SubArrays

    One of the major design goals is to ensure high performance for views of both and IndexCartesian arrays. Furthermore, views of IndexLinear arrays should themselves be IndexLinear to the extent that it is possible.

    Consider making 2d slices of a 3d array:

    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,1,(2:3)[j]] and S2[i,j] with A[1,i,(2:3)[j]].

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

    SubArray design

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

    1. struct SubArray{T,N,P,I,L} <: AbstractArray{T,N}
    2.     parent::P
    3.     indices::I
    4.     offset1::Int       # for linear indexing and pointer, only valid when L==true
    5.     stride1::Int       # used only for linear indexing
    6.     ...
    7. 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{Float64,2,Array{Float64,3},Tuple{Base.Slice{Base.OneTo{Int64}},Int64,UnitRange{Int64}},false}. Note in particular the tuple parameter, which stores the types of the indices used to create S1. Likewise,

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

    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

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

    for any pair of indices (i,j) (except s 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 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:

    1. julia> Base.viewindexing(S1.indices)
    2. IndexCartesian()
    4. julia> Base.viewindexing(S2.indices)
    5. IndexLinear()

    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:

    1. julia> A = reshape(1:5*2, 5, 2)
    2. 5×2 reshape(::UnitRange{Int64}, 5, 2) with eltype Int64:
    3.  1   6
    4.  2   7
    5.  3   8
    6.  4   9
    8. julia> diff(A[2:2:4,:][:])
    9. 3-element Array{Int64,1}:
    10.  2
    12.  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.

    • 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 indices tuple. Neither do user-supplied indices necessarily line up with entries in the indices tuple (e.g., the second user-supplied index might correspond to the third dimension of the parent array, and the third element in the indices 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 indices tuple. Some examples:

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

      Naively, you’d think you could just set S.parent = A and S.indices = (:,:,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.indices, (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:

      Instead, though, we get:

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