Question about symmetric coloring

[amontoison]

For the following matrix, how can I know if I should use the "compressed columns" in the AD context associated to colors[i] or colors[j] to determine the coefficient H[i,j]?

10×10 SparseMatrixCSC{Float64, Int64} with 58 stored entries:
 2.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  2.0  2.0
 1.0  2.0   ⋅    ⋅    ⋅    ⋅    ⋅    ⋅   1.0  1.0
 1.0   ⋅   2.0   ⋅    ⋅    ⋅    ⋅    ⋅   1.0  1.0
 1.0   ⋅    ⋅   2.0   ⋅    ⋅    ⋅    ⋅   1.0  1.0
 1.0   ⋅    ⋅    ⋅   2.0   ⋅    ⋅    ⋅   1.0  1.0
 1.0   ⋅    ⋅    ⋅    ⋅   2.0   ⋅    ⋅   1.0  1.0
 1.0   ⋅    ⋅    ⋅    ⋅    ⋅   2.0   ⋅   1.0  1.0
 1.0   ⋅    ⋅    ⋅    ⋅    ⋅    ⋅   2.0  1.0  1.0
 2.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  2.0  1.0
 2.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  2.0

julia> colors = symmetric_coloring(H, GreedyColoringAlgorithm())
10-element Vector{Int64}:
 1
 2
 2
 2
 2
 2
 2
 2
 3
 4

[gdalle]

It depends which one is the only nonzero coefficient in its group of columns:

help?> ADTypes.symmetric_coloring
  symmetric_coloring(M::AbstractMatrix, ca::ColoringAlgorithm)::AbstractVector{<:Integer}


  Use algorithm ca to construct a symmetrically structurally orthogonal
  partition of the columns (or rows) of the symmetric matrix M.

  The result is a coloring vector c of length size(M, 1) == size(M, 2) such
  that for every non-zero coefficient M[i, j], at least one of the following
  conditions holds:

    •  column j is the only column of its color c[j] with a non-zero
       coefficient in row i;

    •  column i is the only column of its color c[i] with a non-zero
       coefficient in row j.

The star coloring algorithm guarantees that at least one of the conditions will be met.

[amontoison]

Thank you @gdalle!