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sparsematrix.jl
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# This file is a part of Julia. License is MIT: https://julialang.org/license
# Compressed sparse columns data structure
# Assumes that no zeros are stored in the data structure
# Assumes that row values in rowval for each column are sorted
# issorted(rowval[colptr[i]:(colptr[i+1]-1)]) == true
"""
SparseMatrixCSC{Tv,Ti<:Integer} <: AbstractSparseMatrix{Tv,Ti}
Matrix type for storing sparse matrices in the
[Compressed Sparse Column](@ref man-csc) format.
"""
struct SparseMatrixCSC{Tv,Ti<:Integer} <: AbstractSparseMatrix{Tv,Ti}
m::Int # Number of rows
n::Int # Number of columns
colptr::Vector{Ti} # Column i is in colptr[i]:(colptr[i+1]-1)
rowval::Vector{Ti} # Row indices of stored values
nzval::Vector{Tv} # Stored values, typically nonzeros
function SparseMatrixCSC{Tv,Ti}(m::Integer, n::Integer, colptr::Vector{Ti}, rowval::Vector{Ti},
nzval::Vector{Tv}) where {Tv,Ti<:Integer}
@noinline throwsz(str, lbl, k) =
throw(ArgumentError("number of $str ($lbl) must be ≥ 0, got $k"))
m < 0 && throwsz("rows", 'm', m)
n < 0 && throwsz("columns", 'n', n)
new(Int(m), Int(n), colptr, rowval, nzval)
end
end
function SparseMatrixCSC(m::Integer, n::Integer, colptr::Vector, rowval::Vector, nzval::Vector)
Tv = eltype(nzval)
Ti = promote_type(eltype(colptr), eltype(rowval))
SparseMatrixCSC{Tv,Ti}(m, n, colptr, rowval, nzval)
end
size(S::SparseMatrixCSC) = (S.m, S.n)
# Define an alias for views of a SparseMatrixCSC which include all rows and a unit range of the columns.
# Also define a union of SparseMatrixCSC and this view since many methods can be defined efficiently for
# this union by extracting the fields via the get function: getcolptr, getrowval, and getnzval. The key
# insight is that getcolptr on a SparseMatrixCSCView returns an offset view of the colptr of the
# underlying SparseMatrixCSC
const SparseMatrixCSCView{Tv,Ti} =
SubArray{Tv,2,SparseMatrixCSC{Tv,Ti},
Tuple{Base.Slice{Base.OneTo{Int}},I}} where {I<:AbstractUnitRange}
const SparseMatrixCSCUnion{Tv,Ti} = Union{SparseMatrixCSC{Tv,Ti}, SparseMatrixCSCView{Tv,Ti}}
getcolptr(S::SparseMatrixCSC) = S.colptr
getcolptr(S::SparseMatrixCSCView) = view(S.parent.colptr, first(axes(S, 2)):(last(axes(S, 2)) + 1))
getrowval(S::SparseMatrixCSC) = S.rowval
getrowval(S::SparseMatrixCSCView) = S.parent.rowval
getnzval( S::SparseMatrixCSC) = S.nzval
getnzval( S::SparseMatrixCSCView) = S.parent.nzval
nzvalview(S::SparseMatrixCSC) = view(S.nzval, 1:nnz(S))
"""
nnz(A)
Returns the number of stored (filled) elements in a sparse array.
# Examples
```jldoctest
julia> A = sparse(2I, 3, 3)
3×3 SparseMatrixCSC{Int64,Int64} with 3 stored entries:
[1, 1] = 2
[2, 2] = 2
[3, 3] = 2
julia> nnz(A)
3
```
"""
nnz(S::SparseMatrixCSC) = Int(S.colptr[S.n + 1] - 1)
nnz(S::ReshapedArray{T,1,<:SparseMatrixCSC}) where T = nnz(parent(S))
count(pred, S::SparseMatrixCSC) = count(pred, nzvalview(S)) + pred(zero(eltype(S)))*(prod(size(S)) - nnz(S))
"""
nonzeros(A)
Return a vector of the structural nonzero values in sparse array `A`. This
includes zeros that are explicitly stored in the sparse array. The returned
vector points directly to the internal nonzero storage of `A`, and any
modifications to the returned vector will mutate `A` as well. See
[`rowvals`](@ref) and [`nzrange`](@ref).
# Examples
```jldoctest
julia> A = sparse(2I, 3, 3)
3×3 SparseMatrixCSC{Int64,Int64} with 3 stored entries:
[1, 1] = 2
[2, 2] = 2
[3, 3] = 2
julia> nonzeros(A)
3-element Array{Int64,1}:
2
2
2
```
"""
nonzeros(S::SparseMatrixCSC) = S.nzval
"""
rowvals(A::SparseMatrixCSC)
Return a vector of the row indices of `A`. Any modifications to the returned
vector will mutate `A` as well. Providing access to how the row indices are
stored internally can be useful in conjunction with iterating over structural
nonzero values. See also [`nonzeros`](@ref) and [`nzrange`](@ref).
# Examples
```jldoctest
julia> A = sparse(2I, 3, 3)
3×3 SparseMatrixCSC{Int64,Int64} with 3 stored entries:
[1, 1] = 2
[2, 2] = 2
[3, 3] = 2
julia> rowvals(A)
3-element Array{Int64,1}:
1
2
3
```
"""
rowvals(S::SparseMatrixCSC) = S.rowval
"""
nzrange(A::SparseMatrixCSC, col::Integer)
Return the range of indices to the structural nonzero values of a sparse matrix
column. In conjunction with [`nonzeros`](@ref) and
[`rowvals`](@ref), this allows for convenient iterating over a sparse matrix :
A = sparse(I,J,V)
rows = rowvals(A)
vals = nonzeros(A)
m, n = size(A)
for i = 1:n
for j in nzrange(A, i)
row = rows[j]
val = vals[j]
# perform sparse wizardry...
end
end
"""
nzrange(S::SparseMatrixCSC, col::Integer) = S.colptr[col]:(S.colptr[col+1]-1)
function Base.show(io::IO, ::MIME"text/plain", S::SparseMatrixCSC)
xnnz = nnz(S)
print(io, S.m, "×", S.n, " ", typeof(S), " with ", xnnz, " stored ",
xnnz == 1 ? "entry" : "entries")
if xnnz != 0
print(io, ":")
show(io, S)
end
end
Base.show(io::IO, S::SparseMatrixCSC) = Base.show(convert(IOContext, io), S::SparseMatrixCSC)
function Base.show(io::IOContext, S::SparseMatrixCSC)
if nnz(S) == 0
return show(io, MIME("text/plain"), S)
end
limit::Bool = get(io, :limit, false)
rows = displaysize(io)[1] - 4 # -4 from [Prompt, header, newline after elements, new prompt]
will_fit = !limit || rows >= nnz(S) # Will the whole matrix fit when printed?
if rows <= 2 && !will_fit
print(io, "\n \u22ee")
return
end
iob = IOBuffer()
ioc = IOContext(iob, :compact => true)
function _format_line(r, col)
pad = ndigits(max(S.m, S.n))
print(ioc, " [", rpad(S.rowval[r], pad), ", ", lpad(col, pad), "] = ")
if isassigned(S.nzval, Int(r))
show(ioc, S.nzval[r])
else
print(ioc, Base.undef_ref_str)
end
return String(take!(iob))
end
if will_fit
print_count = nnz(S)
else
print_count = div(rows-1, 2)
end
count = 0
for col = 1:S.n, r = nzrange(S, col)
count += 1
print(io, "\n", _format_line(r, col))
count == print_count && break
end
if !will_fit
print(io, "\n \u22ee")
# find the column to start printing in for the last print_count elements
nextcol = searchsortedfirst(S.colptr, nnz(S) - print_count + 1)
for r = (nnz(S) - print_count + 1) : (S.colptr[nextcol] - 1)
print(io, "\n", _format_line(r, nextcol - 1))
end
# print all of the remaining columns
for col = nextcol:S.n, r = nzrange(S, col)
print(io, "\n", _format_line(r, col))
end
end
end
## Reshape
function sparse_compute_reshaped_colptr_and_rowval(colptrS::Vector{Ti}, rowvalS::Vector{Ti},
mS::Int, nS::Int, colptrA::Vector{Ti},
rowvalA::Vector{Ti}, mA::Int, nA::Int) where Ti
lrowvalA = length(rowvalA)
maxrowvalA = (lrowvalA > 0) ? maximum(rowvalA) : zero(Ti)
((length(colptrA) == (nA+1)) && (maximum(colptrA) <= (lrowvalA+1)) && (maxrowvalA <= mA)) || throw(BoundsError())
colptrS[1] = 1
colA = 1
colS = 1
ptr = 1
@inbounds while colA <= nA
offsetA = (colA - 1) * mA
while ptr <= colptrA[colA+1]-1
rowA = rowvalA[ptr]
i = offsetA + rowA - 1
colSn = div(i, mS) + 1
rowS = mod(i, mS) + 1
while colS < colSn
colptrS[colS+1] = ptr
colS += 1
end
rowvalS[ptr] = rowS
ptr += 1
end
colA += 1
end
@inbounds while colS <= nS
colptrS[colS+1] = ptr
colS += 1
end
end
function copy(ra::ReshapedArray{<:Any,2,<:SparseMatrixCSC})
mS,nS = size(ra)
a = parent(ra)
mA,nA = size(a)
numnz = nnz(a)
colptr = similar(a.colptr, nS+1)
rowval = similar(a.rowval)
nzval = copy(a.nzval)
sparse_compute_reshaped_colptr_and_rowval(colptr, rowval, mS, nS, a.colptr, a.rowval, mA, nA)
return SparseMatrixCSC(mS, nS, colptr, rowval, nzval)
end
## Alias detection and prevention
using Base: dataids, unaliascopy
Base.dataids(S::SparseMatrixCSC) = (dataids(S.colptr)..., dataids(S.rowval)..., dataids(S.nzval)...)
Base.unaliascopy(S::SparseMatrixCSC) = typeof(S)(S.m, S.n, unaliascopy(S.colptr), unaliascopy(S.rowval), unaliascopy(S.nzval))
## Constructors
copy(S::SparseMatrixCSC) =
SparseMatrixCSC(S.m, S.n, copy(S.colptr), copy(S.rowval), copy(S.nzval))
function copyto!(A::SparseMatrixCSC, B::SparseMatrixCSC)
# If the two matrices have the same length then all the
# elements in A will be overwritten.
if length(A) == length(B)
resize!(A.nzval, length(B.nzval))
resize!(A.rowval, length(B.rowval))
if size(A) == size(B)
# Simple case: we can simply copy the internal fields of B to A.
copyto!(A.colptr, B.colptr)
copyto!(A.rowval, B.rowval)
else
# This is like a "reshape B into A".
sparse_compute_reshaped_colptr_and_rowval(A.colptr, A.rowval, A.m, A.n, B.colptr, B.rowval, B.m, B.n)
end
else
length(A) >= length(B) || throw(BoundsError())
lB = length(B)
nnzA = nnz(A)
nnzB = nnz(B)
# Up to which col, row, and ptr in rowval/nzval will A be overwritten?
lastmodcolA = div(lB - 1, A.m) + 1
lastmodrowA = mod(lB - 1, A.m) + 1
lastmodptrA = A.colptr[lastmodcolA]
while lastmodptrA < A.colptr[lastmodcolA+1] && A.rowval[lastmodptrA] <= lastmodrowA
lastmodptrA += 1
end
lastmodptrA -= 1
if lastmodptrA >= nnzB
# A will have fewer non-zero elements; unmodified elements are kept at the end.
deleteat!(A.rowval, nnzB+1:lastmodptrA)
deleteat!(A.nzval, nnzB+1:lastmodptrA)
else
# A will have more non-zero elements; unmodified elements are kept at the end.
resize!(A.rowval, nnzB + nnzA - lastmodptrA)
resize!(A.nzval, nnzB + nnzA - lastmodptrA)
copyto!(A.rowval, nnzB+1, A.rowval, lastmodptrA+1, nnzA-lastmodptrA)
copyto!(A.nzval, nnzB+1, A.nzval, lastmodptrA+1, nnzA-lastmodptrA)
end
# Adjust colptr accordingly.
@inbounds for i in 2:length(A.colptr)
A.colptr[i] += nnzB - lastmodptrA
end
sparse_compute_reshaped_colptr_and_rowval(A.colptr, A.rowval, A.m, lastmodcolA-1, B.colptr, B.rowval, B.m, B.n)
end
copyto!(A.nzval, B.nzval)
return A
end
## similar
#
# parent method for similar that preserves stored-entry structure (for when new and old dims match)
function _sparsesimilar(S::SparseMatrixCSC, ::Type{TvNew}, ::Type{TiNew}) where {TvNew,TiNew}
newcolptr = copyto!(similar(S.colptr, TiNew), S.colptr)
newrowval = copyto!(similar(S.rowval, TiNew), S.rowval)
return SparseMatrixCSC(S.m, S.n, newcolptr, newrowval, similar(S.nzval, TvNew))
end
# parent methods for similar that preserves only storage space (for when new and old dims differ)
_sparsesimilar(S::SparseMatrixCSC, ::Type{TvNew}, ::Type{TiNew}, dims::Dims{2}) where {TvNew,TiNew} =
SparseMatrixCSC(dims..., fill(one(TiNew), last(dims)+1), similar(S.rowval, TiNew), similar(S.nzval, TvNew))
# parent method for similar that allocates an empty sparse vector (when new dims are single)
_sparsesimilar(S::SparseMatrixCSC, ::Type{TvNew}, ::Type{TiNew}, dims::Dims{1}) where {TvNew,TiNew} =
SparseVector(dims..., similar(S.rowval, TiNew, 0), similar(S.nzval, TvNew, 0))
#
# The following methods hook into the AbstractArray similar hierarchy. The first method
# covers similar(A[, Tv]) calls, which preserve stored-entry structure, and the latter
# methods cover similar(A[, Tv], shape...) calls, which preserve storage space when the shape
# calls for a two-dimensional result.
similar(S::SparseMatrixCSC{<:Any,Ti}, ::Type{TvNew}) where {Ti,TvNew} = _sparsesimilar(S, TvNew, Ti)
similar(S::SparseMatrixCSC{<:Any,Ti}, ::Type{TvNew}, dims::Union{Dims{1},Dims{2}}) where {Ti,TvNew} =
_sparsesimilar(S, TvNew, Ti, dims)
# The following methods cover similar(A, Tv, Ti[, shape...]) calls, which specify the
# result's index type in addition to its entry type, and aren't covered by the hooks above.
# The calls without shape again preserve stored-entry structure, whereas those with shape
# preserve storage space when the shape calls for a two-dimensional result.
similar(S::SparseMatrixCSC, ::Type{TvNew}, ::Type{TiNew}) where{TvNew,TiNew} =
_sparsesimilar(S, TvNew, TiNew)
similar(S::SparseMatrixCSC, ::Type{TvNew}, ::Type{TiNew}, dims::Union{Dims{1},Dims{2}}) where {TvNew,TiNew} =
_sparsesimilar(S, TvNew, TiNew, dims)
similar(S::SparseMatrixCSC, ::Type{TvNew}, ::Type{TiNew}, m::Integer) where {TvNew,TiNew} =
_sparsesimilar(S, TvNew, TiNew, (m,))
similar(S::SparseMatrixCSC, ::Type{TvNew}, ::Type{TiNew}, m::Integer, n::Integer) where {TvNew,TiNew} =
_sparsesimilar(S, TvNew, TiNew, (m, n))
# converting between SparseMatrixCSC types
AbstractMatrix{Tv}(A::SparseMatrixCSC) where {Tv} = SparseMatrixCSC{Tv}(A)
SparseMatrixCSC{Tv}(S::SparseMatrixCSC{Tv}) where {Tv} = copy(S)
SparseMatrixCSC{Tv}(S::SparseMatrixCSC) where {Tv} = SparseMatrixCSC{Tv,eltype(S.colptr)}(S)
SparseMatrixCSC{Tv,Ti}(S::SparseMatrixCSC{Tv,Ti}) where {Tv,Ti} = copy(S)
function SparseMatrixCSC{Tv,Ti}(S::SparseMatrixCSC) where {Tv,Ti}
eltypeTicolptr = convert(Vector{Ti}, S.colptr)
eltypeTirowval = convert(Vector{Ti}, S.rowval)
eltypeTvnzval = convert(Vector{Tv}, S.nzval)
return SparseMatrixCSC(S.m, S.n, eltypeTicolptr, eltypeTirowval, eltypeTvnzval)
end
# converting from other matrix types to SparseMatrixCSC (also see sparse())
SparseMatrixCSC(M::Matrix) = sparse(M)
SparseMatrixCSC(M::AbstractMatrix{Tv}) where {Tv} = SparseMatrixCSC{Tv,Int}(M)
SparseMatrixCSC{Tv}(M::AbstractMatrix{Tv}) where {Tv} = SparseMatrixCSC{Tv,Int}(M)
function SparseMatrixCSC{Tv,Ti}(M::AbstractMatrix) where {Tv,Ti}
@assert !has_offset_axes(M)
I = findall(x -> x != 0, M)
eltypeTiI = Ti[i[1] for i in I]
eltypeTiJ = Ti[i[2] for i in I]
eltypeTvV = Tv[M[i] for i in I]
return sparse_IJ_sorted!(eltypeTiI, eltypeTiJ, eltypeTvV, size(M)...)
end
function SparseMatrixCSC{Tv,Ti}(M::StridedMatrix) where {Tv,Ti}
nz = count(t -> t != 0, M)
colptr = zeros(Ti, size(M, 2) + 1)
nzval = Vector{Tv}(undef, nz)
rowval = Vector{Ti}(undef, nz)
colptr[1] = 1
cnt = 1
@inbounds for j in 1:size(M, 2)
for i in 1:size(M, 1)
v = M[i, j]
if v != 0
rowval[cnt] = i
nzval[cnt] = v
cnt += 1
end
end
colptr[j+1] = cnt
end
return SparseMatrixCSC(size(M, 1), size(M, 2), colptr, rowval, nzval)
end
SparseMatrixCSC(M::Adjoint{<:Any,<:SparseMatrixCSC}) = copy(M)
SparseMatrixCSC(M::Transpose{<:Any,<:SparseMatrixCSC}) = copy(M)
# converting from SparseMatrixCSC to other matrix types
function Matrix(S::SparseMatrixCSC{Tv}) where Tv
# Handle cases where zero(Tv) is not defined but the array is dense.
A = length(S) == nnz(S) ? Matrix{Tv}(undef, S.m, S.n) : zeros(Tv, S.m, S.n)
for Sj in 1:S.n
for Sk in nzrange(S, Sj)
Si = S.rowval[Sk]
Sv = S.nzval[Sk]
A[Si, Sj] = Sv
end
end
return A
end
Array(S::SparseMatrixCSC) = Matrix(S)
convert(T::Type{<:SparseMatrixCSC}, m::AbstractMatrix) = m isa T ? m : T(m)
float(S::SparseMatrixCSC) = SparseMatrixCSC(S.m, S.n, copy(S.colptr), copy(S.rowval), float.(S.nzval))
complex(S::SparseMatrixCSC) = SparseMatrixCSC(S.m, S.n, copy(S.colptr), copy(S.rowval), complex(copy(S.nzval)))
"""
sparse(A)
Convert an AbstractMatrix `A` into a sparse matrix.
# Examples
```jldoctest
julia> A = Matrix(1.0I, 3, 3)
3×3 Array{Float64,2}:
1.0 0.0 0.0
0.0 1.0 0.0
0.0 0.0 1.0
julia> sparse(A)
3×3 SparseMatrixCSC{Float64,Int64} with 3 stored entries:
[1, 1] = 1.0
[2, 2] = 1.0
[3, 3] = 1.0
```
"""
sparse(A::AbstractMatrix{Tv}) where {Tv} = convert(SparseMatrixCSC{Tv,Int}, A)
sparse(S::SparseMatrixCSC) = copy(S)
sparse_IJ_sorted!(I,J,V,m,n) = sparse_IJ_sorted!(I,J,V,m,n,+)
sparse_IJ_sorted!(I,J,V::AbstractVector{Bool},m,n) = sparse_IJ_sorted!(I,J,V,m,n,|)
function sparse_IJ_sorted!(I::AbstractVector{Ti}, J::AbstractVector{Ti},
V::AbstractVector,
m::Integer, n::Integer, combine::Function) where Ti<:Integer
@assert !has_offset_axes(I, J, V)
m = m < 0 ? 0 : m
n = n < 0 ? 0 : n
if isempty(V); return spzeros(eltype(V),Ti,m,n); end
cols = zeros(Ti, n+1)
cols[1] = 1 # For cumsum purposes
cols[J[1] + 1] = 1
lastdup = 1
ndups = 0
I_lastdup = I[1]
J_lastdup = J[1]
L = length(I)
@inbounds for k=2:L
if I[k] == I_lastdup && J[k] == J_lastdup
V[lastdup] = combine(V[lastdup], V[k])
ndups += 1
else
cols[J[k] + 1] += 1
lastdup = k-ndups
I_lastdup = I[k]
J_lastdup = J[k]
if ndups != 0
I[lastdup] = I_lastdup
V[lastdup] = V[k]
end
end
end
colptr = cumsum!(similar(cols), cols)
# Allow up to 20% slack
if ndups > 0.2*L
numnz = L-ndups
deleteat!(I, (numnz+1):L)
deleteat!(V, (numnz+1):length(V))
end
return SparseMatrixCSC(m, n, colptr, I, V)
end
"""
sparse(I, J, V,[ m, n, combine])
Create a sparse matrix `S` of dimensions `m x n` such that `S[I[k], J[k]] = V[k]`. The
`combine` function is used to combine duplicates. If `m` and `n` are not specified, they
are set to `maximum(I)` and `maximum(J)` respectively. If the `combine` function is not
supplied, `combine` defaults to `+` unless the elements of `V` are Booleans in which case
`combine` defaults to `|`. All elements of `I` must satisfy `1 <= I[k] <= m`, and all
elements of `J` must satisfy `1 <= J[k] <= n`. Numerical zeros in (`I`, `J`, `V`) are
retained as structural nonzeros; to drop numerical zeros, use [`dropzeros!`](@ref).
For additional documentation and an expert driver, see `Base.SparseArrays.sparse!`.
# Examples
```jldoctest
julia> Is = [1; 2; 3];
julia> Js = [1; 2; 3];
julia> Vs = [1; 2; 3];
julia> sparse(Is, Js, Vs)
3×3 SparseMatrixCSC{Int64,Int64} with 3 stored entries:
[1, 1] = 1
[2, 2] = 2
[3, 3] = 3
```
"""
function sparse(I::AbstractVector{Ti}, J::AbstractVector{Ti}, V::AbstractVector{Tv}, m::Integer, n::Integer, combine) where {Tv,Ti<:Integer}
@assert !has_offset_axes(I, J, V)
coolen = length(I)
if length(J) != coolen || length(V) != coolen
throw(ArgumentError(string("the first three arguments' lengths must match, ",
"length(I) (=$(length(I))) == length(J) (= $(length(J))) == length(V) (= ",
"$(length(V)))")))
end
if m == 0 || n == 0 || coolen == 0
if coolen != 0
if n == 0
throw(ArgumentError("column indices J[k] must satisfy 1 <= J[k] <= n"))
elseif m == 0
throw(ArgumentError("row indices I[k] must satisfy 1 <= I[k] <= m"))
end
end
SparseMatrixCSC(m, n, fill(one(Ti), n+1), Vector{Ti}(), Vector{Tv}())
else
# Allocate storage for CSR form
csrrowptr = Vector{Ti}(undef, m+1)
csrcolval = Vector{Ti}(undef, coolen)
csrnzval = Vector{Tv}(undef, coolen)
# Allocate storage for the CSC form's column pointers and a necessary workspace
csccolptr = Vector{Ti}(undef, n+1)
klasttouch = Vector{Ti}(undef, n)
# Allocate empty arrays for the CSC form's row and nonzero value arrays
# The parent method called below automagically resizes these arrays
cscrowval = Vector{Ti}()
cscnzval = Vector{Tv}()
sparse!(I, J, V, m, n, combine, klasttouch,
csrrowptr, csrcolval, csrnzval,
csccolptr, cscrowval, cscnzval)
end
end
sparse(I::AbstractVector, J::AbstractVector, V::AbstractVector, m::Integer, n::Integer, combine) =
sparse(AbstractVector{Int}(I), AbstractVector{Int}(J), V, m, n, combine)
"""
sparse!(I::AbstractVector{Ti}, J::AbstractVector{Ti}, V::AbstractVector{Tv},
m::Integer, n::Integer, combine, klasttouch::Vector{Ti},
csrrowptr::Vector{Ti}, csrcolval::Vector{Ti}, csrnzval::Vector{Tv},
[csccolptr::Vector{Ti}], [cscrowval::Vector{Ti}, cscnzval::Vector{Tv}] ) where {Tv,Ti<:Integer}
Parent of and expert driver for [`sparse`](@ref);
see [`sparse`](@ref) for basic usage. This method
allows the user to provide preallocated storage for `sparse`'s intermediate objects and
result as described below. This capability enables more efficient successive construction
of [`SparseMatrixCSC`](@ref)s from coordinate representations, and also enables extraction
of an unsorted-column representation of the result's transpose at no additional cost.
This method consists of three major steps: (1) Counting-sort the provided coordinate
representation into an unsorted-row CSR form including repeated entries. (2) Sweep through
the CSR form, simultaneously calculating the desired CSC form's column-pointer array,
detecting repeated entries, and repacking the CSR form with repeated entries combined;
this stage yields an unsorted-row CSR form with no repeated entries. (3) Counting-sort the
preceding CSR form into a fully-sorted CSC form with no repeated entries.
Input arrays `csrrowptr`, `csrcolval`, and `csrnzval` constitute storage for the
intermediate CSR forms and require `length(csrrowptr) >= m + 1`,
`length(csrcolval) >= length(I)`, and `length(csrnzval >= length(I))`. Input
array `klasttouch`, workspace for the second stage, requires `length(klasttouch) >= n`.
Optional input arrays `csccolptr`, `cscrowval`, and `cscnzval` constitute storage for the
returned CSC form `S`. `csccolptr` requires `length(csccolptr) >= n + 1`. If necessary,
`cscrowval` and `cscnzval` are automatically resized to satisfy
`length(cscrowval) >= nnz(S)` and `length(cscnzval) >= nnz(S)`; hence, if `nnz(S)` is
unknown at the outset, passing in empty vectors of the appropriate type (`Vector{Ti}()`
and `Vector{Tv}()` respectively) suffices, or calling the `sparse!` method
neglecting `cscrowval` and `cscnzval`.
On return, `csrrowptr`, `csrcolval`, and `csrnzval` contain an unsorted-column
representation of the result's transpose.
You may reuse the input arrays' storage (`I`, `J`, `V`) for the output arrays
(`csccolptr`, `cscrowval`, `cscnzval`). For example, you may call
`sparse!(I, J, V, csrrowptr, csrcolval, csrnzval, I, J, V)`.
For the sake of efficiency, this method performs no argument checking beyond
`1 <= I[k] <= m` and `1 <= J[k] <= n`. Use with care. Testing with `--check-bounds=yes`
is wise.
This method runs in `O(m, n, length(I))` time. The HALFPERM algorithm described in
F. Gustavson, "Two fast algorithms for sparse matrices: multiplication and permuted
transposition," ACM TOMS 4(3), 250-269 (1978) inspired this method's use of a pair of
counting sorts.
"""
function sparse!(I::AbstractVector{Ti}, J::AbstractVector{Ti},
V::AbstractVector{Tv}, m::Integer, n::Integer, combine, klasttouch::Vector{Ti},
csrrowptr::Vector{Ti}, csrcolval::Vector{Ti}, csrnzval::Vector{Tv},
csccolptr::Vector{Ti}, cscrowval::Vector{Ti}, cscnzval::Vector{Tv}) where {Tv,Ti<:Integer}
@assert !has_offset_axes(I, J, V)
# Compute the CSR form's row counts and store them shifted forward by one in csrrowptr
fill!(csrrowptr, Ti(0))
coolen = length(I)
@inbounds for k in 1:coolen
Ik = I[k]
if 1 > Ik || m < Ik
throw(ArgumentError("row indices I[k] must satisfy 1 <= I[k] <= m"))
end
csrrowptr[Ik+1] += Ti(1)
end
# Compute the CSR form's rowptrs and store them shifted forward by one in csrrowptr
countsum = Ti(1)
csrrowptr[1] = Ti(1)
@inbounds for i in 2:(m+1)
overwritten = csrrowptr[i]
csrrowptr[i] = countsum
countsum += overwritten
end
# Counting-sort the column and nonzero values from J and V into csrcolval and csrnzval
# Tracking write positions in csrrowptr corrects the row pointers
@inbounds for k in 1:coolen
Ik, Jk = I[k], J[k]
if Ti(1) > Jk || Ti(n) < Jk
throw(ArgumentError("column indices J[k] must satisfy 1 <= J[k] <= n"))
end
csrk = csrrowptr[Ik+1]
csrrowptr[Ik+1] = csrk + Ti(1)
csrcolval[csrk] = Jk
csrnzval[csrk] = V[k]
end
# This completes the unsorted-row, has-repeats CSR form's construction
# Sweep through the CSR form, simultaneously (1) calculating the CSC form's column
# counts and storing them shifted forward by one in csccolptr; (2) detecting repeated
# entries; and (3) repacking the CSR form with the repeated entries combined.
#
# Minimizing extraneous communication and nonlocality of reference, primarily by using
# only a single auxiliary array in this step, is the key to this method's performance.
fill!(csccolptr, Ti(0))
fill!(klasttouch, Ti(0))
writek = Ti(1)
newcsrrowptri = Ti(1)
origcsrrowptri = Ti(1)
origcsrrowptrip1 = csrrowptr[2]
@inbounds for i in 1:m
for readk in origcsrrowptri:(origcsrrowptrip1-Ti(1))
j = csrcolval[readk]
if klasttouch[j] < newcsrrowptri
klasttouch[j] = writek
if writek != readk
csrcolval[writek] = j
csrnzval[writek] = csrnzval[readk]
end
writek += Ti(1)
csccolptr[j+1] += Ti(1)
else
klt = klasttouch[j]
csrnzval[klt] = combine(csrnzval[klt], csrnzval[readk])
end
end
newcsrrowptri = writek
origcsrrowptri = origcsrrowptrip1
origcsrrowptrip1 != writek && (csrrowptr[i+1] = writek)
i < m && (origcsrrowptrip1 = csrrowptr[i+2])
end
# Compute the CSC form's colptrs and store them shifted forward by one in csccolptr
countsum = Ti(1)
csccolptr[1] = Ti(1)
@inbounds for j in 2:(n+1)
overwritten = csccolptr[j]
csccolptr[j] = countsum
countsum += overwritten
end
# Now knowing the CSC form's entry count, resize cscrowval and cscnzval if necessary
cscnnz = countsum - Ti(1)
length(cscrowval) < cscnnz && resize!(cscrowval, cscnnz)
length(cscnzval) < cscnnz && resize!(cscnzval, cscnnz)
# Finally counting-sort the row and nonzero values from the CSR form into cscrowval and
# cscnzval. Tracking write positions in csccolptr corrects the column pointers.
@inbounds for i in 1:m
for csrk in csrrowptr[i]:(csrrowptr[i+1]-Ti(1))
j = csrcolval[csrk]
x = csrnzval[csrk]
csck = csccolptr[j+1]
csccolptr[j+1] = csck + Ti(1)
cscrowval[csck] = i
cscnzval[csck] = x
end
end
SparseMatrixCSC(m, n, csccolptr, cscrowval, cscnzval)
end
function sparse!(I::AbstractVector{Ti}, J::AbstractVector{Ti},
V::AbstractVector{Tv}, m::Integer, n::Integer, combine, klasttouch::Vector{Ti},
csrrowptr::Vector{Ti}, csrcolval::Vector{Ti}, csrnzval::Vector{Tv},
csccolptr::Vector{Ti}) where {Tv,Ti<:Integer}
sparse!(I, J, V, m, n, combine, klasttouch,
csrrowptr, csrcolval, csrnzval,
csccolptr, Vector{Ti}(), Vector{Tv}())
end
function sparse!(I::AbstractVector{Ti}, J::AbstractVector{Ti},
V::AbstractVector{Tv}, m::Integer, n::Integer, combine, klasttouch::Vector{Ti},
csrrowptr::Vector{Ti}, csrcolval::Vector{Ti}, csrnzval::Vector{Tv}) where {Tv,Ti<:Integer}
sparse!(I, J, V, m, n, combine, klasttouch,
csrrowptr, csrcolval, csrnzval,
Vector{Ti}(undef, n+1), Vector{Ti}(), Vector{Tv}())
end
dimlub(I) = isempty(I) ? 0 : Int(maximum(I)) #least upper bound on required sparse matrix dimension
sparse(I,J,v::Number) = sparse(I, J, fill(v,length(I)))
sparse(I,J,V::AbstractVector) = sparse(I, J, V, dimlub(I), dimlub(J))
sparse(I,J,v::Number,m,n) = sparse(I, J, fill(v,length(I)), Int(m), Int(n))
sparse(I,J,V::AbstractVector,m,n) = sparse(I, J, V, Int(m), Int(n), +)
sparse(I,J,V::AbstractVector{Bool},m,n) = sparse(I, J, V, Int(m), Int(n), |)
sparse(I,J,v::Number,m,n,combine::Function) = sparse(I, J, fill(v,length(I)), Int(m), Int(n), combine)
function sparse(T::SymTridiagonal)
m = length(T.dv)
return sparse([1:m;2:m;1:m-1],[1:m;1:m-1;2:m],[T.dv;T.ev;T.ev], Int(m), Int(m))
end
function sparse(T::Tridiagonal)
m = length(T.d)
return sparse([1:m;2:m;1:m-1],[1:m;1:m-1;2:m],[T.d;T.dl;T.du], Int(m), Int(m))
end
function sparse(B::Bidiagonal)
m = length(B.dv)
B.uplo == 'U' || return sparse([1:m;2:m],[1:m;1:m-1],[B.dv;B.ev], Int(m), Int(m)) # lower bidiagonal
return sparse([1:m;1:m-1],[1:m;2:m],[B.dv;B.ev], Int(m), Int(m)) # upper bidiagonal
end
function sparse(D::Diagonal{T}) where T
m = length(D.diag)
return SparseMatrixCSC(m, m, Vector(1:(m+1)), Vector(1:m), Vector{T}(D.diag))
end
## Transposition and permutation methods
"""
halfperm!(X::SparseMatrixCSC{Tv,Ti}, A::SparseMatrixCSC{Tv,Ti},
q::AbstractVector{<:Integer}, f::Function = identity) where {Tv,Ti}
Column-permute and transpose `A`, simultaneously applying `f` to each entry of `A`, storing
the result `(f(A)Q)^T` (`map(f, transpose(A[:,q]))`) in `X`.
`X`'s dimensions must match those of `transpose(A)` (`X.m == A.n` and `X.n == A.m`), and `X`
must have enough storage to accommodate all allocated entries in `A` (`length(X.rowval) >= nnz(A)`
and `length(X.nzval) >= nnz(A)`). Column-permutation `q`'s length must match `A`'s column
count (`length(q) == A.n`).
This method is the parent of several methods performing transposition and permutation
operations on [`SparseMatrixCSC`](@ref)s. As this method performs no argument checking,
prefer the safer child methods (`[c]transpose[!]`, `permute[!]`) to direct use.
This method implements the `HALFPERM` algorithm described in F. Gustavson, "Two fast
algorithms for sparse matrices: multiplication and permuted transposition," ACM TOMS 4(3),
250-269 (1978). The algorithm runs in `O(A.m, A.n, nnz(A))` time and requires no space
beyond that passed in.
"""
function halfperm!(X::SparseMatrixCSC{Tv,Ti}, A::SparseMatrixCSC{Tv,Ti},
q::AbstractVector{<:Integer}, f::Function = identity) where {Tv,Ti}
_computecolptrs_halfperm!(X, A)
_distributevals_halfperm!(X, A, q, f)
return X
end
"""
Helper method for `halfperm!`. Computes `transpose(A[:,q])`'s column pointers, storing them
shifted one position forward in `X.colptr`; `_distributevals_halfperm!` fixes this shift.
"""
function _computecolptrs_halfperm!(X::SparseMatrixCSC{Tv,Ti}, A::SparseMatrixCSC{Tv,Ti}) where {Tv,Ti}
# Compute `transpose(A[:,q])`'s column counts. Store shifted forward one position in X.colptr.
fill!(X.colptr, 0)
@inbounds for k in 1:nnz(A)
X.colptr[A.rowval[k] + 1] += 1
end
# Compute `transpose(A[:,q])`'s column pointers. Store shifted forward one position in X.colptr.
X.colptr[1] = 1
countsum = 1
@inbounds for k in 2:(A.m + 1)
overwritten = X.colptr[k]
X.colptr[k] = countsum
countsum += overwritten
end
end
"""
Helper method for `halfperm!`. With `transpose(A[:,q])`'s column pointers shifted one
position forward in `X.colptr`, computes `map(f, transpose(A[:,q]))` by appropriately
distributing `A.rowval` and `f`-transformed `A.nzval` into `X.rowval` and `X.nzval`
respectively. Simultaneously fixes the one-position-forward shift in `X.colptr`.
"""
@noinline function _distributevals_halfperm!(X::SparseMatrixCSC{Tv,Ti},
A::SparseMatrixCSC{Tv,Ti}, q::AbstractVector{<:Integer}, f::Function) where {Tv,Ti}
@inbounds for Xi in 1:A.n
Aj = q[Xi]
for Ak in nzrange(A, Aj)
Ai = A.rowval[Ak]
Xk = X.colptr[Ai + 1]
X.rowval[Xk] = Xi
X.nzval[Xk] = f(A.nzval[Ak])
X.colptr[Ai + 1] += 1
end
end
return # kill potential type instability
end
function ftranspose!(X::SparseMatrixCSC{Tv,Ti}, A::SparseMatrixCSC{Tv,Ti}, f::Function) where {Tv,Ti}
# Check compatibility of source argument A and destination argument X
if X.n != A.m
throw(DimensionMismatch(string("destination argument `X`'s column count, ",
"`X.n (= $(X.n))`, must match source argument `A`'s row count, `A.m (= $(A.m))`")))
elseif X.m != A.n
throw(DimensionMismatch(string("destination argument `X`'s row count,
`X.m (= $(X.m))`, must match source argument `A`'s column count, `A.n (= $(A.n))`")))
elseif length(X.rowval) < nnz(A)
throw(ArgumentError(string("the length of destination argument `X`'s `rowval` ",
"array, `length(X.rowval) (= $(length(X.rowval)))`, must be greater than or ",
"equal to source argument `A`'s allocated entry count, `nnz(A) (= $(nnz(A)))`")))
elseif length(X.nzval) < nnz(A)
throw(ArgumentError(string("the length of destination argument `X`'s `nzval` ",
"array, `length(X.nzval) (= $(length(X.nzval)))`, must be greater than or ",
"equal to source argument `A`'s allocated entry count, `nnz(A) (= $(nnz(A)))`")))
end
halfperm!(X, A, 1:A.n, f)
end
transpose!(X::SparseMatrixCSC{Tv,Ti}, A::SparseMatrixCSC{Tv,Ti}) where {Tv,Ti} = ftranspose!(X, A, identity)
adjoint!(X::SparseMatrixCSC{Tv,Ti}, A::SparseMatrixCSC{Tv,Ti}) where {Tv,Ti} = ftranspose!(X, A, conj)
function ftranspose(A::SparseMatrixCSC{Tv,Ti}, f::Function) where {Tv,Ti}
X = SparseMatrixCSC(A.n, A.m,
Vector{Ti}(undef, A.m+1),
Vector{Ti}(undef, nnz(A)),
Vector{Tv}(undef, nnz(A)))
halfperm!(X, A, 1:A.n, f)
end
adjoint(A::SparseMatrixCSC) = Adjoint(A)
transpose(A::SparseMatrixCSC) = Transpose(A)
Base.copy(A::Adjoint{<:Any,<:SparseMatrixCSC}) = ftranspose(A.parent, x -> copy(adjoint(x)))
Base.copy(A::Transpose{<:Any,<:SparseMatrixCSC}) = ftranspose(A.parent, x -> copy(transpose(x)))
function Base.permutedims(A::SparseMatrixCSC, (a,b))
(a, b) == (2, 1) && return ftranspose(A, identity)
(a, b) == (1, 2) && return copy(A)
throw(ArgumentError("no valid permutation of dimensions"))
end
"""
unchecked_noalias_permute!(X::SparseMatrixCSC{Tv,Ti},
A::SparseMatrixCSC{Tv,Ti}, p::AbstractVector{<:Integer},
q::AbstractVector{<:Integer}, C::SparseMatrixCSC{Tv,Ti}) where {Tv,Ti}
See [`permute!`](@ref) for basic usage. Parent of `permute[!]`
methods operating on `SparseMatrixCSC`s that assume none of `X`, `A`, and `C` alias each
other. As this method performs no argument checking, prefer the safer child methods
(`permute[!]`) to direct use.
This method consists of two major steps: (1) Column-permute (`Q`,`I[:,q]`) and transpose `A`
to generate intermediate result `(AQ)^T` (`transpose(A[:,q])`) in `C`. (2) Column-permute
(`P^T`, I[:,p]) and transpose intermediate result `(AQ)^T` to generate result
`((AQ)^T P^T)^T = PAQ` (`A[p,q]`) in `X`.
The first step is a call to `halfperm!`, and the second is a variant on `halfperm!` that
avoids an unnecessary length-`nnz(A)` array-sweep and associated recomputation of column
pointers. See [`halfperm!`](:func:Base.SparseArrays.halfperm!) for additional algorithmic
information.
See also: `unchecked_aliasing_permute!`
"""
function unchecked_noalias_permute!(X::SparseMatrixCSC{Tv,Ti},
A::SparseMatrixCSC{Tv,Ti}, p::AbstractVector{<:Integer},
q::AbstractVector{<:Integer}, C::SparseMatrixCSC{Tv,Ti}) where {Tv,Ti}
halfperm!(C, A, q)
_computecolptrs_permute!(X, A, q, X.colptr)
_distributevals_halfperm!(X, C, p, identity)
return X
end
"""
unchecked_aliasing_permute!(A::SparseMatrixCSC{Tv,Ti},
p::AbstractVector{<:Integer}, q::AbstractVector{<:Integer},
C::SparseMatrixCSC{Tv,Ti}, workcolptr::Vector{Ti}) where {Tv,Ti}
See [`permute!`](@ref) for basic usage. Parent of `permute!`
methods operating on [`SparseMatrixCSC`](@ref)s where the source and destination matrices
are the same. See `unchecked_noalias_permute!`
for additional information; these methods are identical but for this method's requirement of
the additional `workcolptr`, `length(workcolptr) >= A.n + 1`, which enables efficient
handling of the source-destination aliasing.
"""
function unchecked_aliasing_permute!(A::SparseMatrixCSC{Tv,Ti},
p::AbstractVector{<:Integer}, q::AbstractVector{<:Integer},
C::SparseMatrixCSC{Tv,Ti}, workcolptr::Vector{Ti}) where {Tv,Ti}
halfperm!(C, A, q)
_computecolptrs_permute!(A, A, q, workcolptr)
_distributevals_halfperm!(A, C, p, identity)
return A
end
"""
Helper method for `unchecked_noalias_permute!` and `unchecked_aliasing_permute!`.
Computes `PAQ`'s column pointers, storing them shifted one position forward in `X.colptr`;
`_distributevals_halfperm!` fixes this shift. Saves some work relative to
`_computecolptrs_halfperm!` as described in `uncheckednoalias_permute!`'s documentation.
"""
function _computecolptrs_permute!(X::SparseMatrixCSC{Tv,Ti},
A::SparseMatrixCSC{Tv,Ti}, q::AbstractVector{<:Integer}, workcolptr::Vector{Ti}) where {Tv,Ti}
# Compute `A[p,q]`'s column counts. Store shifted forward one position in workcolptr.
@inbounds for k in 1:A.n
workcolptr[k+1] = A.colptr[q[k] + 1] - A.colptr[q[k]]
end
# Compute `A[p,q]`'s column pointers. Store shifted forward one position in X.colptr.
X.colptr[1] = 1
countsum = 1
@inbounds for k in 2:(X.n + 1)
overwritten = workcolptr[k]
X.colptr[k] = countsum
countsum += overwritten
end
end
"""
Helper method for `permute` and `permute!` methods operating on `SparseMatrixCSC`s.
Checks compatibility of source argument `A`, row-permutation argument `p`, and
column-permutation argument `q`.
"""
function _checkargs_sourcecompatperms_permute!(A::SparseMatrixCSC,
p::AbstractVector{<:Integer}, q::AbstractVector{<:Integer})
@assert !has_offset_axes(p, q)
if length(q) != A.n
throw(DimensionMismatch(string("the length of column-permutation argument `q`, ",
"`length(q) (= $(length(q)))`, must match source argument `A`'s column ",
"count, `A.n (= $(A.n))`")))
elseif length(p) != A.m
throw(DimensionMismatch(string("the length of row-permutation argument `p`, ",
"`length(p) (= $(length(p)))`, must match source argument `A`'s row count, ",
"`A.m (= $(A.m))`")))
end
end
"""
Helper method for `permute` and `permute!` methods operating on `SparseMatrixCSC`s.
Checks whether row- and column- permutation arguments `p` and `q` are valid permutations.
"""
function _checkargs_permutationsvalid_permute!(
p::AbstractVector{<:Integer}, pcheckspace::Vector{Ti},
q::AbstractVector{<:Integer}, qcheckspace::Vector{Ti}) where Ti<:Integer
if !_ispermutationvalid_permute!(p, pcheckspace)
throw(ArgumentError("row-permutation argument `p` must be a valid permutation"))
elseif !_ispermutationvalid_permute!(q, qcheckspace)
throw(ArgumentError("column-permutation argument `q` must be a valid permutation"))
end
end
function _ispermutationvalid_permute!(perm::AbstractVector{<:Integer},
checkspace::Vector{<:Integer})
@assert !has_offset_axes(perm)
n = length(perm)
checkspace[1:n] .= 0
for k in perm
(0 < k ≤ n) && ((checkspace[k] ⊻= 1) == 1) || return false
end
return true
end
"""
Helper method for `permute` and `permute!` methods operating on `SparseMatrixCSC`s.
Checks compatibility of source argument `A` and destination argument `X`.
"""
function _checkargs_sourcecompatdest_permute!(A::SparseMatrixCSC{Tv,Ti},
X::SparseMatrixCSC{Tv,Ti}) where {Tv,Ti}
if X.m != A.m
throw(DimensionMismatch(string("destination argument `X`'s row count, ",
"`X.m (= $(X.m))`, must match source argument `A`'s row count, `A.m (= $(A.m))`")))
elseif X.n != A.n