P0009R8: <code>mdspan</code>: A Non-Owning Multidimensional Array Reference

1. Revision History

1.1. P0009r8: Pre 2018-11-SanDiego Mailing

Refinement based upon updated prototype / reference implementation

1.2. 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

1.3. 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=...>

SF	F	N	A	SA
1	1	1	2	8

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...>

SF	F	N	A	SA
3	4	5	1	0

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...>

SF	F	N	A	SA
1	2	2	6	2

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.

1.4. 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).

SF	F	N	A	SA
2	11	1	1	0

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

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

SF	F	N	A	SA
0	8	7	0	0

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.

1.5. 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?

SF	F	N	A	SA
1	6	2	6	3

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.

1.6. 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?

SF	F	N	A	SA
0	2	4	3	1

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

SF	F	N	A	SA
0	1	3	2	3

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

SF	F	N	A	SA
9	1	0	0	0

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.

1.7. 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.

1.8. P0009r1 : Pre 2016-02-Jacksonville Mailing

LEWG review at 2015-10-Kona.

LEWG Poll: What should this feature be called?

Name	#
`view`	5
`span`	9
`array_ref`	6
`slice`	6
`array_view`	6
`ref`	0
`array_span`	7
`basic_span`	1
`object_span`	3
`field`	0

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

SF	F	N	A	SA
3	4	2	3	0

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

Syntax	#
`view<int[3][0][][5], property1>`	12
`view<int, dimension<3, 0, dynamic_extent, 5>, property1>`	4
`view<int[3][0][dynamic_extent][5], property1>`	5
`view<int, 3, 0, dynamic_extent, 5, property1>`	4
`view<int, 3, 0, dynamic_extent, 5, properties<property1>>`	2
`view<arr<int, 3, 0, dynamic_extent, 5>, property1>`	4
`view<int[3][0][][5], properties<property1>>`	9

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.

SF	F	N	A	SA
3	6	3	0	0

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

SF	F	N	A	SA
3	4	1	2	1

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.

1.9. P0009r0 : Pre 2015-10-Kona Mailing

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

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

SF	F	N	A	SA
5	3	2	2	0

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

SF	F	N	A	SA
0	10	1	5	0

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

2. Description

The proposed polymorphic multidimensional array reference (mdspan) defines types and functions for mapping multidimensional indices in its domain, a multidimensional index space, to the mdspan's 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. A multidimensional index or multi-index is a member of a multidimensional index space. An mdspan has two policies: the layout mapping and the accessor. The layout mapping specifies the formula, and properties of the formula, for mapping a multi-index from the domain to an element in the codomain. The accessor is an extension point that allows modification of how elements are accessed. For example, P0367 proposed a rich set of potential access properties.

A multidimensional array is not an array-of-array-of-array-of...

The multidimensional array abstraction has been fundamental to numerical computations for over five decades. However, the C/C++ language provides only a one-dimensional array abstraction which can be composed into array-of-array-of-array-of... types. While such types have some similarity to multidimensional arrays, they do not provide adequate multidimensional array functionality of this proposal. Two critical functionality differences are (1) multiple dynamic extents and (2) polymorphic mapping of multi-indices to element objects.

Optimized Implementation of Layout Mapping

We intend the layout mapping of a multi-index to be a constant-time constexpr operation that is trivially inlined and optimized when possible. Compiler vendors may apply optimizations such as loop invariant code motion, including partial evaluation of multi-index layout mappings when indices are loop invariant.

3. Editing Notes

The proposed changes are relative to the working draft of the standard as of N4750.

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].

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

4. Wording

Text in blockquotes is not proposed wording.
The � character is used to denote a placeholder section number which the editor shall determine.

Copy the entire [views] subclause from the current draft, since it is needed for basic_mdspan.

Add the following paragraphs to [views.general]:

�. The header <mdspan> defines the view basic_mdspan, the type alias mdspan, and other facilities for interacting with these views. The basic_mdspan object maps a multidimensional index within its domain to a reference of an element in the codomain. The domain of a basic_mdspan object is a multidimensional index space. 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. A multidimensional index or multi-index is a member of a multidimensional index space. The codomain of a basic_mdspan object is a span of elements.

�. The subspan function generates a basic_mdspan with a domain contained within the input basic_mdspan domain, and a codomain contained within the input basic_mdspan codomain.

Add the following subclauses to the end of the [views] subclause in the current draft:

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

namespace std {
namespace experimental {
namespace fundamentals_v3 {
  // [mdspan.extents], class template extents
  template<ptrdiff_t... StaticExtents>
    class extents;

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

  // [mdspan.accessor.basic]
  template<class ElementType>
  class accessor_basic;

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

  template<class T, ptrdiff_t... Extents>
    using mdspan = basic_mdspan<T, extents<Extents...>>;

  // [mdspan.extents.compare], extents comparison operators
  template<ptrdiff_t... LHS, ptrdiff_t... RHS>
    constexpr bool operator==(const extents<LHS...>& lhs, const extents<RHS...>& rhs) noexcept;
  template<ptrdiff_t... LHS, ptrdiff_t... RHS>
    constexpr bool operator!=(const extents<LHS...>& lhs, const extents<RHS...>& rhs) noexcept;

  // [mdspan.subspan], subspan creation
  template<class ElementType, class Extents, class LayoutPolicy,
           class AccessorPolicy, class... SliceSpecifiers>
  struct mdspan_subspan { // exposition only
    using extents_t = see below;
    using layout_t = see below;
    using type = basic_mdspan<ElementType, extents_t, layout_t,
                              typename AccessorPolicy::offset_policy>;
  };

  template<class ElementType, class Extents, class LayoutPolicy,
           class AccessorPolicy, class... SliceSpecifiers>
  using mdspan_subspan_t = // exposition only
    typename mdspan_subspan<ElementType, Extents, LayoutPolicy,
                            AccessorPolicy, SliceSpecifiers...>::type;

  template<class ElementType, class Extents, class LayoutPolicy,
           class AccessorPolicy, class... SliceSpecifiers>
    mdspan_subspan_t<ElementType, Extents, LayoutPolicy,
                     AccessorPolicy, SliceSpecifiers...>
      subspan(const basic_mdspan<ElementType, Extents, LayoutPolicy,
                                 AccessorPolicy>& src, SliceSpecifiers ... slices) noexcept;

  // tag supporting subspan
  struct all_type { explicit all_type() = default; };
  inline constexpr all_type all = all_type{};
}}}

26.7.� Class template extents [mdspan.extents]

26.7.�.1 Overview [mdspan.extents.overview]

An extents object defines a multidimensional index space.
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.
The dynamic extents of an extents object correspond to the template arguments in the Extents parameter pack that equal dynamic_extent. Let DynamicRank[i] denote the index of the ith dynamic extent in the Extents template parameter pack, and let DynamicIndex[r] indicate the number of dynamic extents in the first r entries of the Extents parameter pack.
An extents object is expected to store dynamic extents. [Note: An implementation should not consume storage for static extents. — end note]
If any template arguments in Extents are negative and not equal to dynamic_extent, the program is ill-formed.

namespace std {
namespace experimental {
namespace fundamentals_v3 {

template<ptrdiff_t... Extents>
class extents {
public:
  // types
  using index_type = ptrdiff_t;

  // [mdspan.extents.cons], Constructors and assignment
  constexpr extents() noexcept;
  constexpr extents(const extents&) noexcept;
  constexpr extents(extents&&) noexcept;
  template<class... IndexType>
  constexpr extents(IndexType... dynamic_extents) noexcept;
  template<class IndexType, size_t rank_dynamic>
  constexpr extents(const array<IndexType, rank_dynamic>&) noexcept;
  template<ptrdiff_t... OtherExtents>
  constexpr extents(const extents<OtherExtents...>& other);
  ~extents() = default;

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

  // [mdspan.extents.obs], Observers of the domain multidimensional index space
  static constexpr size_t rank() noexcept;
  static constexpr size_t rank_dynamic() noexcept;
  static constexpr index_type static_extent(size_t) noexcept;
  constexpr index_type extent(size_t) const noexcept;

private:
  array<index_type, rank_dynamic()> dynamic_extents_; // exposition only
};

}}}

26.7.�.2 Constructors and assignment [mdspan.extents.cons]

constexpr extents() noexcept;

Effects: Aggregate-initializes dynamic_extents_ to { }.
Ensures: Let $R$ equal rank(). $∀$ r ∈ [0, R), if static_extent(r) equals dynamic_extent, then extent(r) equals 0.

constexpr extents(const extents& other);
constexpr extents(extents&& other);

Effects: Initializes dynamic_extents_ with other.dynamic_extents_.
Ensures: Let $R$ equal rank(). $∀$ r ∈ [0, R), extent(r) equals other.extent(r).

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

Requires: Let $R$ equal rank(). $∀$ r ∈ [0, R), if static_extent(r) does not equal dynamic_extent, then static_extent(r) equals other.extent(r).
Constraints: sizeof...(Extents) equals sizeof...(OtherExtents).
Effects: Let $R$ equal rank(). $∀$ r ∈ [0, R), if static_extent(r) equals dynamic_extent, then this initializes dynamic_extents_[DynamicRank[r]] with other.extent(r).
Ensures: *this == other is true.
Throws: Nothing.

template<class... IndexType>
constexpr extents(IndexType... dynamic_extents) noexcept;

Requires: (dynamic_extents>=0) && ... is true.
Constraints:
- is_convertible_v<IndexType, index_type> && ... is true, and
- sizeof...(dynamic_extents)==rank_dynamic() is true.
Effects: Aggregate-initializes dynamic_extents_ to {dynamic_extents...}.
Ensures: extent(DynamicRank[i]) is equal to the ith entry in the parameter pack dynamic_extents.

template<class IndexType, size_t rank_dynamic>
constexpr extents(const array<IndexType, rank_dynamic> & dynamic_extents) noexcept;

Requires: Let $D$ equal rank_dynamic(). $∀$ i ∈ [0, D), dynamic_extents[i] >= 0 is true.
Constraints: is_convertible_v<IndexType, index_type> is true.
Effects: Initializes dynamic_extents_ with dynamic_extents.
Ensures: extent(DynamicRank[i]) equals dynamic_extents[i].

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

Requires: Let $R$ equal rank(). $∀$ r ∈ [0, R), static_extent(r) != dynamic_extent || static_extent(r) == other.extent(r) is true.
Constraints: sizeof...(Extents)==sizeof...(OtherExtents) is true.
Effects: Let $R$ equal rank(). $∀$ r ∈ [0, R), if static_extent(r) equals dynamic_extent, assigns dynamic_extents_[DynamicRank[r]] to other.extent(r).
Ensures: *this==other is true.
Returns: *this.
Throws: Nothing.

26.7.�.3 Observers of the domain multidimensional index space [mdspan.extents.obs]

static constexpr size_t rank() const noexcept;

Returns: sizeof...(Extents)

static constexpr size_t rank_dynamic() const noexcept;

Returns:
- If sizeof...(Extents)==0 is true, then zero;
- Otherwise, ((Extents==dynamic_extent)+...).

[Note: This is the number of dynamic extents. —end note]

static constexpr index_type static_extent(size_t r) const noexcept;

Returns:
- If 0 <= r < rank(), then the rth entry in the Extents parameter pack;
- Otherwise, 1.

constexpr index_type extent(size_t r) const noexcept;

Returns:
- If static_extent(r)==dynamic_extent, then dynamic_extents_[DynamicRank[r]];
- Otherwise, static_extent(r).

26.7.�.4 extents comparison operators [mdspan.extents.compare]

template<ptrdiff_t... LHS, ptrdiff_t... RHS>
  constexpr bool operator==(const extents<LHS...>& lhs, const extents<RHS...>& rhs) noexcept;

Returns:
- If lhs.rank()==rhs.rank() && lhs.extents(r)==rhs.extents(r) is true for all r in the range [0, lhs.rank()), then true;
- Otherwise, false.

template<ptrdiff_t... LHS, ptrdiff_t... RHS>
  constexpr bool operator!=(const extents<LHS...>& lhs, const extents<RHS...>& rhs) noexcept;

Returns: !(lhs==rhs).

26.7.� Layout mapping policy [mdspan.layout]

26.7.�.1 Layout mapping requirements [mdspan.layout.reqs]

A layout mapping policy is a class that contains a layout mapping, a nested class template.
A layout mapping policy meets the requirements in Table �.
A layout mapping meets the requirements of Cpp17DefaultConstructible, Cpp17CopyAssignable, Cpp17EqualityComparable, and the requirements in Table �.
In Table �:
- MP denotes a layout mapping policy.
- E denotes a specialization of extents.
- e denotes an object of type E defining a multidimensional index space.
- r is a value of an integral type such that 0 <= r < e.rank().
- If k... is a pack of an integer type, we say that "k... is in e" if k... denotes a multi-index in e. Let the packs of integer type i... and j... be in e. Denote the $r$ -th member of packs i... and j... by $i$ _r and $j$ _r, respectively. Let r equal $r$ . sizeof...(i)==E::rank() is true, 0 <= $i$ _r < e.extent(r), sizeof...(j)==E::rank(), and 0 <= $j$ _r < e.extent(r).
- M denotes a layout mapping class.
- m denotes an object of type M that maps a pack i... denoting a multi-index to an integral value.

Table � — Layout mapping policy and layout mapping requirements

Expression	Return Type	Returns	Requires
`MP::template mapping<E>`	`M`
`m.extents()`	`E`	Returns: `e`.
`m(i...)`	`E::index_type`	Returns: Mapping of a multi-index `i...` .	Requires: 0 <= `m(i...)`.
`m.required_span_size()`	`E::index_type`	Returns: If the multidimensional index space that `e` defines is empty, then zero, else 1 plus the maximum value of `m(i...)` for all `i...` in `e`.
`m.is_unique()`	`bool`	Returns: `true` if for every `i...!=j...`, `m(i...)!=m(j...)` is `true`.
`m.is_contiguous()`	`bool`	Returns: `true` if the set defined by `m(i)...` equals the set {0, ..., `m.required_span_size()-1`}.
`m.is_strided()`	`bool`	Returns: `true` if there exists an integer $σ$ _r such that, for all `j...` and `i...` in `e`, if all members of `j...` and `i...` are equal except for exactly one r-th member where $j$ _r equals $i$ _r + 1, then `m(j...) - m(i...)` equals $σ$ _r.
`M::is_always_unique()`	`bool`	Returns: `true` if `m.is_unique()` is `true` for any object of type `M`.
`M::is_always_contiguous()`	`bool`	Returns: `true` if `m.is_contiguous()` is `true` for any object of type `M`.
`M::is_always_strided()`	`bool`	Returns: true if `m.is_strided()` is `true` for any object of type `M`.
`m.stride(r)`	`E::index_type`	Returns: $σ$ _r, as defined for `m.is_strided()` above. We say that $σ$ _r is the stride of ordinate `r`.	Requires: `m.is_strided()` is `true`.

26.7.�.2 Class layout_left [mdspan.layout.left]

layout_left meets the requirements of layout mapping policy. [Note: Thus, any well-formed specialization of layout_left::template mapping meets the requirements of layout mapping. —end note]
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.
If Extents is not a (possibly cv-qualified) specialization of extents, then the program is ill-formed.

namespace std {
namespace experimental {
namespace fundamentals_v3 {

struct layout_left {
  template<class Extents>
  class mapping {
  public:
    // [mdspan.layout.left.cons], layout_left::mapping constructors
    constexpr mapping() noexcept;
    constexpr mapping(const mapping& other) noexcept;
    constexpr mapping(mapping&& other) noexcept;
    constexpr mapping(const Extents& e) noexcept;
    template<class OtherExtents>
      constexpr mapping(const mapping<OtherExtents>& other);

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

    // [mdspan.layout.left.ops], layout_left::mapping operations
    Extents extents() const noexcept;

    constexpr typename Extents::index_type required_span_size() const noexcept;

    template<class... Indices>
      typename Extents::index_type operator()(Indices... is) const;

    static constexpr bool is_always_unique();
    static constexpr bool is_always_contiguous();
    static constexpr bool is_always_strided();

    constexpr bool is_unique() const;
    constexpr bool is_contiguous() const;
    constexpr bool is_strided() const;

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

    typename Extents::index_type stride(size_t rank) const noexcept;

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

}}}

26.7.�.2.1 layout_left::mapping constructors [mdspan.layout.left.cons]

constexpr mapping() noexcept;

Effects: Default-initializes extents_.
Ensures: extents()==Extents() is true.

constexpr mapping(const mapping& other) noexcept;

Effects: Initializes extents_ with other.extents_.
Ensures: extents()==other.extents() is true.

constexpr mapping(mapping&& other) noexcept;

Effects: Initializes extents_ with move(other.extents_).
Ensures: extents() equals the result of other.extents() before the invocation of the move.

constexpr mapping(const Extents & e) noexcept;

Effects: Initializes extents_ with e.
Ensures: extents()==e is true.

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

Requires: other.extents() meets the requirements for use in the initialization of extents_.
Effects: Initializes extents_ with other.extents().
Ensures: extents()==other.extents() is true.
Throws: Nothing.

26.7.�.2.2 layout_left::mapping operations [mdspan.layout.left.ops]

Extents extents() const noexcept;

Returns: extents_.

typename Extents::index_type required_span_size() const noexcept;

Returns: The product of extents().extent(r) for all r where 0 <= r < extents().rank().

template<class... Indices>
  typename Extents::index_type operator()(Indices... i) const;

Constraints:
- sizeof...(Indices)==extents().rank() is true, and
- is_convertible_v<Indices, typename Extents::index_type> && ... is true.
Requires: 0 <= array{i...}[r] < extents().extent(r) for all r in the range [0, extents().rank()).
Returns: Let $i$ _k denote the k-th member of i..., and let R equal extents().rank():
- If R is zero, then zero;
- Otherwise, the sum of $i$ _k ⨯ stride(k) for 0 <= k < R.

static constexpr bool is_always_unique();
static constexpr bool is_always_contiguous();
static constexpr bool is_always_strided();
constexpr bool is_unique() const;
constexpr bool is_contiguous() const;
constexpr bool is_strided() const;

Returns: true.

typename Extents::index_type stride(size_t r) const;

Returns:
- If r is zero, then one;
- Otherwise, the product of extents().extent(k) for 0 <= k < r, where $i$ _k denotes the k-th member of i... .

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

Returns: extents()==other.extents() is true.

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

Returns: extents()!=other.extents() is true.

26.7.�.3 Class layout_right [mdspan.layout.right]

layout_right meets the requirements of layout mapping policy. [Note: Thus, any well-formed specialization of layout_right::template mapping meets the requirements of layout mapping. —end note]
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.
If Extents is not a (possibly cv-qualified) specialization of extents, the program is ill-formed.

namespace std {
namespace experimental {
namespace fundamentals_v3 {

struct layout_right {
  template<class Extents>
  class mapping {
  public:
    // [mdspan.layout.right.cons], layout_left::mapping constructors
    constexpr mapping() noexcept;
    constexpr mapping(const mapping& other) noexcept;
    constexpr mapping(mapping&& other) noexcept;
    constexpr mapping(Extents e) noexcept;
    template<class OtherExtents>
      constexpr mapping(const mapping<OtherExtents>& other);

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

    // [mdspan.layout.right.ops], layout_right::mapping operations
    Extents extents() const noexcept;

    constexpr typename Extents::index_type required_span_size() const noexcept;

    template<class... Indices>
      typename Extents::index_type operator()(Indices... is) const;

    static constexpr bool is_always_unique() noexcept;
    static constexpr bool is_always_contiguous() noexcept;
    static constexpr bool is_always_strided() noexcept;

    constexpr bool is_unique() const noexcept;
    constexpr bool is_contiguous() const noexcept;
    constexpr bool is_strided() const noexcept;

    typename Extents::index_type stride(size_t rank) const noexcept;

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

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

26.7.�.3.1 layout_right::mapping constructors [mdspan.layout.right.cons]

constexpr mapping() noexcept;

Effects: Default-initializes extents_.
Ensures: extents()==Extents() is true.

constexpr mapping(const mapping& other) noexcept;

Effects: Initializes extents_ with other.extents_.
Ensures: extents()==other.extents() is true.

constexpr mapping(mapping&& other) noexcept;

Effects: Initializes extents_ with move(other.extents_).
Ensures: extents() equals the result of other.extents() before the invocation of the move.

constexpr mapping(Extents e) noexcept;

Effects: Initializes extents_ with e.
Ensures: extents()==e is true.

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

Requires: other.extents() meets the requirements for use in the initialization of extents_.
Effects: Initializes extents_ with other.extents().
Ensures: extents()==other.extents() is true.
Throws: Nothing.

26.7.�.3.2 layout_right::mapping operations [mdspan.layout.right.ops]

Extents extents() const noexcept;

Returns: extents_.

typename Extents::index_type required_span_size() const noexcept;

Returns: The product of extents().extent(r) for all r where 0 <= r < extents().rank().

template<class... Indices>
  typename Extents::index_type operator()(Indices... i) const noexcept;

Constraints:
- sizeof...(Indices)==extents().rank() is true, and
- is_convertible_v<Indices, typename Extents::index_type> && ... is true.
Requires: 0 <= array{i...}[r] < extents().extent(r) for all r in the range [0, extents().rank()).
Returns: Let $i$ _k denote the k-th member of i..., and let R equal extents().rank():
- If R is zero, then zero;
- Otherwise, the sum of $i$ _k ⨯ stride(k) for 0 <= k < R.

static constexpr bool is_always_unique() noexcept;
static constexpr bool is_always_contiguous() noexcept;
static constexpr bool is_always_strided() noexcept;
constexpr bool is_unique() const noexcept;
constexpr bool is_contiguous() const noexcept;
constexpr bool is_strided() const noexcept;

Returns: true.

typename Extents::index_type stride(size_t r) const noexcept;

Returns: Equivalent to s in the following:

Extents::index_type s = 1;
for(size_t k=r+1; k<extents().rank(); ++k)
  s *= extents().extent(r);

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

Returns: extents()==other.extents() is true.

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

Returns: extents()!=other.extents() is true.

26.7.�.4 Class layout_stride [mdspan.layout.stride]

layout_stride meets the requirements of layout mapping policy.
The layout mapping property layout_stride gives a layout mapping where the strides are user defined.
If Extents is not a (possibly cv-qualified) specialization of extents, the program is ill-formed.

namespace std {
namespace experimental {
namespace fundamentals_v3 {

struct layout_stride {
  template<class Extents>
  class mapping {
  public:
    // [mdspan.layout.stride.cons], layout_stride::mapping constructors
    constexpr mapping() noexcept;
    constexpr mapping(mapping const& other) noexcept;
    constexpr mapping(mapping&& other) noexcept;
    constexpr mapping(const Extents& e, const array<typename Extents::index_type, Extents::rank()>& s) noexcept;
    template<class OtherExtents>
      constexpr mapping(const mapping<OtherExtents>& other);

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

    // [mdspan.layout.stride.ops], layout_stride::mapping operations
    Extents extents() const noexcept;
    array<typename Extents::index_type, Extents::rank()> strides() const noexcept;

    constexpr typename Extents::index_type required_span_size() const noexcept;

    template<class... Indices>
      typename Extents::index_type operator()(Indices... is) const;

    static constexpr bool is_always_unique() noexcept;
    static constexpr bool is_always_contiguous() noexcept;
    static constexpr bool is_always_strided() noexcept;

    constexpr bool is_unique() const noexcept;
    constexpr bool is_contiguous() const noexcept;
    constexpr bool is_strided() const noexcept;

    typename Extents::index_type stride(size_t rank) const noexcept;

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

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

26.7.�.4.1 layout_stride::mapping constructors [mdspan.layout.stride.cons]

constexpr mapping() noexcept;

Effects: Default-initializes extents_.
Ensures:
- extents()==Extents() is true, and
- strides()==array<typename Extents::index_type, Extents::rank()>() is true.

constexpr mapping(const mapping& other) noexcept;

Effects: Initializes extents_ with other.extents_.
Ensures:
- extents()==other.extents() is true, and
- strides()==other.strides() is true.

constexpr mapping(mapping&& other) noexcept;

Effects: Initializes extents_ with move(other.extents_).
Ensures:
- extents() equals the result of other.extents() before the invocation of the move, and
- strides() equals the result of other.strides() before the invocation of the move.

constexpr mapping(Extents e, array<typename Extents::index_type, Extents::rank()> s) noexcept;

Let R equal Extents::rank(). If

p

is a permutation of the integers 0, ..., R-1, then let p be an exposition-only array of size_t of length R such that p[i] equals

p

_i for 0 <= i < R.

Requires:
- s[i] > 0 is true for 0 < i <= R.
- If R is nonzero, then there exists a permutation $p$ of the integers 0, ..., R-1 such that s(p[i]) >= s(p[i-1])*e.extent(p[i-1]) is true for 1 <= i < R.
Effects:
- Initializes extents_ with e, and
- initializes strides_ with s.
Ensures:
- extents()==e is true, and
- strides()==s is true.
Throws: Nothing.

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

Requires: other.extents() meets the requirements for use in the initialization of extents_.
Effects:
- Initializes extents_ with other.extents(), and
- initializes strides_ with other.strides().
Ensures:
- extents()==other.extents() is true, and
- strides()==other.strides() is true.
Throws: Nothing.

26.7.�.4.2 layout_stride::mapping operations [mdspan.layout.stride.ops]

Extents extents() const noexcept;

Returns: extents_.

array<typename Extents::index_type, Extents::rank()> strides() const noexcept;

Returns: strides_.

typename Extents::index_type required_span_size() const noexcept;

Returns: The maximum of extents().extent(r)*stride(r) for all r where 0 <= r < extents().rank().

template <class... Indices>
  typename Extents::index_type operator()(Indices... i) const;

Constraints:
- sizeof...(Indices)==Extents::rank() is true, and
- is_convertible_v<Indices, typename Extents::index_type> && ... is true.
Requires: 0 <= array{i...}[r] < extents().extent(r) for all r in the range [0, extents().rank()).
Returns: Let $i$ _k denote the k-th member of i..., let R equal Extents::rank(), and let s be strides():
- If R is zero, then zero;
- Otherwise, the sum of $i$ _k ⨯ s[k] for 0 <= k < R.
Throws: Nothing.

static constexpr bool is_always_unique() noexcept;
static constexpr bool is_always_strided() noexcept;
constexpr bool is_unique() const noexcept;
constexpr bool is_strided() const noexcept;

Returns: true.

static constexpr bool is_always_contiguous() noexcept;

Returns: false.

constexpr bool is_contiguous() const noexcept;

Let R equal Extents::rank(). If

p

is a permutation of the integers 0, ..., R-1, then let p be an exposition-only array of size_t of length R such that p[i] equals

p

_i for 0 <= i < R.

Returns:
- If R is zero, then true;
- Else, if there is a permutation $p$ of the integers 0, ..., R-1 such that min(stride(p[i]) equals 1 for 0 <= i < R, and stride(p[i]) equals stride(p[i-1])*extents().extent(p[i-1]) for 1 <= i < R, then true;
- Otherwise, false.

typename Extents::index_type stride(size_t r) const noexcept;

Returns: strides_[r].

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

Returns: extents()==other.extents().

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

Returns: extents()!=other.extents().

26.7.� Accessor Policy [mdspan.accessor]

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

26.7.�.1 Accessor policy requirements [mdspan.accessor.reqs]

An accessor policy defines:

a handle to a single element of type element_type;
a handle to a contiguous set of elements of type element_type, accessible through the policy’s access function;
conversion of a handle to a contiguous set of elements, to a pointer [conv.array]; and
getting a handle to the contiguous subset of elements beginning at an integer offset value.

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

[Note: The constructor of the object representing the contiguous set of objects may impose additional restrictions on the contiguous set of objects, such as restricting element_type to be trivially copyable or requiring a larger alignment than alignment_of_v<element_type>. — end note]

In Table �:

A denotes an accessor policy.
a denotes an object of type A.
p denotes an object of type A::pointer.
i and j each denote a ptrdiff_t value.

Table �: Accessor policy requirements

Expression	Return Type	Notes/Returns/Requires
`A::element_type`		Type of each element in the contiguous set of elements.
`A::pointer`		Type through which the range of elements are accessed. Requires: Cpp17DefaultConstructible, Cpp17CopyConstructible, and Cpp17CopyAssignable.
`A::reference`		Type through which an element is accessed.
`A::offset_policy`		Accessor policy for accessing a pointer returned by `a.offset(p,i)`. Requires: `A::offset_policy` meets the requirements of an accessor policy in Table �. `is_convertible_v<A, A::offset_policy>` is `true` and `A::offset_policy(a)` is valid.
`a.decay(p)`	`A::element_type*`	Returns: A pointer that references the same location as `p`.
`a.access(p,i)`	`A::reference`	Returns: An object which provides access to the `i`-th element in the range of elements that starts at `p`.
`a.offset(p,i)`	`A::offset_policy::pointer`	Returns: A pointer that references the same location as `a.decay(p)+i`. Requires: `a.decay(p)+i` equals `A::offset_policy(a).decay(a.offset(p,i))` `A::offset_policy(a).access(a.offset(p,i),j)` is valid if `a.access(p,i+j)` is valid.

26.7.�.2 Class accessor_basic [mdspan.accessor.basic]

accessor_basic meets the requirements of accessor policy.
If T is not an object type or is an array type, the program is ill-formed.

namespace std {
namespace experimental {
namespace fundamentals_v3 {

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

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

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

    constexpr pointer decay(pointer p) const noexcept;
  };

}}}

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

Requires: p + i is dereferenceable.
Returns: p + i.

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

Requires: p + i is dereferenceable.
Returns: p[i].

constexpr pointer decay(pointer p) const noexcept;

Returns: p.

26.7.� Class template basic_mdspan [mdspan.basic]

The basic_mdspan class template maps a multi-index within a multi-index domain to a reference to an element in the codomain span.
The multi-index domain space is the Cartesian product of the extents: $[0,$ extent(0)) ⨯ [0, extent(1)) ⨯ ... ⨯ [0, extent(rank()-1)). Each extent may be statically or dynamically specified.
As with span, the storage of the objects in the codomain span of a basic_mdspan is owned by some other object.
ElementType is required to be a complete object type that is not an abstract class type or an array type.
Extents is required to be a (cv-unqualified) specialization of extents.
LayoutPolicy is required to meet the layout mapping policy requirements, otherwise the program is ill-formed.
AccessorPolicy is required to meet the accessor policy requirements and std::is_same_v<typename AccessorPolicy::element_type,ElementType> must be false, otherwise the program is ill-formed.

namespace std {
namespace experimental {
namespace fundamentals_v3 {

template<class ElementType, class Extents, class LayoutPolicy, class AccessorPolicy>
class basic_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 = typename accessor_type::element_type;
  using value_type = remove_cv_t<element_type>;
  using index_type = ptrdiff_t ;
  using difference_type = ptrdiff_t;
  using pointer = typename accessor_type::pointer;
  using reference = typename accessor_type::reference;

  // [mdspan.basic.cons], basic_mdspan constructors, assignment, and destructor
  constexpr basic_mdspan() noexcept = default;
  constexpr basic_mdspan(const basic_mdspan&) noexcept = default;
  constexpr basic_mdspan(basic_mdspan&&) noexcept;
  template<class... IndexType>
    explicit constexpr basic_mdspan(pointer p, IndexType... dynamic_extents);
  template<class IndexType, size_t N>
    explicit constexpr basic_mdspan(pointer p, const array<IndexType, N>& dynamic_extents);
  constexpr basic_mdspan(pointer p, const mapping_type& m);
  constexpr basic_mdspan(pointer p, const mapping_type& m, const accessor_type& a);
  template<class OtherElementType, class OtherExtents, class OtherLayoutPolicy, class OtherAccessorPolicy>
    constexpr basic_mdspan(const basic_mdspan<OtherElementType, OtherExtents, OtherLayoutPolicy, OtherAccessorPolicy>& other);

  ~basic_mdspan() = default;

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

  // [mdspan.basic.mapping], basic_mdspan mapping domain multi-index to access codomain element
  constexpr reference operator[](index_type) const noexcept;
  template<class... IndexType>
    constexpr reference operator()(IndexType... indices) const noexcept;
  template<class IndexType, size_t N>
    constexpr reference operator()(const array<IndexType, N>& indices) const noexcept;
  accessor_type accessor() const;

  // [mdspan.basic.domobs], basic_mdspan observers of the domain multidimensional index space
  static constexpr int rank() noexcept;
  static constexpr int rank_dynamic() noexcept;
  static constexpr index_type static_extent(size_t) noexcept;

  constexpr Extents extents() const noexcept;
  constexpr index_type extent(size_t) const noexcept;
  constexpr index_type size() const noexcept;
  constexpr index_type unique_size() const noexcept;

  // [mdspan.basic.codomain], basic_mdspan observers of the codomain
  constexpr span<element_type> span() const noexcept;
  constexpr pointer data() const noexcept;

  // [mdspan.basic.obs], basic_mdspan observers of the mapping
  static constexpr bool is_always_unique() noexcept;
  static constexpr bool is_always_contiguous() noexcept;
  static constexpr bool is_always_strided() noexcept;

  constexpr mapping_type mapping() const noexcept;
  constexpr bool is_unique() const noexcept;
  constexpr bool is_contiguous() const noexcept;
  constexpr bool is_strided() const noexcept;
  constexpr index_type stride(size_t) const;

private:
  accessor_type acc_; // exposition only
  mapping_type map_; // exposition only
  pointer ptr_; // exposition only
};

}}}

26.7.�.1 basic_mdspan constructors, assignment, and destructor [mdspan.basic.cons] // Mapping domain multi-index to access codomain element

constexpr basic_mdspan() noexcept = default;

Effects:
- Zero-initializes ptr_,
- Value-initializes map_, and
- Value-initializes acc_.
Ensures:
- size()==0 is true,
- extents()==Extents() is true, and
- mapping()==mapping_type() is true.

constexpr basic_mdspan(const basic_mdspan& other) noexcept = default;

Effects:
- Initializes ptr_ with other.ptr_,
- Initializes map_ with other.map_, and
- Initializes acc_ with other.acc_.
Ensures:
- size()==other.size() is true,
- extents()==other.extents() is true, and
- mapping()==other.mapping() is true.

constexpr basic_mdspan(basic_mdspan&& other) noexcept;

Effects:
- Initializes ptr_ with move(other.ptr_),
- Initializes map_ with move(other.map_), and
- Initializes acc_ with move(other.acc_).

template<class... IndexType>
  explicit constexpr basic_mdspan(pointer ptr, IndexType... dynamic_extents);

Constraints:
- (is_convertible_v<IndexType, index_type> && ...) is true,
- sizeof...(dynamic_extents)==rank_dynamic() is true,
- is_constructible_v<mapping_type, Extents> is true, and
- is_default_constructible_v<accessor_type> is true.
Effects:
- Initializes ptr_ with ptr,
- Initializes map_ with Extents(dynamic_extents...), and
- Value-initializes acc_.
Ensures:
- extents()==Extents(dynamic_extents...) is true, and
- mapping()==mapping_type(Extents(dynamic_extents...)) is true.
Throws: Nothing.

template<class IndexType, size_t N>
  explicit constexpr basic_mdspan(pointer p, const array<IndexType, N>& dynamic_extents);

Constraints:
- is_convertible_v<IndexType, index_type> is true,
- N==rank_dynamic() is true,
- is_constructible_v<mapping_type, Extents> is true, and
- is_default_constructible_v<accessor_type> is true.
Effects: Equivalent to basic_mdspan(p, dynamic_extents[Rs]...), with Rs... from index_sequence<Rs...> matching make_index_sequence<N>.
Throws: Nothing.

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

Constraints: is_default_constructible_v<accessor_type> is true.
Effects:
- Initializes ptr_ with p,
- Initializes map_ with m, and
- Value-initializes acc_
Ensures:
- extents()==m.extents() is true, and
- mapping()==m is true.
Throws: Nothing.

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

Effects:
- Initializes ptr_ with p,
- Initializes map_ with m, and
- Initializes acc_ with a.
Ensures:
- extents()==m.extents() is true, and
- mapping()==m is true.
Throws: Nothing.

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

Constraints:
- is_convertible_v<OtherLayoutPolicy::template mapping<OtherExtents>, mapping_type is true,
- is_convertible_v<OtherAccessor, Accessor> is true, and
- is_convertible_v<OtherAccessor::pointer, pointer> is true.
- is_convertible_v<OtherExtents, extents_type> is true
- 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.
Requires:
- 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.
Effects:
- Initializes ptr_ with other.ptr_,
- Initializes map_ with other.map_, and
- Initializes acc_ with other.acc_.
Ensures:
- extents()==Extents(other.extents()) is true, and
- mapping()==mapping_type(other.mapping()) is true.
Throws: Nothing.

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

Constraints:
- is_assignable_v<mapping_type, OtherLayoutPolicy::template mapping<OtherExtents>> is true,
- is_assignable_v<Accessor, OtherAccessor> is true, and
- is_assignable_v<pointer, OtherAccessor::pointer> is true.
- OtherExtents::rank() == rank() is true
- 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.
Requires:
- 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.
Effects:
- Assigns other.ptr_ to ptr_,
- Assigns other.map_ to map_, and
- Assigns other.acc_ to acc_.
Ensures:
- extents()==Extents(other.extents()) is true, and
- mapping()==mapping_type(other.mapping()) is true.
Throws: Nothing.

26.7.�.2 basic_mdspan mapping domain multi-index to access codomain element [mdspan.basic.mapping]

constexpr reference operator[](index_type i) const;

Constraints: rank()==1 is true.
Requires: acc_.access(ptr_, map_(i)) shall be valid.
Effects: Equivalent to return (*this)(i);.

template<class... IndexType>
  constexpr reference operator()(IndexType... indices) const;

Constraints:
- (is_convertible_v<IndexType, index_type> && ...) is true, and
- sizeof...(IndexType)==rank() is true.
Requires: acc_.access(ptr_, map_(indices...)) shall be valid.
Effects: Equivalent to return acc_.access(ptr_, map_(indices...));.
Throws: Nothing.

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

Constraints:
- is_convertible_v<IndexType, index_type> is true, and
- rank()==N is true.
Requires: acc_.access(ptr_, std::apply(map_, indices)) shall be valid.
Effects: Equivalent to return std::apply(*this, indices);.
Throws: nothing.

accessor_type accessor() const;

Returns: acc_.

26.7.�.3 basic_mdspan observers of the domain multidimensional index space [mdspan.basic.domobs]

static constexpr int rank() noexcept;

Returns: Extents::rank().

static constexpr int rank_dynamic() noexcept;

Returns: Extents::rank_dynamic().

static constexpr index_type static_extent(size_t r) noexcept;

Returns: Extents::static_extent(r).

constexpr Extents extents() const noexcept;

Returns: mapping().extents().

constexpr index_type extent(size_t r) const noexcept;

Returns: extents().extent(r).

constexpr index_type size() const noexcept;

Returns: Product of extent(r) for all r where 0 <= r < extents().rank().

constexpr index_type unique_size() const noexcept;

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

26.7.�.3 basic_mdspan observers of the codomain [mdspan.basic.codomain]

constexpr span<element_type> span() const noexcept;

Returns: Equivalent to span<element_type>(acc_.decay(ptr_), required_span_size()).

constexpr pointer data() const noexcept;

Returns: ptr_.

26.7.�.4 basic_mdspan observers of the mapping [mdspan.basic.obs]

static constexpr bool is_always_unique() noexcept;

Returns: mapping_type::is_always_unique().

static constexpr bool is_always_contiguous() noexcept;

Returns: mapping_type::is_always_contiguous().

static constexpr bool is_always_strided() noexcept;

Returns: mapping_type::is_always_strided().

constexpr mapping_type mapping() const noexcept;

Returns: map_.

constexpr bool is_unique() const noexcept;

Returns: mapping().is_unique().

constexpr bool is_contiguous() const noexcept;

Returns: mapping().is_contiguous().

constexpr bool is_strided() const noexcept;

Returns: mapping().is_strided().

constexpr index_type stride(size_t r) const;

Returns: mapping().stride(r).

26.7.� subspan [mdspan.subspan]

namespace std {
namespace experimental {
namespace fundamentals_v3 {

  // [mdspan.subspan], subspan creation
  template<class ElementType, class Extents, class LayoutPolicy,
           class AccessorPolicy, class... SliceSpecifiers>
  struct mdspan_subspan { // exposition only
    using extents_t = see below;
    using layout_t = see below;
    using type = basic_mdspan<ElementType, extents_t, layout_t,
                              typename AccessorPolicy::offset_policy>;
  };

  template<class ElementType, class Extents, class LayoutPolicy,
           class AccessorPolicy, class... SliceSpecifiers>
  using mdspan_subspan_t = // exposition only
    typename mdspan_subspan<ElementType, Extents, LayoutPolicy,
                            AccessorPolicy, SliceSpecifiers...>::type;

  template<class ElementType, class Extents, class LayoutPolicy,
           class AccessorPolicy, class... SliceSpecifiers>
    mdspan_subspan_t<ElementType, Extents, LayoutPolicy,
                     AccessorPolicy, SliceSpecifiers...>
      subspan(const basic_mdspan<ElementType, Extents, LayoutPolicy,
                                 AccessorPolicy>& src,
         SliceSpecifiers ... slices) noexcept;

}}}

The subspan function creates a basic_mdspan that is a view of a (potentially trivial) subset of another basic_mdspan. The SliceSpecifier template argument(s) and the corresponding value(s) of the arguments of subspan after src determine the subset of src that the return value views.

Let sub be the return value of subspan(src, slices...). Denote the value of the $k$ -th member of slices... by $s$ _k. Denote the type of the $k$ -th member of slices... by $S$ _k.

Let ρ be sub.rank(). ρ equals the number of k such that is_convertible_v< $S$ _k, pair<ptrdiff_t, ptrdiff_t>> || is_convertible_v< $S$ _k, all_type> is true.

Define rank_map as the length ρ std::integer_sequence of ptrdiff_t consisting of the unique values of k, in increasing order, such that is_convertible_v< $S$ _k, pair<ptrdiff_t, ptrdiff_t>> || is_convertible_v< $S$ _k, all_type> is true. Define the exposition-only constexpr function get_seq<k>(seq), where k is an integer, as the function returning the k-th entry of the std::integer_sequence seq. We say that the integer r "is not in rank_map" when, for 0 <= k < ρ, r does not equal get_seq<k>(rank_map).

In the exposition-only struct mdspan_subspan,

The extents_t type alias is a specialization of extents [mdspan.extents], and is determined by the description of the semantics of subspan given below;
The layout_t type alias meets the requirements of a layout mapping policy [mdspan.layout.reqs] and is implementation defined; and
The type type alias specifies the type returned by subspan.

[Note: High-quality implementations will avoid layout_stride whenever possible, if the input layout is layout_left or layout_right. — end note]

Let first and last be exposition-only array<ptrdiff_t, sizeof...(SliceSpecifier)>, where SliceSpecifier... is the same parameter pack as subspan's. For 0 <= r < sizeof...(SliceSpecifier), define the values of first[r] and last[r] as follows, where src and slices... are the arguments of subspan.

If is_convertible_v< $S$ _r, ptrdiff_t>, then first[r] equals $s$ _r, and last[r] equals first[r] + 1;
Else, if is_convertible_v< $S$ _r, pair<ptrdiff_t, ptrdiff_t>>, then first[r] equals p.first, and last[r] equals p.second, where p is the result of converting $s$ _r to pair<ptrdiff_t, ptrdiff_t>;
Else, if is_convertible_v< $S$ _r, all_type>, then first[r] = 0, and last[r] equals src.extent(r).

Requires:
- sizeof(slices...) equals src.rank().
- For 0 <= r < sizeof...(slices), 0 <= first[r] && first[r] < last[r] && last[r] <= src.extent(r) is true.
Constraints:
- 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: Valid and useful layout mapping policies exist, for which taking an arbitrary subspan does not make sense. — end note]
- For 0 <= k < sizeof...(slices), is_convertible_v< $S$ _k, ptrdiff_t> || is_convertible_v< $S$ _k, pair<ptrdiff_t, ptrdiff_t>> || is_convertible_v< $S$ _k, all_type> is true.
Ensures:
- sub.rank() equals ρ.
- mdspan_subspan<ElementType, Extents, LayoutPolicy, AccessorPolicy, SliceSpecifiers...>::extents_t::rank() equals ρ.
- Let the pack i... denote a multi-index in the domain of sub such that $i$ _k < sub.extents(k) for 0 <= k < ρ, where $i$ _k denotes the k-th member of i... . Let the pack j... denote a multi-index in the domain of src where the get_seq<k>(rank_map)-th member of j... equals first[get_seq<k>(rank_map)] + $i$ _k for 0 <= k < $ρ$ , and the r-th member of j... equals first[r] for all r such that 0 <= r < src.rank() and r is not in rank_map. Then, sub(i...) and src(j...) refer to the same element.
- For 0 <= k < ρ, sub.extent(k) equals last[get_seq<k>(rank_map)] - first[get_seq<k>(rank_map)].
- If src.is_strided(), then sub.is_strided() is true, and for 0 <= k < ρ, sub.stride(k) equals src.stride(get_seq<k>(rank_map)).
- For 0 <= k < ρ, let rmk equal get_seq<k>(rank_map); then, if src.static_extent(rmk) does not equal dynamic_extent and is_convertible_v< $S$ _rmk, all_type> is true, then sub.static_extent(k) equals src.static_extent(rmk).

[Note: Example of subspan use:

// Create a mapping
typedef extents<3,dynamic_extent,7> Extents3D;
layout_right::template mapping<Extents3D> map_right(10);

// Allocate a basic_mdspan
int* ptr = new int[3*8*10];
basic_mdspan<int,Extents3D,layout_right> a(ptr,map_right);

// Initialize the span
for(int i0=0; i0<a.extent(0); i0++)
  for(int i1=0; i1<a.extent(1); i1++)
    for(int i2=0; i2<a.extent(2); i2++)
      a(i0,i1,i2) = 10000*i0+100*i1+i2;

// Create Subspan
auto a_sub = subspan(a,1,std::pair<int,int>(4,6),std::pair<int,int>(1,6));

// Print values of subspan
for(int i0=0; i0<a_sub.extent(0); i0++) {
  for(int i1=0; i1<a_sub.extent(1); i1++) {
    std::cout << a_sub(i0,i1) << " ";
  }
  std::cout << std::endl;
}

/* Output
10401 10402 10403 10404 10405
10501 10502 10503 10504 10505
*/

- end note]

5. Next Steps

Wording editing as per guidance from LWG.

LEWG issue

Previous paper:

[N4355]

P0860 : Access Policy Generating Proxy Reference

The reference type may be a proxy for accessing an element_type object. For example, the atomic AccessorPolicy in P0860 defines AccessorPolicy::template accessor_type<T>::reference to be atomic_ref<T> from P0019.

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

P0009R8
`mdspan`: A Non-Owning Multidimensional Array Reference

Draft Proposal, 2018-10-07

1. Revision History

1.1. P0009r8: Pre 2018-11-SanDiego Mailing

1.2. P0009r7: Post 2018-06-Rapperswil Mailing

1.3. P0009r6 : Pre 2018-06-Rapperswil Mailing

1.4. P0009r5 : Pre 2018-03-Jacksonville Mailing

1.5. P0009r4 : Pre 2017-11-Albuquerque Mailing

1.6. P0009r3 : Post 2016-06-Oulu Mailing

1.7. P0009r2 : Pre 2016-06-Oulu Mailing

1.8. P0009r1 : Pre 2016-02-Jacksonville Mailing

1.9. P0009r0 : Pre 2015-10-Kona Mailing

2. Description

3. Editing Notes

4. Wording

5. Next Steps

References

Informative References

P0009R8mdspan: A Non-Owning Multidimensional Array Reference

Draft Proposal, 2018-10-07

1. Revision History

1.1. P0009r8: Pre 2018-11-SanDiego Mailing

1.2. P0009r7: Post 2018-06-Rapperswil Mailing

1.3. P0009r6 : Pre 2018-06-Rapperswil Mailing

1.4. P0009r5 : Pre 2018-03-Jacksonville Mailing

1.5. P0009r4 : Pre 2017-11-Albuquerque Mailing

1.6. P0009r3 : Post 2016-06-Oulu Mailing

1.7. P0009r2 : Pre 2016-06-Oulu Mailing

1.8. P0009r1 : Pre 2016-02-Jacksonville Mailing

1.9. P0009r0 : Pre 2015-10-Kona Mailing

1.10. Related Activity

2. Description

3. Editing Notes

4. Wording

5. Next Steps

6. Related Work

References

Informative References

P0009R8
`mdspan`: A Non-Owning Multidimensional Array Reference