Defining a multiple-dimensional array of arbitrary dimension in Julia

Question

Context

This question is related to this one.

In Julia, I wanted to make a 2-dimensional array of 5 x 5 with the (i, j) element having [i,j] like this:

5×5 Matrix{Vector{Int64}}:
 [1, 1]  [1, 2]  [1, 3]  [1, 4]  [1, 5]
 [2, 1]  [2, 2]  [2, 3]  [2, 4]  [2, 5]
 [3, 1]  [3, 2]  [3, 3]  [3, 4]  [3, 5]
 [4, 1]  [4, 2]  [4, 3]  [4, 4]  [4, 5]
 [5, 1]  [5, 2]  [5, 3]  [5, 4]  [5, 5]

I tried this with using array comprehension:

N = 5
L_2 = [[x1,x2] for x1 = 1:N, x2 = 1:N]

What I want to do

I want to generalize this definition for arbitrary dimension D.

L_1 = [[x1] for x1 = 1:N] # 1-dimensional
L_2 = [[x1,x2] for x1 = 1:N, x2 = 1:N] # 2-dimensional
L_3 = [[x1,x2,x3] for x1 = 1:N, x2 = 1:N,x3 = 1:N] # 3-dimensional
...

#L_D = ??? # D-dimensional

How can I define?

It is okay without using array comprehension.

Any information would be appreciated.

Answer 1

It doesn't seem like you want CartesianIndices for this, but for the record, CartesianIndices can take any Tuple of Int (more accurately Dims aka NTuple{N,Int} where N) to represent an Array's size. CartesianIndices((5,5)) for a 5x5, CartesianIndices((2,8,3)) for a 2x8x3, etc. You can make a quick NxNxNx... Tuple representing a size with D dimensions with NtotheD(N,D) = ntuple(i -> N, D).

Answer 2

You can generalize the vcat approach I have posted in the other answer like this:

julia> lattice(N, D) = vcat.((reshape(1:N, ntuple(j -> j == i ? N : 1, D)) for i in 1:D)...)
lattice (generic function with 1 method)

julia> lattice(2, 1)
2-element Vector{Vector{Int64}}:
 [1]
 [2]

julia> lattice(2, 2)
2×2 Matrix{Vector{Int64}}:
 [1, 1]  [1, 2]
 [2, 1]  [2, 2]

julia> lattice(2, 3)
2×2×2 Array{Vector{Int64}, 3}:
[:, :, 1] =
 [1, 1, 1]  [1, 2, 1]
 [2, 1, 1]  [2, 2, 1]

[:, :, 2] =
 [1, 1, 2]  [1, 2, 2]
 [2, 1, 2]  [2, 2, 2]

julia> lattice(2, 4)
2×2×2×2 Array{Vector{Int64}, 4}:
[:, :, 1, 1] =
 [1, 1, 1, 1]  [1, 2, 1, 1]
 [2, 1, 1, 1]  [2, 2, 1, 1]

[:, :, 2, 1] =
 [1, 1, 2, 1]  [1, 2, 2, 1]
 [2, 1, 2, 1]  [2, 2, 2, 1]

[:, :, 1, 2] =
 [1, 1, 1, 2]  [1, 2, 1, 2]
 [2, 1, 1, 2]  [2, 2, 1, 2]

[:, :, 2, 2] =
 [1, 1, 2, 2]  [1, 2, 2, 2]
 [2, 1, 2, 2]  [2, 2, 2, 2]

julia> lattice(2, 5)
2×2×2×2×2 Array{Vector{Int64}, 5}:
[:, :, 1, 1, 1] =
 [1, 1, 1, 1, 1]  [1, 2, 1, 1, 1]
 [2, 1, 1, 1, 1]  [2, 2, 1, 1, 1]

[:, :, 2, 1, 1] =
 [1, 1, 2, 1, 1]  [1, 2, 2, 1, 1]
 [2, 1, 2, 1, 1]  [2, 2, 2, 1, 1]

[:, :, 1, 2, 1] =
 [1, 1, 1, 2, 1]  [1, 2, 1, 2, 1]
 [2, 1, 1, 2, 1]  [2, 2, 1, 2, 1]

[:, :, 2, 2, 1] =
 [1, 1, 2, 2, 1]  [1, 2, 2, 2, 1]
 [2, 1, 2, 2, 1]  [2, 2, 2, 2, 1]

[:, :, 1, 1, 2] =
 [1, 1, 1, 1, 2]  [1, 2, 1, 1, 2]
 [2, 1, 1, 1, 2]  [2, 2, 1, 1, 2]

[:, :, 2, 1, 2] =
 [1, 1, 2, 1, 2]  [1, 2, 2, 1, 2]
 [2, 1, 2, 1, 2]  [2, 2, 2, 1, 2]

[:, :, 1, 2, 2] =
 [1, 1, 1, 2, 2]  [1, 2, 1, 2, 2]
 [2, 1, 1, 2, 2]  [2, 2, 1, 2, 2]

[:, :, 2, 2, 2] =
 [1, 1, 2, 2, 2]  [1, 2, 2, 2, 2]
 [2, 1, 2, 2, 2]  [2, 2, 2, 2, 2]