r equals Extents::rank()-1, otherwise, the product of extents().extent(k) for all k in the range [ r+1 , Extents::rank() ).

template<class OtherExtents>
  friend constexpr bool operator==(const mapping& x, const mapping<OtherExtents>& y) noexcept;

• 10 Effects: Equivalent to: return x.extents() == y.extents();.

22.7.�.4 Class template layout_stride [mdspan.layout.stride]

1 layout_stride meets the requirements of layout mapping policy.

2 layout_stride is a trivially copyable type. layout_stride::mapping<Extents> is a trivially copyable type.

3 The layout mapping property layout_stride gives a layout mapping where the strides are user defined.

4 If Extents is not a (possibly cv-qualified) specialization of extents, then the program is ill-formed.

namespace std {

struct layout_stride {
  template<class Extents>
  class mapping {
  public:
    using size_type = typename Extents::size_type;
    using extents_type = Extents;
    using layout_type = layout_stride;

    constexpr mapping() noexcept = default;
    constexpr mapping(const mapping&) noexcept = default;
    constexpr mapping(mapping&&) noexcept = default;
    constexpr mapping(const Extents&,
                      const array<size_type, Extents::rank()>&) noexcept;
    template<class OtherExtents>
      constexpr mapping(const mapping<OtherExtents>&) noexcept;
    template<class LayoutPolicy, class OtherExtents>
      constexpr mapping(const LayoutPolicy::template mapping<OtherExtents>&) noexcept;

    constexpr mapping& operator=(const mapping&) noexcept = default;
    constexpr mapping& operator=(mapping&&) noexcept = default;
    template<class OtherExtents>
      constexpr mapping& operator=(const mapping<OtherExtents>&) noexcept;

    constexpr Extents extents() const noexcept { return extents_; }
    constexpr array<typename size_type, Extents::rank()> strides() const noexcept
    { return strides_; }

    constexpr size_type required_span_size() const noexcept;

    template<class... Indices>
      constexpr size_type operator()(Indices...) const noexcept ;

    static constexpr bool is_always_unique() noexcept { return true; }
    static constexpr bool is_always_contiguous() noexcept { return false; }
    static constexpr bool is_always_strided() noexcept { return true; }

    constexpr bool is_unique() const noexcept { return true; }
    constexpr bool is_contiguous() const noexcept;
    constexpr bool is_strided() const noexcept { return true; }

    constexpr size_type stride(size_t) const noexcept;

    template<class OtherExtents>
      friend constexpr bool operator==(const mapping&, const mapping<OtherExtents>&) noexcept;

  private:
    Extents extents_{}; // exposition only
    array<size_type, Extents::rank()> strides_{}; // exposition only
  };
};
}

22.7.�.4.1 layout_stride::mapping members [mdspan.layout.layout_stride]

constexpr mapping(const Extents& e, array<size_type, Extents::rank()> s) noexcept;

1 Let P be a permutation of the integers 0, ..., Extents::rank()-1 and let p_i be the i^th element of P.

• 2 Preconditions:

  • (2.1)s[i] > 0 is true for all i in the range [0, Extents::rank() ).

  • (2.2) If Extents::rank() is greater than zero, then there exists a permutation P such that s[ p_i ] >= s( p_i − 1 ]) * e.extent( p_i − 1 ) is true for all i in the range [1, Extents::rank() ).

• 3 Effects: Initializes extents_ with e, and initializes strides_ with s.

template<class OtherExtents>
  constexpr mapping(const mapping<OtherExtents>& other) noexcept;

• 4 Constraints: is_convertible_v<OtherExtents,Extents> is true.

• 5 Effects: Initializes extents_ with other.extents(), and initializes strides_ with other.strides().

template<class LayoutPolicy, class OtherExtents>
  constexpr mapping(const LayoutPolicy::template mapping<OtherExtents>& other) noexcept;

• 6 Constraints:

  • (6.1) is_convertible_v<OtherExtents,Extents> is true.

  • (6.2) LayoutPolicy meets the layout mapping policy requirements.

  • (6.3) other.is_always_unique() is true.

  • (6.4) other.is_always_strided() is true.

• 7 Effects: Initializes extents_ with other.extents(), and initializes strides_ such that strides_[r] == other.stride(r) is true for all r in the range [0, Extents::rank() ).

template<class OtherExtents>
  constexpr mapping& operator=(const mapping<OtherExtents>& other) noexcept;

• 8 Effects: Equivalent to:

  extents_ = other.extents();
  return *this;

constexpr size_type required_span_size() const noexcept;

• 9 Returns: If (Extents::rank() > 0) is true and (extents().extent(r) > 0) is true for all r in the range [0, Extents::rank() ), the maximum of extents().extent(r) * stride(r) for all r in the range [0, Extents::rank() ). Otherwise, Extents::rank() == 0 ? 1 : 0.

template<class... Indices> 
  constexpr size_type operator()(Indices... i) const noexcept;

• 10 Constraints:

  • (10.1) sizeof...(Indices) == Extents::rank() is true, and

  • (10.1) (is_convertible_v<Indices, size_type> && ...) is true.

• 11 Preconditions: 0 ≤ array{i...}[r] < extents().extent(r) for all r in the range [0, Extents::rank() ).

• 12 Effects: Let P... be the parameter pack such that is_same_v<make_index_sequence<size_type, sizeof...(Indices)>, integer_sequence<size_type, P...>> is true.
  Equivalent to: return Extents::rank() > 0 ? (i*stride(P()) + ...) : 0;

constexpr bool is_contiguous() const noexcept;

• 13 Let P be a permutation of the integers 0, ..., Extents::rank()-1 and let p_i be the i^th element of P.

• 14Returns:

  • (14.1) true if Extents::ranks() is zero.

  • (14.2) Otherwise, true if there is a permutation P such that min(stride( p_i ) equals one for i in the range [0, Extents::rank() ), and stride( p_i ) equals stride( p_i − 1 ) * extents().extent( p_i − 1 ) for i in the range [1, Extents::rank() ).

  • (14.3) Otherwise, false.

template<class OtherExtents>
  friend constexpr bool operator==(const mapping& x, const mapping<OtherExtents>& y) noexcept;

• 15 Effects: Equivalent to: return x.extents() == y.extents();.

22.7.� Accessor Policy [mdspan.accessor]

1 An accessor policy defines types and operations by which a contiguous set of objects are accessed.

22.7.�.1 Accessor policy requirements [mdspan.accessor.reqs]

2 An accessor policy defines:

• (2.1) a handle to a single element of type element_type;

• (2.2) a handle to a contiguous set of elements of type element_type, accessible through the policy’s access method;

• (2.3) conversion of a handle to a contiguous set of elements, to a pointer [conv.array]; and

• (2.4) getting a handle to the contiguous subset of elements beginning at an integer offset value.

3 [Note: The type of reference need not be element_type&. The type of pointer need not be element_type*. — end note]

4 An accessor policy meets the requirements of Cpp17DefaultConstructible, Cpp17CopyAssignable, and Cpp17EqualityComparable.

5 In Table �:

• (5.1) A denotes an accessor policy.

• (5.2) a denotes an object of type A.

• (5.3) p denotes an object of type A::pointer.

• (5.4) i and j each denote a size_t value.

22.7.�.2 Class template default_accessor [mdspan.accessor.default]

1 default_accessor meets the requirements of accessor policy.

2 ElementType is required to be a complete object type that is neither an abstract class type nor an array type.

namespace std {
template<class ElementType>
  struct default_accessor {
    using offset_policy = default_accessor;
    using element_type = ElementType;
    using reference = ElementType&;
    using pointer = ElementType*;

    constexpr default_accessor() noexcept = default;

    template<class OtherElementType>
    constexpr default_accessor(default_accessor<OtherElementType>) noexcept {}

    constexpr typename offset_policy::pointer
      offset(pointer p, size_t i) const noexcept;

    constexpr reference access(pointer p, size_t i) const noexcept;
  };
}

22.7.�.2 Class template default_accessor members [mdspan.accessor.members]

template<class OtherElementType>
constexpr default_accessor(default_accessor<OtherElementType>) noexcept {}

• 1 Constraints:

  • (1.1) is_convertible_v<typename default_accessor<OtherElementType>::pointer, pointer> is true.

constexpr typename offset_policy::pointer
  offset(pointer p, size_t i) const noexcept;

• 2 Preconditions: p + i is dereferenceable.

• 3 Returns: p + i.

constexpr reference access(pointer p, size_t i) const noexcept;

• 4 Preconditions: p + i is dereferenceable.

• 5 Returns: p[i].

22.7.� Class template mdspan [mdspan.basic]

22.7.�.1 mdspan overview [mdspan.basic.overview]

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

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

3 The codomain of an mdspan object is a contiguous span of elements.

4 As with span, the storage of the objects in the codomain of an mdspan is owned by some other object.

namespace std {

template<class ElementType, class Extents, class LayoutPolicy, class AccessorPolicy>
class mdspan {
public:

  // Domain and codomain types
  using extents_type = Extents;
  using layout_type = LayoutPolicy;
  using accessor_type = AccessorPolicy;
  using mapping_type = typename layout_type::template mapping_type<extents_type>;
  using element_type = ElementType;
  using value_type = remove_cv_t<element_type>;
  using size_type = size_t ;
  using difference_type = ptrdiff_t;
  using pointer = typename accessor_type::pointer;
  using reference = typename accessor_type::reference;

  // [mdspan.basic.cons], mdspan constructors, assignment, and destructor
  constexpr mdspan() = default;
  constexpr mdspan(const mdspan& rhs) = default;
  constexpr mdspan(mdspan&& rhs) = default;

  template<class... SizeTypes>
    explicit constexpr mdspan(pointer ptr, SizeTypes... dynamic_extents);
  template<class SizeType, size_t N>
  constexpr mdspan(pointer p, const array<SizeType, N>& dynamic_extents);
  constexpr mdspan(pointer p, const Extents& ext);
  constexpr mdspan(pointer p, const mapping_type& m);
  constexpr mdspan(pointer p, const mapping_type& m, const accessor_type& a);

  template<class OtherElementType, class OtherExtents, class OtherLayoutPolicy, class OtherAccessorPolicy>
    constexpr mdspan(
      const mdspan<OtherElementType, OtherExtents, OtherLayoutPolicy, OtherAccessorPolicy>& other);

  constexpr mdspan& operator=(const mdspan& rhs) = default;
  constexpr mdspan& operator=(mdspan&& rhs) = default;
  template<class OtherElementType, class OtherExtents, class OtherLayoutPolicy, class OtherAccessorPolicy>
    constexpr mdspan& operator=(
      const mdspan<OtherElementType, OtherExtents, OtherLayoutPolicy, OtherAccessorPolicy>& other);

  // [mdspan.basic.mapping], mdspan mapping domain multidimensional index to access codomain element
  template<class... SizeTypes>
    constexpr reference operator[](SizeTypes... indices) const;
  template<class SizeType, size_t N>
    constexpr reference operator[](const array<SizeType, N>& indices) const;

  constexpr accessor_type accessor() const { return acc_; }

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

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

  // [mdspan.basic.codomain], mdspan observers of the codomain
  constexpr pointer data() const { return ptr_; }

  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 mapping_type mapping() const { return map_; }
  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:
  accessor_type acc_{}; // exposition only
  mapping_type map_{}; // exposition only
  pointer ptr_{}; // exposition only
};

}

5 mdspan<ElementType, Extents, LayoutPolicy, AccessorPolicy> is a trivially copyable type if AccessorPolicy, LayoutPolicy::mapping_type<Extents> and AccessorPolicy::pointer are trivially copyable types. mdspan<ElementType, Extents, LayoutPolicy, AccessorPolicy> is a trivially default constructible type if AccessorPolicy, LayoutPolicy::mapping_type<Extents> and AccessorPolicy::pointer are trivially default constructible types.

6 ElementType is required to be a complete object type that is neither an abstract class type nor an array type.

7 If Extents is not a (cv-unqualified) specialization of extents, then the program is ill-formed.

8 If LayoutPolicy does not meet the layout mapping policy requirements, then the program is ill-formed.

9 If AccessorPolicy does not meet the accessor policy requirements or if is_same_v<typename AccessorPolicy::element_type,ElementType> is false, then the program is ill-formed.

22.7.�.1 mdspan constructors and assignment operators [mdspan.basic.cons]

template<class... SizeTypes>
  explicit constexpr mdspan(pointer ptr, SizeTypes... dynamic_extents);

• 1 Constraints:

  • (1.1) (is_convertible_v<SizeTypes, size_type> && ...) is true,

  • (1.2) sizeof...(dynamic_extents) == rank_dynamic() is true,

  • (1.3) is_constructible_v<mapping_type, Extents> is true, and

  • (1.4) is_default_constructible_v<accessor_type> is true.

• 2 Effects:

  • (2.1) Initializes ptr_ with ptr, and

  • (2.2) Initializes map_ with Extents(dynamic_extents...).

template<class SizeType, size_t N>
  constexpr mdspan(pointer p, const array<SizeType, N>& dynamic_extents);

• 3 Constraints:

  • (3.1) is_convertible_v<SizeType, size_type> is true,

  • (3.2) N == rank_dynamic() is true,

  • (3.3) is_constructible_v<mapping_type, Extents> is true, and

  • (3.4) is_default_constructible_v<accessor_type> is true.

• 4 Effects: Equivalent to: mdspan(p, dynamic_extents[Rs]...), with Rs... from index_sequence<Rs...> matching make_index_sequence<N>.

constexpr mdspan(pointer p, const Extents& ext);

• 5 Constraints:

  • (5.1) is_constructible_v<mapping_type, Extents> is true, and

  • (5.2) is_default_constructible_v<accessor_type> is true.

• 6 Effects: Equivalent to: mdspan(p, mapping_type(ext)).

constexpr mdspan(pointer p, const mapping_type& m);

• 7 Constraints: is_default_constructible_v<accessor_type> is true.

• 8 Effects:

  • (8.1) Initializes ptr_ with p, and

  • (8.2) Initializes map_ with m.

constexpr mdspan(pointer p, const mapping_type& m, const accessor_type& a);

• 9Effects:

  • (9.1) Initializes ptr_ with p,

  • (9.2) Initializes map_ with m, and

  • (9.3) Initializes acc_ with a.

template<class OtherElementType, class OtherExtents, class OtherLayoutPolicy, class OtherAccessor>
  constexpr mdspan(const mdspan<OtherElementType, OtherExtents, OtherLayoutPolicy, OtherAccessor>& other);

• 10 Constraints:

  • (10.1) is_convertible_v<OtherLayoutPolicy::template mapping<OtherExtents>, mapping_type is true;

  • (10.2) is_convertible_v<OtherAccessor, Accessor> is true;

  • (10.3) is_convertible_v<OtherAccessor::pointer, pointer> is true;

  • (10.4) is_convertible_v<OtherExtents, extents_type> is true; and

  • (10.5) For all r in the range [0, rank()), if other.static_extent(r) != dynamic_extent && static_extent(r) != dynamic_extent is true, then other.static_extent(r) == static_extent(r) is true.

• 11 Preconditions: For all r in the range [0, rank()), if other.static_extent(r) == dynamic_extent || static_extent(r) == dynamic_extent is true, then other.extent(r) == extent(r) is true.

• 12 Effects:

  • (12.1) Initializes ptr_ with other.ptr_,

  • (12.2) initializes map_ with other.map_, and

  • (12.3) initializes acc_ with other.acc_.

template<class OtherElementType, class OtherExtents, class OtherLayoutPolicy, class OtherAccessor>
  constexpr mdspan& operator=(
    const mdspan<OtherElementType, OtherExtents, OtherLayoutPolicy, OtherAccessor>& other);

• 13 Constraints:

  • (13.1) is_assignable_v<mapping_type, OtherLayoutPolicy::template mapping<OtherExtents>> is true;

  • (13.2) is_assignable_v<Accessor, OtherAccessor> is true;

  • (13.3) is_assignable_v<pointer, OtherAccessor::pointer> is true;

  • (13.4) OtherExtents::rank() == rank() is true; and

  • (13.5) For all r in the range [0, rank()), if other.static_extent(r) != dynamic_extent && static_extent(r) != dynamic_extent is true, then other.static_extent(r) == static_extent(r) is true.

• 14 Preconditions: For all r in the range [0, rank()), if other.static_extent(r) == dynamic_extent || static_extent(r) == dynamic_extent is true, then other.extent(r) == extent(r) is true.

• 15 Effects:

  • (15.1) Assigns other.ptr_ to ptr_,

  • (15.2) Assigns other.map_ to map_, and

  • (15.3) Assigns other.acc_ to acc_.

22.7.�.2 mdspan members [mdspan.basic.members]

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

• 1 Constraints:

  • (1.1) (is_convertible_v<SizeTypes, size_type> && ...) is true, and

  • (1.2) sizeof...(SizeTypes) == rank() is true.

• 2 Preconditions: acc_.access(ptr_, map_(indices...)) shall be valid.

• 3 Effects: Equivalent to: return acc_.access(ptr_, map_(indices...));.

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

• 4 Constraints:

  • (4.1) is_convertible_v<SizeType, size_type> is true, and

  • (4.2) rank() == N is true.

• 5 Effects: Equivalent to: return apply(*this, indices);.

constexpr size_type size() const;

• 6 Returns: Product of extent(r) for all r in the range [0, Extents::rank()).

constexpr size_type unique_size() const;

• 7 Returns: The number of unique elements in the codomain. [Note:_ If mapping().is_unique() is true, this is identical to size(). _—end note]

22.7.� submdspan [mdspan.submdspan]

1 submdspan creates an mdspan with a domain that is a subset of the input mdspan’s domain, and a codomain that is a subset of the input mdspan’s codomain.

2 The SliceSpecifier template argument(s) and the corresponding value(s) of the arguments of submdspan after src determine the subset of src that the mdspan returned by submdspan views.

namespace std {

  // [mdspan.submdspan], submdspan creation
  template<class ElementType, class Extents, class LayoutPolicy,
           class AccessorPolicy, class... SliceSpecifiers>
      constexpr mdspan<see below>
      submdspan(const mdspan<ElementType, Extents, LayoutPolicy,
                             AccessorPolicy>& src, SliceSpecifiers... slices);
}

3 Let sub be the return value of submdspan(src, slices...), let s_k be the k-th element of slices..., and let S_k be the type of the k-th element of slices....

4 Define map_rank as an array<size_t,src.rank()> such that map_rank[j] equals dynamic_extent if is_convertible_v<S_j,size_t> is true, or else map_rank[j] equals the number of S_k with k < j such that is_convertible_v<S_k,pair<size_t,size_t>> || is_convertible_v<S_k,full_extent_t> is true.

5 Let first and last be exposition-only variables of type array<size_t,src.rank()>. For r in the range [0, src.rank()), define the values of first[r] and last[r] as follows:

• if is_convertible_v<S_r,size_t>, then first[r] equals s_r, and last[r] equals first[r] + 1;
• otherwise, if is_convertible_v<S_r,pair<size_t,size_t>>, then first[r] equals p.first, and last[r] equals p.second, where p is the result of converting s_r to pair<size_t,size_t>;
• otherwise, if is_convertible_v<S_r,full_extent_t>, then first[r] equals 0, and last[r] equals src.extent(r).

6 Constraints:

• sizeof(slices...) equals src.rank(),
• LayoutPolicy is layout_left, layout_right, layout_stride, or any type in a possibly empty set of implementation-defined types, each of which meets the requirements of a layout mapping policy [mdspan.layout.reqs] [Note: Implementation and user defined layout mapping policies could exist, for which taking an arbitrary submdspan does not make sense. — end note]; and
• For all k in the range [0, src.rank()), is_convertible_v<S_k,size_t> || is_convertible_v<S_k,pair<size_t,size_t>> || is_convertible_v<S_k,full_extent_t> is true.

7 Preconditions:

• For 0 ≤ r < src.rank(), 0 <= first[r] && first[r] <= last[r] && last[r] <= src.extent(r) is true.

8 Ensures: All of the following:

• sub.rank() equals the number of k such that is_convertible_v<S_k,pair<size_t,size_t>> || is_convertible_v<S_k,full_extent_t> is true.
• Let the pack i... denote a multidimensional index in the domain of src with i_k denoting the k-th element of i..., such that i_k is greater than or equal to first[k] and i_k is less than last[k] for all k in the range [0,src.rank()). Let the pack j... denote a multidimensional index in the domain of sub with j_s denoting the s-th element of j..., such that j_s is equal to i_k minus first[k] where map_rank[k] equals s for all s in the range [0,sub.rank()). Then sub(j...) and src(i...) refer to the same element in the codomain of src.
• For 0 ≤ k < src.rank(), if map_rank[k] != -1 is true, then sub.extent(map_rank[k]) equals last[k] - first[k].
• If src.is_strided() is true, then sub.is_strided() is true, and for all k in the range [0, src.rank()), if map_rank[k] != -1 is true, then sub.stride(map_rank[k]) equals src.stride(k).
• For all k in the range [0, src.rank()), if map_rank[k] != -1 is true and src.static_extent(k) does not equal dynamic_extent and is_convertible_v<S_k,full_extent_t> is true, then sub.static_extent(map_rank[k]) equals src.static_extent(k).

[Note:* Example of submdspan use:]

5 Next Steps

We would like LEWG to poll on sending P0009 (‘mdspan’) to LWG for C++23.

6 Implementation

There is an mdspan implementation available at https://github.com/kokkos/mdspan/.

7 Related Work

The original version of this paper, N4355, predates the “P” naming for papers.

Related papers:

• P0122 : span: bounds-safe views for sequences of objects The mdspan codomain concept of span is well-aligned with this paper.
• P0367 : Accessors: The P0367 Accessors proposal includes polymorphic mechanisms for accessing the memory an object or span of objects. The AccessorPolicy extension point in this proposal is intended to include such memory access properties.
• P0331 : Motivation and Examples for Multidimensional Array
• P0332 : Relaxed Incomplete Multidimensional Array Type Declaration
• P0454 : Wording for a Minimal mdspan Included proposed modification of span to better align span with mdspan.
• P0546 : Preparing span for the future Proposed modification of span
• P0856 : Restrict access property for mdspan and span
• P0860 : atomic access policy for mdspan
• P0900 : An Ontology of Properties for mdspan
• P2128 : Multidimensional subscript operator
• P2299 : mdspan and CTAD