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reducedim.jl
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reducedim.jl
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# This file is a part of Julia. License is MIT: http://julialang.org/license
## Functions to compute the reduced shape
# for reductions that expand 0 dims to 1
reduced_dims(a::AbstractArray, region) = reduced_dims(size(a), region)
# for reductions that keep 0 dims as 0
reduced_dims0(a::AbstractArray, region) = reduced_dims0(size(a), region)
function reduced_dims{N}(siz::NTuple{N,Int}, d::Int, rd::Int)
if d < 1
throw(ArgumentError("dimension must be ≥ 1, got $d"))
elseif d == 1
return tuple(rd, siz[d+1:N]...)::typeof(siz)
elseif 1 < d < N
return tuple(siz[1:d-1]..., rd, siz[d+1:N]...)::typeof(siz)
elseif d == N
return tuple(siz[1:N-1]..., rd)::typeof(siz)
else
return siz
end
end
reduced_dims{N}(siz::NTuple{N,Int}, d::Int) = reduced_dims(siz, d, 1)
function reduced_dims0{N}(siz::NTuple{N,Int}, d::Int)
if d < 1
throw(ArgumentError("dimension must be ≥ 1, got $d"))
elseif d <= N
return reduced_dims(siz, d, (siz[d] == 0 ? 0 : 1))
else
return siz
end
end
function reduced_dims{N}(siz::NTuple{N,Int}, region)
rsiz = [siz...]
for i in region
isa(i, Integer) || throw(ArgumentError("reduced dimension(s) must be integers"))
d = convert(Int, i)::Int
if d < 1
throw(ArgumentError("region dimension(s) must be ≥ 1, got $d"))
elseif d <= N
rsiz[d] = 1
end
end
tuple(rsiz...)::typeof(siz)
end
function reduced_dims0{N}(siz::NTuple{N,Int}, region)
rsiz = [siz...]
for i in region
isa(i, Integer) || throw(ArgumentError("reduced dimension(s) must be integers"))
d = convert(Int, i)::Int
if d < 1
throw(ArgumentError("region dimension(s) must be ≥ 1, got $d"))
elseif d <= N
rsiz[d] = (rsiz[d] == 0 ? 0 : 1)
end
end
tuple(rsiz...)::typeof(siz)
end
function regionsize(a, region)
s = 1
for d in region
s *= size(a,d)
end
s
end
###### Generic reduction functions #####
## initialization
for (Op, initfun) in ((:(typeof(+)), :zero), (:(typeof(*)), :one), (:(typeof(scalarmax)), :typemin), (:(typeof(scalarmin)), :typemax), (:(typeof(max)), :typemin), (:(typeof(min)), :typemax))
@eval initarray!{T}(a::AbstractArray{T}, ::$(Op), init::Bool) = (init && fill!(a, $(initfun)(T)); a)
end
for (Op, initval) in ((:(typeof(&)), true), (:(typeof(|)), false))
@eval initarray!(a::AbstractArray, ::$(Op), init::Bool) = (init && fill!(a, $initval); a)
end
reducedim_initarray{R}(A::AbstractArray, region, v0, ::Type{R}) = fill!(similar(A,R,reduced_dims(A,region)), v0)
reducedim_initarray{T}(A::AbstractArray, region, v0::T) = reducedim_initarray(A, region, v0, T)
reducedim_initarray0{R}(A::AbstractArray, region, v0, ::Type{R}) = fill!(similar(A,R,reduced_dims0(A,region)), v0)
reducedim_initarray0{T}(A::AbstractArray, region, v0::T) = reducedim_initarray0(A, region, v0, T)
# TODO: better way to handle reducedim initialization
#
# The current scheme is basically following Steven G. Johnson's original implementation
#
promote_union(T::Union) = promote_type(T.types...)
promote_union(T) = T
function reducedim_init{S}(f, op::typeof(+), A::AbstractArray{S}, region)
_reducedim_init(f, op, zero, sum, A, region)
end
function reducedim_init{S}(f, op::typeof(*), A::AbstractArray{S}, region)
_reducedim_init(f, op, one, prod, A, region)
end
function _reducedim_init(f, op, fv, fop, A, region)
T = promote_union(eltype(A))
if method_exists(zero, Tuple{Type{T}})
x = f(zero(T))
z = op(fv(x), fv(x))
Tr = typeof(z) == typeof(x) && !isbits(T) ? T : typeof(z)
else
z = fv(fop(f, A))
Tr = typeof(z)
end
return reducedim_initarray(A, region, z, Tr)
end
reducedim_init{T}(f, op::typeof(max), A::AbstractArray{T}, region) = reducedim_init(f, scalarmax, A, region)
reducedim_init{T}(f, op::typeof(min), A::AbstractArray{T}, region) = reducedim_init(f, scalarmin, A, region)
reducedim_init{T}(f::Union{typeof(abs),typeof(abs2)}, op::typeof(max), A::AbstractArray{T}, region) = reducedim_init(f, scalarmax, A, region)
reducedim_init{T}(f, op::typeof(scalarmax), A::AbstractArray{T}, region) = reducedim_initarray0(A, region, typemin(f(zero(T))))
reducedim_init{T}(f, op::typeof(scalarmin), A::AbstractArray{T}, region) = reducedim_initarray0(A, region, typemax(f(zero(T))))
reducedim_init{T}(f::Union{typeof(abs),typeof(abs2)}, op::typeof(scalarmax), A::AbstractArray{T}, region) =
reducedim_initarray(A, region, zero(f(zero(T))))
reducedim_init(f, op::typeof(&), A::AbstractArray, region) = reducedim_initarray(A, region, true)
reducedim_init(f, op::typeof(|), A::AbstractArray, region) = reducedim_initarray(A, region, false)
# specialize to make initialization more efficient for common cases
for (IT, RT) in ((CommonReduceResult, :(eltype(A))), (SmallSigned, :Int), (SmallUnsigned, :UInt))
T = Union{[AbstractArray{t} for t in IT.types]..., [AbstractArray{Complex{t}} for t in IT.types]...}
@eval begin
reducedim_init(f::typeof(identity), op::typeof(+), A::$T, region) =
reducedim_initarray(A, region, zero($RT))
reducedim_init(f::typeof(identity), op::typeof(*), A::$T, region) =
reducedim_initarray(A, region, one($RT))
reducedim_init(f::Union{typeof(abs),typeof(abs2)}, op::typeof(+), A::$T, region) =
reducedim_initarray(A, region, real(zero($RT)))
reducedim_init(f::Union{typeof(abs),typeof(abs2)}, op::typeof(*), A::$T, region) =
reducedim_initarray(A, region, real(one($RT)))
end
end
reducedim_init(f::Union{typeof(identity),typeof(abs),typeof(abs2)}, op::typeof(+), A::AbstractArray{Bool}, region) =
reducedim_initarray(A, region, 0)
## generic (map)reduction
has_fast_linear_indexing(a::AbstractArray) = false
has_fast_linear_indexing(a::Array) = true
function check_reducedims(R, A)
# Check whether R has compatible dimensions w.r.t. A for reduction
#
# It returns an integer value (useful for choosing implementation)
# - If it reduces only along leading dimensions, e.g. sum(A, 1) or sum(A, (1, 2)),
# it returns the length of the leading slice. For the two examples above,
# it will be size(A, 1) or size(A, 1) * size(A, 2).
# - Otherwise, e.g. sum(A, 2) or sum(A, (1, 3)), it returns 0.
#
ndims(R) <= ndims(A) || throw(DimensionMismatch("Cannot reduce $(ndims(A))-dimensional array to $(ndims(R)) dimensions"))
lsiz = 1
had_nonreduc = false
for i = 1:ndims(A)
sRi = size(R, i)
sAi = size(A, i)
if sRi == 1
first(indices(R, i)) == first(indices(A, i)) ||
throw(DimensionMismatch("Reduction along dimension $i must use lower indices"))
if sAi > 1
if had_nonreduc
lsiz = 0 # to reduce along i, but some previous dimensions were non-reducing
else
lsiz *= sAi # if lsiz was set to zero, it will stay to be zero
end
end
else
indices(R, i) == indices(A, i) ||
throw(DimensionMismatch("Reduction on array with indices $(indices(A)) with output with indices $(indices(R))"))
had_nonreduc = true
end
end
return lsiz
end
function _mapreducedim!{T,N}(f, op, R::AbstractArray, A::AbstractArray{T,N})
lsiz = check_reducedims(R,A)
isempty(A) && return R
sizA1 = size(A, 1)
if has_fast_linear_indexing(A) && lsiz > 16
# use mapreduce_impl, which is probably better tuned to achieve higher performance
nslices = div(length(A), lsiz)
ibase = first(linearindices(A))-1
for i = 1:nslices
@inbounds R[i] = op(R[i], mapreduce_impl(f, op, A, ibase+1, ibase+lsiz))
ibase += lsiz
end
return R
end
IRmax = dims_tail(map(last, indices(R)), A)
if size(R, 1) == 1 && sizA1 > 1
# keep the accumulator as a local variable when reducing along the first dimension
i1 = first(indices(A, 1))
@inbounds for IA in CartesianRange(tail(indices(A)))
IR = min(IA, IRmax)
r = R[i1,IR]
@simd for i in indices(A, 1)
r = op(r, f(A[i, IA]))
end
R[i1,IR] = r
end
else
@inbounds for IA in CartesianRange(tail(indices(A)))
IR = min(IA, IRmax)
@simd for i in indices(A, 1)
R[i,IR] = op(R[i,IR], f(A[i,IA]))
end
end
end
return R
end
mapreducedim!(f, op, R::AbstractArray, A::AbstractArray) =
(_mapreducedim!(f, op, R, A); R)
reducedim!{RT}(op, R::AbstractArray{RT}, A::AbstractArray) =
mapreducedim!(identity, op, R, A, zero(RT))
mapreducedim(f, op, A::AbstractArray, region, v0) =
mapreducedim!(f, op, reducedim_initarray(A, region, v0), A)
mapreducedim{T}(f, op, A::AbstractArray{T}, region) =
mapreducedim!(f, op, reducedim_init(f, op, A, region), A)
reducedim(op, A::AbstractArray, region, v0) = mapreducedim(identity, op, A, region, v0)
reducedim(op, A::AbstractArray, region) = mapreducedim(identity, op, A, region)
##### Specific reduction functions #####
for (fname, op) in [(:sum, :+), (:prod, :*),
(:maximum, :scalarmax), (:minimum, :scalarmin),
(:all, :&), (:any, :|)]
fname! = Symbol(fname, '!')
@eval begin
$(fname!)(f::Function, r::AbstractArray, A::AbstractArray; init::Bool=true) =
mapreducedim!(f, $(op), initarray!(r, $(op), init), A)
$(fname!)(r::AbstractArray, A::AbstractArray; init::Bool=true) = $(fname!)(identity, r, A; init=init)
$(fname)(f::Function, A::AbstractArray, region) =
mapreducedim(f, $(op), A, region)
$(fname)(A::AbstractArray, region) = $(fname)(identity, A, region)
end
end
for (fname, fbase, fun) in [(:sumabs, :sum, :abs),
(:sumabs2, :sum, :abs2),
(:maxabs, :maximum, :abs),
(:minabs, :minimum, :abs)]
fname! = Symbol(fname, '!')
fbase! = Symbol(fbase, '!')
@eval begin
$(fname!)(r::AbstractArray, A::AbstractArray; init::Bool=true) =
$(fbase!)($(fun), r, A; init=init)
$(fname)(A::AbstractArray, region) = $(fbase)($(fun), A, region)
end
end
##### findmin & findmax #####
function findminmax!{T,N}(f, Rval, Rind, A::AbstractArray{T,N})
(isempty(Rval) || isempty(A)) && return Rval, Rind
check_reducedims(Rval, A)
for i = 1:N
indices(Rval, i) == indices(Rind, i) || throw(DimensionMismatch("Find-reduction: outputs must have the same indices"))
end
# If we're reducing along dimension 1, for efficiency we can make use of a temporary.
# Otherwise, keep the result in Rval/Rind so that we traverse A in storage order.
IRmax = dims_tail(map(last, indices(Rval)), A)
k = 0
if size(Rval, 1) < size(A, 1)
i1 = first(indices(A, 1))
@inbounds for IA in CartesianRange(tail(indices(A)))
IR = min(IRmax, IA)
tmpRv = Rval[i1,IR]
tmpRi = Rind[i1,IR]
for i in indices(A,1)
k += 1
tmpAv = A[i,IA]
if f(tmpAv, tmpRv)
tmpRv = tmpAv
tmpRi = k
end
end
Rval[i1,IR] = tmpRv
Rind[i1,IR] = tmpRi
end
else
@inbounds for IA in CartesianRange(tail(indices(A)))
IR = min(IRmax, IA)
for i in indices(A, 1)
k += 1
tmpAv = A[i,IA]
if f(tmpAv, Rval[i,IR])
Rval[i,IR] = tmpAv
Rind[i,IR] = k
end
end
end
end
Rval, Rind
end
"""
findmin!(rval, rind, A, [init=true]) -> (minval, index)
Find the minimum of `A` and the corresponding linear index along singleton
dimensions of `rval` and `rind`, and store the results in `rval` and `rind`.
"""
function findmin!{R}(rval::AbstractArray{R},
rind::AbstractArray,
A::AbstractArray;
init::Bool=true)
findminmax!(<, initarray!(rval, scalarmin, init), rind, A)
end
function findmin{T}(A::AbstractArray{T}, region)
if isempty(A)
return (similar(A, reduced_dims0(A, region)),
zeros(Int, reduced_dims0(A, region)))
end
return findminmax!(<, reducedim_initarray0(A, region, typemax(T)),
zeros(Int, reduced_dims0(A, region)), A)
end
"""
findmax!(rval, rind, A, [init=true]) -> (maxval, index)
Find the maximum of `A` and the corresponding linear index along singleton
dimensions of `rval` and `rind`, and store the results in `rval` and `rind`.
"""
function findmax!{R}(rval::AbstractArray{R},
rind::AbstractArray,
A::AbstractArray;
init::Bool=true)
findminmax!(>, initarray!(rval, scalarmax, init), rind, A)
end
function findmax{T}(A::AbstractArray{T}, region)
if isempty(A)
return (similar(A, reduced_dims0(A,region)),
zeros(Int, reduced_dims0(A,region)))
end
return findminmax!(>, reducedim_initarray0(A, region, typemin(T)),
zeros(Int, reduced_dims0(A, region)), A)
end
dims_tail{T}(dims::Tuple{}, Aref::AbstractArray{T,0}) = CartesianIndex(())
dims_tail{T,N}(dims::NTuple{N,Int}, Aref::AbstractArray{T,N}) = CartesianIndex(tail(dims))
@inline dims_tail{T,M,N}(dims::NTuple{M,Int}, Aref::AbstractArray{T,N}) = dims_tail(tuple(dims..., 1), Aref)