SubArrays

Julia’s SubArray type is a container encoding a “view” of a parent AbstractArray. This page documents some of the design principles and implementation of SubArrays.

Indexing: cartesian vs. linear indexing

Broadly speaking, there are two main ways to access data in an array. The first, often called cartesian indexing, uses N indexes for an N -dimensional AbstractArray. For example, a matrix A (2-dimensional) can be indexed in cartesian style as A[i,j]. The second indexing method, referred to as linear indexing, uses a single index even for higher-dimensional objects. For example, if A = reshape(1:12, 3, 4), then the expression A[5] returns the value 5. Julia allows you to combine these styles of indexing: for example, a 3d array A3 can be indexed as A3[i,j], in which case i is interpreted as a cartesian index for the first dimension, and j is a linear index over dimensions 2 and 3.

For Arrays, linear indexing appeals to the underlying storage format: an array is laid out as a contiguous block of memory, and hence the linear index is just the offset (+1) of the corresponding entry relative to the beginning of the array. However, this is not true for many other AbstractArray types: examples include SparseMatrixCSC, arrays that require some kind of computation (such as interpolation), and the type under discussion here, SubArray. For these types, the underlying information is more naturally described in terms of cartesian indexes.

You can manually convert from a cartesian index to a linear index with sub2ind, and vice versa using ind2sub. getindex and setindex! functions for AbstractArray types may include similar operations.

While converting from a cartesian index to a linear index is fast (it’s just multiplication and addition), converting from a linear index to a cartesian index is very slow: it relies on the div operation, which is one of the slowest low-level operations you can perform with a CPU. For this reason, any code that deals with AbstractArray types is best designed in terms of cartesian, rather than linear, indexing.

Index replacement

Consider making 2d slices of a 3d array:

S1 = view(A, :, 5, 2:6)
S2 = view(A, 5, :, 2:6)

view drops “singleton” dimensions (ones that are specified by an Int), so both S1 and S2 are two-dimensional SubArrays. Consequently, the natural way to index these is with S1[i,j]. To extract the value from the parent array A, the natural approach is to replace S1[i,j] with A[i,5,(2:6)[j]] and S2[i,j] with A[5,i,(2:6)[j]].

The key feature of the design of SubArrays is that this index replacement can be performed without any runtime overhead.

SubArray design

Type parameters and fields

The strategy adopted is first and foremost expressed in the definition of the type:

immutable SubArray{T,N,P,I,L} <: AbstractArray{T,N}
    parent::P
    indexes::I
    offset1::Int       # for linear indexing and pointer, only valid when L==true
    stride1::Int       # used only for linear indexing
    ...
end

SubArray has 5 type parameters. The first two are the standard element type and dimensionality. The next is the type of the parent AbstractArray. The most heavily-used is the fourth parameter, a Tuple of the types of the indices for each dimension. The final one, L, is only provided as a convenience for dispatch; it’s a boolean that represents whether the index types support fast linear indexing. More on that later.

If in our example above A is a Array{Float64, 3}, our S1 case above would be a SubArray{Int64,2,Array{Int64,3},Tuple{Colon,Int64,UnitRange{Int64}},false}. Note in particular the tuple parameter, which stores the types of the indices used to create S1. Likewise,

julia> S1.indexes
(Colon(),5,2:6)

Storing these values allows index replacement, and having the types encoded as parameters allows one to dispatch to efficient algorithms.

Index translation

Performing index translation requires that you do different things for different concrete SubArray types. For example, for S1, one needs to apply the i,j indices to the first and third dimensions of the parent array, whereas for S2 one needs to apply them to the second and third. The simplest approach to indexing would be to do the type-analysis at runtime:

parentindexes = Array{Any}(0)
for thisindex in S.indexes
    ...
    if isa(thisindex, Int)
        # Don't consume one of the input indexes
        push!(parentindexes, thisindex)
    elseif isa(thisindex, AbstractVector)
        # Consume an input index
        push!(parentindexes, thisindex[inputindex[j]])
        j += 1
    elseif isa(thisindex, AbstractMatrix)
        # Consume two input indices
        push!(parentindexes, thisindex[inputindex[j], inputindex[j+1]])
        j += 2
    elseif ...
end
S.parent[parentindexes...]

Unfortunately, this would be disastrous in terms of performance: each element access would allocate memory, and involves the running of a lot of poorly-typed code.

The better approach is to dispatch to specific methods to handle each type of stored index. That’s what reindex does: it dispatches on the type of the first stored index and consumes the appropriate number of input indices, and then it recurses on the remaining indices. In the case of S1, this expands to

Base.reindex(S1, S1.indexes, (i, j)) == (i, S1.indexes[2], S1.indexes[3][j])

for any pair of indices (i,j) (except CartesianIndexs and arrays thereof, see below).

This is the core of a SubArray; indexing methods depend upon reindex to do this index translation. Sometimes, though, we can avoid the indirection and make it even faster.

Linear indexing

Linear indexing can be implemented efficiently when the entire array has a single stride that separates successive elements, starting from some offset. This means that we can pre-compute these values and represent linear indexing simply as an addition and multiplication, avoiding the indirection of reindex and (more importantly) the slow computation of the cartesian coordinates entirely.

For SubArray types, the availability of efficient linear indexing is based purely on the types of the indices, and does not depend on values like the size of the parent array. You can ask whether a given set of indices supports fast linear indexing with the internal Base.viewindexing function:

julia> Base.viewindexing(S1.indexes)
Base.LinearSlow()

julia> Base.viewindexing(S2.indexes)
Base.LinearFast()

This is computed during construction of the SubArray and stored in the L type parameter as a boolean that encodes fast linear indexing support. While not strictly necessary, it means that we can define dispatch directly on SubArray{T,N,A,I,true} without any intermediaries.

Since this computation doesn’t depend on runtime values, it can miss some cases in which the stride happens to be uniform:

julia> A = reshape(1:4*2, 4, 2)
4×2 Base.ReshapedArray{Int64,2,UnitRange{Int64},Tuple{}}:
 1  5
 2  6
 3  7
 4  8

julia> diff(A[2:2:4,:][:])
3-element Array{Int64,1}:
 2
 2
 2

A view constructed as view(A, 2:2:4, :) happens to have uniform stride, and therefore linear indexing indeed could be performed efficiently. However, success in this case depends on the size of the array: if the first dimension instead were odd,

julia> A = reshape(1:5*2, 5, 2)
5×2 Base.ReshapedArray{Int64,2,UnitRange{Int64},Tuple{}}:
 1   6
 2   7
 3   8
 4   9
 5  10

julia> diff(A[2:2:4,:][:])
3-element Array{Int64,1}:
 2
 3
 2

then A[2:2:4,:] does not have uniform stride, so we cannot guarantee efficient linear indexing. Since we have to base this decision based purely on types encoded in the parameters of the SubArray, S = view(A, 2:2:4, :) cannot implement efficient linear indexing.

A few details

  • Note that the Base.reindex function is agnostic to the types of the input indices; it simply determines how and where the stored indices should be reindexed. It not only supports integer indices, but it supports non-scalar indexing, too. This means that views of views don’t need two levels of indirection; they can simply re-compute the indices into the original parent array!

  • Hopefully by now it’s fairly clear that supporting slices means that the dimensionality, given by the parameter N, is not necessarily equal to the dimensionality of the parent array or the length of the indexes tuple. Neither do user-supplied indices necessarily line up with entries in the indexes tuple (e.g., the second user-supplied index might correspond to the third dimension of the parent array, and the third element in the indexes tuple).

    What might be less obvious is that the dimensionality of the stored parent array must be equal to the number of effective indices in the indexes tuple. Some examples:

    A = reshape(1:35, 5, 7) # A 2d parent Array
    S = view(A, 2:7)         # A 1d view created by linear indexing
    S = view(A, :, :, 1:1)   # Appending extra indices is supported
    

    Naively, you’d think you could just set S.parent = A and S.indexes = (:,:,1:1), but supporting this dramatically complicates the reindexing process, especially for views of views. Not only do you need to dispatch on the types of the stored indices, but you need to examine whether a given index is the final one and “merge” any remaining stored indices together. This is not an easy task, and even worse: it’s slow since it implicitly depends upon linear indexing.

    Fortunately, this is precisely the computation that ReshapedArray performs, and it does so linearly if possible. Consequently, view ensures that the parent array is the appropriate dimensionality for the given indices by reshaping it if needed. The inner SubArray constructor ensures that this invariant is satisfied.

  • CartesianIndex and arrays thereof throw a nasty wrench into the reindex scheme. Recall that reindex simply dispatches on the type of the stored indices in order to determine how many passed indices should be used and where they should go. But with CartesianIndex, there’s no longer a one-to-one correspondence between the number of passed arguments and the number of dimensions that they index into. If we return to the above example of Base.reindex(S1, S1.indexes, (i, j)), you can see that the expansion is incorrect for i, j = CartesianIndex(), CartesianIndex(2,1). It should skip the CartesianIndex() entirely and return:

    (CartesianIndex(2,1)[1], S1.indexes[2], S1.indexes[3][CartesianIndex(2,1)[2]])
    

    Instead, though, we get:

    (CartesianIndex(), S1.indexes[2], S1.indexes[3][CartesianIndex(2,1)])
    

    Doing this correctly would require combined dispatch on both the stored and passed indices across all combinations of dimensionalities in an intractable manner. As such, reindex must never be called with CartesianIndex indices. Fortunately, the scalar case is easily handled by first flattening the CartesianIndex arguments to plain integers. Arrays of CartesianIndex, however, cannot be split apart into orthogonal pieces so easily. Before attempting to use reindex, view must ensure that there are no arrays of CartesianIndex in the argument list. If there are, it can simply “punt” by avoiding the reindex calculation entirely, constructing a nested SubArray with two levels of indirection instead.