MultiArray Concept

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MultiArray Concept

The MultiArray concept defines an interface to hierarchically nested containers. It specifies operations for accessing elements, traversing containers, and creating views of array data. MultiArray defines a flexible memory model that accomodates a variety of data layouts.

At each level (or dimension) of a MultiArray's container hierarchy lie a set of ordered containers, each of which contains the same number and type of values. The depth of this container hierarchy is the MultiArray's dimensionality. MultiArray is recursively defined; the containers at each level of the container hierarchy model MultiArray as well. While each dimension of a MultiArray has its own size, the list of sizes for all dimensions defines the shape of the entire MultiArray. At the base of this hierarchy lie 1-dimensional MultiArrays. Their values are the contained objects of interest and not part of the container hierarchy. These are the MultiArray's elements.

Like other container concepts, MultiArray exports iterators to traverse its values. In addition, values can be addressed directly using the familiar bracket notation.

MultiArray also specifies routines for creating specialized views. A view lets you treat a subset of the underlying elements in a MultiArray as though it were a separate MultiArray. Since a view refers to the same underlying elements, changes made to a view's elements will be reflected in the original MultiArray. For example, given a 3-dimensional "cube" of elements, a 2-dimensional slice can be viewed as if it were an independent MultiArray. Views are created using index_gen and index_range objects. index_ranges denote elements from a certain dimension that are to be included in a view. index_gen aggregates range data and performs bookkeeping to determine the view type to be returned. MultiArray's operator[] must be passed the result of N chained calls to index_gen::operator[], i.e.

indices[a0][a1]...[aN];

where N is the MultiArray's dimensionality and indices an object of type index_gen. The view type is dependent upon the number of degenerate dimensions specified to index_gen. A degenerate dimension occurs when a single-index is specified to index_gen for a certain dimension. For example, if indices is an object of type index_gen, then the following example:

indices[index_range(0,5)][2][index_range(0,4)];

has a degenerate second dimension. The view generated from the above specification will have 2 dimensions with shape 5 x 4. If the "2" above were replaced with another index_range object, for example:

indices[index_range(0,5)][index_range(0,2)][index_range(0,4)];

then the view would have 3 dimensions.

MultiArray exports information regarding the memory layout of its contained elements. Its memory model for elements is completely defined by 4 properties: the origin, shape, index bases, and strides. The origin is the address in memory of the element accessed as a[0][0]...[0], where a is a MultiArray. The shape is a list of numbers specifying the size of containers at each dimension. For example, the first extent is the size of the outermost container, the second extent is the size of its subcontainers, and so on. The index bases are a list of signed values specifying the index of the first value in a container. All containers at the same dimension share the same index base. Note that since positive index bases are possible, the origin need not exist in order to determine the location in memory of the MultiArray's elements. The strides determine how index values are mapped to memory offsets. They accomodate a number of possible element layouts. For example, the elements of a 2 dimensional array can be stored by row (i.e., the elements of each row are stored contiguously) or by column (i.e., the elements of each column are stored contiguously).

Two concept checking classes for the MultiArray concepts (ConstMultiArrayConcept and MutableMultiArrayConcept) are in the namespace boost::multi_array_concepts in <boost/multi_array/concept_checks.hpp>.

Notation

What follows are the descriptions of symbols that will be used to describe the MultiArray interface.

Table 27.1. Notation

| A | A type that is a model of MultiArray | | a,b | Objects of type A | | NumDims | The numeric dimension parameter associated with A. | | Dims | Some numeric dimension parameter such that 0<Dims<NumDims. | | indices | An object created by some number of chained calls to index_gen::operator[](index_range). | | index_list | An object whose type models Collection | | idx | A signed integral value. | | tmp | An object of type boost::array<index,NumDims> |

Associated Types

Table 27.2. Associated Types

Valid expressions

Table 27.3. Valid Expressions

std::accumulate(a.shape(),a.shape+a.num_dimensions(),
    size_type(1),std::multiplies<size_type>());

| | a.size() | size_type | This returns the number of values contained in a. It is equivalent to a.shape()[0]; | | a(index_list) | element&; if a is mutable, const element& otherwise. | This expression accesses a specific element of a.index_list is the unique set of indices that address the element returned. It is equivalent to the following code (disregarding intermediate temporaries):

// multiply indices by strides
    std::transform(index_list.begin(), index_list.end(),
      a.strides(), tmp.begin(), std::multiplies<index>()),

    // add the sum of the products to the origin
    *std::accumulate(tmp.begin(), tmp.end(), a.origin());

| | a.begin() | iterator if a is mutable, const_iterator otherwise. | This returns an iterator pointing to the beginning of a. | | a.end() | iterator if a is mutable, const_iterator otherwise. | This returns an iterator pointing to the end of a. | | a.rbegin() | reverse_iterator if a is mutable, const_reverse_iterator otherwise. | This returns a reverse iterator pointing to the beginning of a reversed. | | a.rend() | reverse_iterator if a is mutable, const_reverse_iterator otherwise. | This returns a reverse iterator pointing to the end of a reversed. | | a[idx] | reference if a is mutable, const_reference otherwise. | This returns a reference type that is bound to the index idx value of a. Note that if i is the index base for this dimension, the above expression returns the (idx-i)th element (counting from zero). The expression is equivalent to *(a.begin()+idx-a.index_bases()[0]);. | | a[indices] | array_view<Dims>::type if a is mutable, const_array_view<Dims>::type otherwise. | This expression generates a view of the array determined by the index_range and index values used to construct indices. | | a == b | bool | This performs a lexicographical comparison of the values of a and b. The element type must model EqualityComparable for this expression to be valid. | | a < b | bool | This performs a lexicographical comparison of the values of a and b. The element type must model LessThanComparable for this expression to be valid. | | a <= b | bool | This performs a lexicographical comparison of the values of a and b. The element type must model EqualityComparable and LessThanComparable for this expression to be valid. | | a > b | bool | This performs a lexicographical comparison of the values of a and b. The element type must model EqualityComparable and LessThanComparable for this expression to be valid. | | a >= b | bool | This performs a lexicographical comparison of the values of a and b. The element type must model LessThanComparable for this expression to be valid. |

Complexity guarantees

begin() and end() execute in amortized constant time. size() executes in at most linear time in the MultiArray's size.

Invariants

Table 27.4. Invariants

| Valid range | [a.begin(),a.end()) is a valid range. | | Range size | a.size() == std::distance(a.begin(),a.end());. | | Completeness | Iteration through the range [a.begin(),a.end()) will traverse across every value_type of a. | | Accessor Equivalence | Calling a[a1][a2]...[aN] where N==NumDims yields the same result as calling a(index_list), where index_list is a Collection containing the values a1...aN. |

Associated Types for Views

The following MultiArray associated types define the interface for creating views of existing MultiArrays. Their interfaces and roles in the concept are described below.

index_range

index_range objects represent half-open strided intervals. They are aggregated (using an index_gen object) and passed to a MultiArray's operator[] to create an array view. When creating a view, each index_range denotes a range of valid indices along one dimension of a MultiArray. Elements that are accessed through the set of ranges specified will be included in the constructed view. In some cases, an index_range is created without specifying start or finish values. In those cases, the object is interpreted to start at the beginning of a MultiArray dimension and end at its end.

index_range objects can be constructed and modified several ways in order to allow convenient and clear expression of a range of indices. To specify ranges, index_range supports a set of constructors, mutating member functions, and a novel specification involving inequality operators. Using inequality operators, a half open range [5,10) can be specified as follows:

5 <= index_range() < 10;

or

4 < index_range() <= 9;

and so on. The following describes the index_range interface.

Table 27.5. Notation

| i | An object of type index_range. | | idx,idx1,idx2,idx3 | Objects of type index. |

Table 27.6. Associated Types

Table 27.7. Valid Expressions

index_gen

index_gen aggregates index_range objects in order to specify view parameters. Chained calls to operator[] store range and dimension information used to instantiate a new view into a MultiArray.

Table 27.8. Notation

| Dims,Ranges | Unsigned integral values. | | x | An object of type template gen_type<Dims,Ranges>::type. | | i | An object of type index_range. | | idx | Objects of type index. |

Table 27.9. Associated Types

Table 27.10. Valid Expressions

Models

• multi_array
• multi_array_ref
• const_multi_array_ref
• template array_view<Dims>::type
• template const_array_view<Dims>::type
• template subarray<Dims>::type
• template const_subarray<Dims>::type

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