multidimensional.jl

# This file is a part of Julia. License is MIT: http:https://julialang.org/license

### Multidimensional iterators
module IteratorsMD
    import Base: eltype, length, size, start, done, next, last, in, getindex,
                 setindex!, linearindexing, min, max, zero, one, isless, eachindex,
                 ndims, iteratorsize, to_index

    importall ..Base.Operators
    import Base: simd_outer_range, simd_inner_length, simd_index
    using Base: LinearFast, LinearSlow, AbstractCartesianIndex, fill_to_length, tail

    export CartesianIndex, CartesianRange

    # CartesianIndex
    immutable CartesianIndex{N} <: AbstractCartesianIndex{N}
        I::NTuple{N,Int}
        CartesianIndex(index::NTuple{N,Integer}) = new(index)
    end

    CartesianIndex{N}(index::NTuple{N,Integer}) = CartesianIndex{N}(index)
    (::Type{CartesianIndex})(index::Integer...) = CartesianIndex(index)
    (::Type{CartesianIndex{N}}){N}(index::Vararg{Integer,N}) = CartesianIndex{N}(index)
    # Allow passing tuples smaller than N
    (::Type{CartesianIndex{N}}){N}(index::Tuple) = CartesianIndex{N}(fill_to_length(index, 1, Val{N}))
    (::Type{CartesianIndex{N}}){N}(index::Integer...) = CartesianIndex{N}(index)
    (::Type{CartesianIndex{N}}){N}() = CartesianIndex{N}(())
    # Un-nest passed CartesianIndexes
    CartesianIndex(index::Union{Integer, CartesianIndex}...) = CartesianIndex(flatten(index))
    flatten(I::Tuple{}) = I
    flatten(I::Tuple{Any}) = I
    flatten{N}(I::Tuple{CartesianIndex{N}}) = I[1].I
    @inline flatten(I) = _flatten(I...)
    @inline _flatten() = ()
    @inline _flatten(i, I...)                 = (i, _flatten(I...)...)
    @inline _flatten(i::CartesianIndex, I...) = (i.I..., _flatten(I...)...)
    CartesianIndex(index::Tuple{Vararg{Union{Integer, CartesianIndex}}}) = CartesianIndex(index...)

    # length
    length{N}(::CartesianIndex{N})=N
    length{N}(::Type{CartesianIndex{N}})=N

    # indexing
    getindex(index::CartesianIndex, i::Integer) = index.I[i]

    # zeros and ones
    zero{N}(::CartesianIndex{N}) = zero(CartesianIndex{N})
    zero{N}(::Type{CartesianIndex{N}}) = CartesianIndex(ntuple(x -> 0, Val{N}))
    one{N}(::CartesianIndex{N}) = one(CartesianIndex{N})
    one{N}(::Type{CartesianIndex{N}}) = CartesianIndex(ntuple(x -> 1, Val{N}))

    # arithmetic, min/max
    (-){N}(index::CartesianIndex{N}) = CartesianIndex{N}(map(-, index.I))
    (+){N}(index1::CartesianIndex{N}, index2::CartesianIndex{N}) = CartesianIndex{N}(map(+, index1.I, index2.I))
    (-){N}(index1::CartesianIndex{N}, index2::CartesianIndex{N}) = CartesianIndex{N}(map(-, index1.I, index2.I))
    min{N}(index1::CartesianIndex{N}, index2::CartesianIndex{N}) = CartesianIndex{N}(map(min, index1.I, index2.I))
    max{N}(index1::CartesianIndex{N}, index2::CartesianIndex{N}) = CartesianIndex{N}(map(max, index1.I, index2.I))

    (+){N}(index::CartesianIndex{N}, i::Integer) = CartesianIndex{N}(map(x->x+i, index.I))
    (+){N}(i::Integer, index::CartesianIndex{N}) = index+i
    (-){N}(index::CartesianIndex{N}, i::Integer) = CartesianIndex{N}(map(x->x-i, index.I))
    (-){N}(i::Integer, index::CartesianIndex{N}) = CartesianIndex{N}(map(x->i-x, index.I))
    (*){N}(a::Integer, index::CartesianIndex{N}) = CartesianIndex{N}(map(x->a*x, index.I))
    (*)(index::CartesianIndex,a::Integer)=*(a,index)

    # comparison
    @inline isless{N}(I1::CartesianIndex{N}, I2::CartesianIndex{N}) = _isless(0, I1.I, I2.I)
    @inline function _isless{N}(ret, I1::NTuple{N,Int}, I2::NTuple{N,Int})
        newret = ifelse(ret==0, icmp(I1[N], I2[N]), ret)
        _isless(newret, Base.front(I1), Base.front(I2))
    end
    _isless(ret, ::Tuple{}, ::Tuple{}) = ifelse(ret==1, true, false)
    icmp(a, b) = ifelse(isless(a,b), 1, ifelse(a==b, 0, -1))

    # Iteration
    immutable CartesianRange{I<:CartesianIndex}
        start::I
        stop::I
    end

    CartesianRange{N}(index::CartesianIndex{N}) = CartesianRange(one(index), index)
    CartesianRange(::Tuple{}) = CartesianRange{CartesianIndex{0}}(CartesianIndex{0}(()),CartesianIndex{0}(()))
    CartesianRange{N}(sz::NTuple{N,Int}) = CartesianRange(CartesianIndex(sz))
    CartesianRange{N}(rngs::NTuple{N,Union{Integer,AbstractUnitRange}}) = CartesianRange(CartesianIndex(map(first, rngs)), CartesianIndex(map(last, rngs)))

    ndims(R::CartesianRange) = length(R.start)
    ndims{I<:CartesianIndex}(::Type{CartesianRange{I}}) = length(I)

    eachindex(::LinearSlow, A::AbstractArray) = CartesianRange(indices(A))

    @inline eachindex(::LinearSlow, A::AbstractArray, B::AbstractArray...) = CartesianRange(maxsize((), A, B...))
    maxsize(sz) = sz
    @inline maxsize(sz, A, B...) = maxsize(maxt(sz, size(A)), B...)
    @inline maxt(a::Tuple{}, b::Tuple{}) = ()
    @inline maxt(a::Tuple{}, b::Tuple)   = b
    @inline maxt(a::Tuple,   b::Tuple{}) = a
    @inline maxt(a::Tuple,   b::Tuple)   = (max(a[1], b[1]), maxt(tail(a), tail(b))...)

    eltype{I}(::Type{CartesianRange{I}}) = I
    iteratorsize{I}(::Type{CartesianRange{I}}) = Base.HasShape()

    @inline function start{I<:CartesianIndex}(iter::CartesianRange{I})
        if any(map(>, iter.start.I, iter.stop.I))
            return iter.stop+1
        end
        iter.start
    end
    @inline function next{I<:CartesianIndex}(iter::CartesianRange{I}, state)
        state, I(inc(state.I, iter.start.I, iter.stop.I))
    end
    # increment & carry
    @inline inc(::Tuple{}, ::Tuple{}, ::Tuple{}) = ()
    @inline inc(state::Tuple{Int}, start::Tuple{Int}, stop::Tuple{Int}) = (state[1]+1,)
    @inline function inc(state, start, stop)
        if state[1] < stop[1]
            return (state[1]+1,tail(state)...)
        end
        newtail = inc(tail(state), tail(start), tail(stop))
        (start[1], newtail...)
    end
    @inline done{I<:CartesianIndex}(iter::CartesianRange{I}, state) = state.I[end] > iter.stop.I[end]

    # 0-d cartesian ranges are special-cased to iterate once and only once
    start{I<:CartesianIndex{0}}(iter::CartesianRange{I}) = false
    next{I<:CartesianIndex{0}}(iter::CartesianRange{I}, state) = iter.start, true
    done{I<:CartesianIndex{0}}(iter::CartesianRange{I}, state) = state

    size{I<:CartesianIndex}(iter::CartesianRange{I}) = map(dimlength, iter.start.I, iter.stop.I)
    dimlength(start, stop) = stop-start+1

    length(iter::CartesianRange) = prod(size(iter))

    last(iter::CartesianRange) = iter.stop

    to_index(c::CartesianIndex) = c

    @inline function in{I<:CartesianIndex}(i::I, r::CartesianRange{I})
        _in(true, i.I, r.start.I, r.stop.I)
    end
    _in(b, ::Tuple{}, ::Tuple{}, ::Tuple{}) = b
    @inline _in(b, i, start, stop) = _in(b & (start[1] <= i[1] <= stop[1]), tail(i), tail(start), tail(stop))

    simd_outer_range(iter::CartesianRange{CartesianIndex{0}}) = iter
    function simd_outer_range{I}(iter::CartesianRange{I})
        start = CartesianIndex(tail(iter.start.I))
        stop  = CartesianIndex(tail(iter.stop.I))
        CartesianRange(start, stop)
    end

    simd_inner_length{I<:CartesianIndex{0}}(iter::CartesianRange{I}, ::CartesianIndex) = 1
    simd_inner_length(iter::CartesianRange, I::CartesianIndex) = iter.stop[1]-iter.start[1]+1

    simd_index{I<:CartesianIndex{0}}(iter::CartesianRange{I}, ::CartesianIndex, I1::Int) = iter.start
    @inline function simd_index{N}(iter::CartesianRange, Ilast::CartesianIndex{N}, I1::Int)
        CartesianIndex((I1+iter.start[1], Ilast.I...))
    end

    # Split out the first N elements of a tuple
    @inline split{N}(t, V::Type{Val{N}}) = _split((), t, V)
    @inline _split(tN, trest, V) = _split((tN..., trest[1]), tail(trest), V)
    # exit either when we've exhausted the input tuple or when tN has length N
    @inline _split{N}(tN::NTuple{N}, ::Tuple{}, ::Type{Val{N}}) = tN, ()  # ambig.
    @inline _split{N}(tN,            ::Tuple{}, ::Type{Val{N}}) = tN, ()
    @inline _split{N}(tN::NTuple{N},  trest,    ::Type{Val{N}}) = tN, trest
end  # IteratorsMD


using .IteratorsMD

## Support for SubArray with arrays of CartesianIndex
function _indices_sub{N}(S::SubArray, pinds, i1::AbstractArray{CartesianIndex{N}}, I...)
    @_inline_meta
    (unsafe_indices(i1)..., _indices_sub(S, IteratorsMD.split(pinds, Val{N})[2], I...)...)
end

## Bounds-checking with CartesianIndex
@inline checkbounds_indices(::Type{Bool}, ::Tuple{}, I::Tuple{CartesianIndex,Vararg{Any}}) =
    checkbounds_indices(Bool, (), (I[1].I..., tail(I)...))
@inline checkbounds_indices(::Type{Bool}, IA::Tuple{Any}, I::Tuple{CartesianIndex,Vararg{Any}}) =
    checkbounds_indices(Bool, IA, (I[1].I..., tail(I)...))
@inline checkbounds_indices(::Type{Bool}, IA::Tuple, I::Tuple{CartesianIndex,Vararg{Any}}) =
    checkbounds_indices(Bool, IA, (I[1].I..., tail(I)...))

# Support indexing with an array of CartesianIndex{N}s
# Here we try to consume N of the indices (if there are that many available)
# The first two simply handle ambiguities
@inline function checkbounds_indices{N}(::Type{Bool}, ::Tuple{}, I::Tuple{AbstractArray{CartesianIndex{N}},Vararg{Any}})
    checkindex(Bool, (), I[1]) & checkbounds_indices(Bool, (), tail(I))
end
@inline function checkbounds_indices{N}(::Type{Bool}, IA::Tuple{Any}, I::Tuple{AbstractArray{CartesianIndex{N}},Vararg{Any}})
    checkindex(Bool, IA, I[1]) & checkbounds_indices(Bool, (), tail(I))
end
@inline function checkbounds_indices{N}(::Type{Bool}, IA::Tuple, I::Tuple{AbstractArray{CartesianIndex{N}},Vararg{Any}})
    IA1, IArest = IteratorsMD.split(IA, Val{N})
    checkindex(Bool, IA1, I[1]) & checkbounds_indices(Bool, IArest, tail(I))
end

function checkindex{N}(::Type{Bool}, inds::Tuple, I::AbstractArray{CartesianIndex{N}})
    b = true
    for i in I
        b &= checkbounds_indices(Bool, inds, (i,))
    end
    b
end

# combined count of all indices, including CartesianIndex and
# AbstractArray{CartesianIndex}
# rather than returning N, it returns an NTuple{N,Bool} so the result is inferrable
@inline index_ndims(i1, I...) = (true, index_ndims(I...)...)
@inline function index_ndims{N}(i1::CartesianIndex{N}, I...)
    (map(x->true, i1.I)..., index_ndims(I...)...)
end
@inline function index_ndims{N}(i1::AbstractArray{CartesianIndex{N}}, I...)
    (ntuple(x->true, Val{N})..., index_ndims(I...)...)
end
index_ndims() = ()

# Recursively compute the lengths of a list of indices, without dropping scalars
# These need to be inlined for more than 3 indexes
# Trailing CartesianIndex{0}s and arrays thereof are strange when used as
# trailing indexes -- they behave as though they were never there for the
# purposes of generalized linear indexing.
typealias CI0 Union{CartesianIndex{0}, AbstractArray{CartesianIndex{0}}}
index_lengths(A::AbstractArray, I::Colon) = (_length(A),)
@inline index_lengths(A::AbstractArray, I...) = index_lengths_dim(A, 1, I...)
index_lengths_dim(A, dim) = ()
index_lengths_dim(A, dim, ::Colon) = (trailingsize(indices(A), dim),)
index_lengths_dim(A, dim, ::Colon, i::CI0, I::CI0...) = (trailingsize(indices(A), dim), index_lengths_dim(A, dim+1, i, I...)...)
@inline index_lengths_dim(A, dim, ::Colon, i, I...) = (_length(indices(A, dim)), index_lengths_dim(A, dim+1, i, I...)...)
@inline index_lengths_dim(A, dim, ::Real, I...) = (1, index_lengths_dim(A, dim+1, I...)...)
@inline index_lengths_dim{N}(A, dim, ::CartesianIndex{N}, I...) = (1, index_lengths_dim(A, dim+N, I...)...)
@inline index_lengths_dim(A, dim, i::AbstractArray, I...) = (length(i), index_lengths_dim(A, dim+1, I...)...)
@inline index_lengths_dim(A, dim, i::AbstractArray{Bool}, I...) = (sum(i), index_lengths_dim(A, dim+1, I...)...)
@inline index_lengths_dim{N}(A, dim, i::AbstractArray{CartesianIndex{N}}, I...) = (length(i), index_lengths_dim(A, dim+N, I...)...)

# shape of array to create for getindex() with indexes I, dropping scalars
# returns a Tuple{Vararg{AbstractUnitRange}} of indices
index_shape(A::AbstractArray,  I::Colon)    = (linearindices(A),)
@inline index_shape(A::AbstractArray, I...) = index_shape_dim(indices(A), I...)
@inline index_shape_dim(inds::Tuple{Any}, ::Colon)          = inds
@inline index_shape_dim(inds,             ::Colon)          = (OneTo(trailingsize(inds)),)
@inline index_shape_dim(inds,             ::Colon, i::CI0, I::CI0...) =
    (OneTo(trailingsize(inds)), index_shape_dim((), i, I...)...)
@inline function index_shape_dim(inds,    ::Colon, i, I...)
    inds1, indstail = IteratorsMD.split(inds, Val{1})
    (inds1..., index_shape_dim(indstail, i, I...)...)
end
@inline index_shape_dim(inds,    ::Real...)             = ()
@inline index_shape_dim(inds,    ::Real, I...)          = index_shape_dim(safe_tail(inds), I...)
@inline index_shape_dim{N}(inds, ::CartesianIndex{N}, I...) =
    index_shape_dim(IteratorsMD.split(inds, Val{N})[2], I...)
@inline index_shape_dim(inds, i::AbstractArray, I...) =
    (indices(i)..., index_shape_dim(safe_tail(inds), I...)...)
@inline index_shape_dim(inds, i::AbstractArray{Bool}, I...) =
    (OneTo(sum(i)), index_shape_dim(safe_tail(inds), I...)...)
# single CartesianIndex version not needed because of call to flatten in _getindex...
# ...but array of CartesianIndex is not covered
@inline function index_shape_dim{N}(inds, i::AbstractArray{CartesianIndex{N}}, I...)
    indsN, indstail = IteratorsMD.split(inds, Val{N})
    (indices(i)..., index_shape_dim(indstail, I...)...)
end

# Convert Colon indices into explicit indices
@inline decolon(A::AbstractArray, ::Colon) = (linearindices(A),)
@inline decolon(A::AbstractArray, I...) = decolon_dim(indices(A), I...)
@inline decolon_dim(inds)  = ()
@inline decolon_dim(inds::Tuple{Any}, ::Colon)       = inds
@inline decolon_dim(inds,             ::Colon)       = (OneTo(trailingsize(inds)),)
@inline decolon_dim(inds,             ::Colon, i::CI0, I::CI0...) = (OneTo(trailingsize(inds)), i, I...)
@inline function decolon_dim(inds,    ::Colon, I...)
    inds1, indstail = IteratorsMD.split(inds, Val{1})
    (maybe_oneto(inds1...), decolon_dim(indstail, I...)...)
end
@inline decolon_dim(inds, i1, I...) = (i1, decolon_dim(safe_tail(inds), I...)...)
@inline function decolon_dim{N}(inds, i1::AbstractArray{CartesianIndex{N}}, I...)
    indsN, indstail = IteratorsMD.split(inds, Val{N})
    (i1, decolon_dim(indstail, I...)...)
end
maybe_oneto(i) = i
maybe_oneto() = OneTo(1)

### From abstractarray.jl: Internal multidimensional indexing definitions ###
getindex(x::Number, i::CartesianIndex{0}) = x
getindex(t::Tuple, I...) = getindex(t, IteratorsMD.flatten(I)...)

# These are not defined on directly on getindex to avoid
# ambiguities for AbstractArray subtypes. See the note in abstractarray.jl

# Note that it's most efficient to call checkbounds first, and then to_index
@inline function _getindex{T,N}(l::LinearIndexing, A::AbstractArray{T,N}, I::Vararg{Union{Real, AbstractArray, Colon},N})
    @boundscheck checkbounds(A, I...)
    _unsafe_getindex(l, A, I...)
end
# Explicitly allow linear indexing with one non-scalar index
@inline function _getindex(l::LinearIndexing, A::AbstractArray, i::Union{Real, AbstractArray, Colon})
    @boundscheck checkbounds(A, i)
    _unsafe_getindex(l, _maybe_reshape(l, A, (i,)), i)
end
# But we can speed up LinearSlow arrays by reshaping them to vectors:
_maybe_reshape(::LinearFast, A::AbstractArray, i) = A
_maybe_reshape(::LinearSlow, A::AbstractVector, i) = A
@inline _maybe_reshape(::LinearSlow, A::AbstractArray, i) = _maybe_reshape(LinearSlow(), index_ndims(i...), A)
@inline _maybe_reshape{T,N}(::LinearIndexing, ::NTuple{N}, A::AbstractArray{T,N}) = A
@inline _maybe_reshape{N}(::LinearIndexing, ::NTuple{N}, A) = reshape(A, Val{N})

@inline function _getindex{N}(l::LinearIndexing, A::AbstractArray, I::Vararg{Union{Real, AbstractArray, Colon},N}) # TODO: DEPRECATE FOR #14770
    @boundscheck checkbounds(A, I...)
    _unsafe_getindex(l, _maybe_reshape(l, A, I), I...)
end

@generated function _unsafe_getindex(::LinearIndexing, A::AbstractArray, I::Union{Real, AbstractArray, Colon}...)
    N = length(I)
    quote
        # This is specifically *not* inlined.
        @nexprs $N d->(I_d = to_index(I[d]))
        shape = @ncall $N index_shape A I
        dest = similar(A, shape)
        map(unsafe_length, indices(dest)) == map(unsafe_length, shape) || throw_checksize_error(dest, shape)
        @ncall $N _unsafe_getindex! dest A I
    end
end

# logical indexing optimization - don't use find (within to_index)
function _unsafe_getindex(::LinearIndexing, src::AbstractArray, I::AbstractArray{Bool})
    shape = index_shape(src, I)
    dest = similar(src, shape)
    map(unsafe_length, indices(dest)) == map(unsafe_length, shape) || throw_checksize_error(dest, shape)

    D = eachindex(dest)
    Ds = start(D)
    for (b, s) in zip(I, eachindex(src))
        @inbounds if b
            d, Ds = next(D, Ds)
            dest[d] = src[s]
        end
    end
    dest
end

# specialized form for LinearFast
function _unsafe_getindex(::LinearFast, src::AbstractArray, I::AbstractArray{Bool})
    shape = index_shape(src, I)
    dest = similar(src, shape)
    map(unsafe_length, indices(dest)) == map(unsafe_length, shape) || throw_checksize_error(dest, shape)

    D = eachindex(dest)
    Ds = start(D)
    s = first(linearindices(src))-1
    for i in eachindex(I)
        s += 1
        @inbounds if I[i]
            d, Ds = next(D, Ds)
            dest[d] = src[s]
        end
    end
    dest
end

# Always index with the exactly indices provided.
@generated function _unsafe_getindex!(dest::AbstractArray, src::AbstractArray, I::Union{Real, AbstractArray, Colon}...)
    N = length(I)
    quote
        $(Expr(:meta, :inline))
        @nexprs $N d->(I_d = I[d])
        J = @ncall $N decolon src I
        @nexprs $N d->(J_d = J[d])
        D = eachindex(dest)
        Ds = start(D)
        @inbounds @nloops $N j d->J_d begin
            d, Ds = next(D, Ds)
            dest[d] = @ncall $N getindex src j
        end
        dest
    end
end

@noinline throw_checksize_error(A, sz) = throw(DimensionMismatch("output array is the wrong size; expected $sz, got $(size(A))"))

## setindex! ##
# For multi-element setindex!, we check bounds, convert the indices (to_index),
# and ensure the value to set is either an AbstractArray or a Repeated scalar
# before redispatching to the _unsafe_batchsetindex!
_iterable(v::AbstractArray) = v
_iterable(v) = Iterators.repeated(v)
@inline function _setindex!{T,N}(l::LinearIndexing, A::AbstractArray{T,N}, x, J::Vararg{Union{Real,AbstractArray,Colon},N})
    @boundscheck checkbounds(A, J...)
    _unsafe_setindex!(l, A, x, J...)
end
@inline function _setindex!(l::LinearIndexing, A::AbstractArray, x, j::Union{Real,AbstractArray,Colon})
    @boundscheck checkbounds(A, j)
    _unsafe_setindex!(l, _maybe_reshape(l, A, (j,)), x, j)
    A
end
@inline function _setindex!{N}(l::LinearIndexing, A::AbstractArray, x, J::Vararg{Union{Real, AbstractArray, Colon},N}) # TODO: DEPRECATE FOR #14770
    @boundscheck checkbounds(A, J...)
    _unsafe_setindex!(l, _maybe_reshape(l, A, J), x, J...)
    A
end

@inline function _unsafe_setindex!(::LinearIndexing, A::AbstractArray, x, J::Union{Real,AbstractArray,Colon}...)
    _unsafe_batchsetindex!(A, _iterable(x), to_indexes(J...)...)
end

# 1-d logical indexing: override the above to avoid calling find (in to_index)
function _unsafe_setindex!(::LinearIndexing, A::AbstractArray, x, I::AbstractArray{Bool})
    X = _iterable(x)
    Xs = start(X)
    c = 0
    @inbounds for (iA, i) in zip(eachindex(A), eachindex(I))
        Ii = I[i]
        if Ii
            done(X, Xs) && throw_setindex_mismatch(x, c+1)
            (v, Xs) = next(X, Xs)
            A[iA] = v
            c += 1
        end
    end
    setindex_shape_check(X, c)
    A
end

# specialized form for LinearFast
function _unsafe_setindex!(::LinearFast, A::AbstractArray, x, I::AbstractArray{Bool})
    X = _iterable(x)
    Xs = start(X)
    iA = 0
    c = 0
    for i in eachindex(I)
        iA += 1
        @inbounds if I[i]
            done(X, Xs) && throw_setindex_mismatch(x, c+1)
            (v, Xs) = next(X, Xs)
            A[iA] = v
            c += 1
        end
    end
    setindex_shape_check(X, c)
    A
end

@generated function _unsafe_batchsetindex!(A::AbstractArray, X, I::Union{Real,AbstractArray,Colon}...)
    N = length(I)
    quote
        @nexprs $N d->(I_d = I[d])
        idxlens = @ncall $N index_lengths A I
        @ncall $N setindex_shape_check X (d->idxlens[d])
        J = @ncall $N decolon A I
        @nexprs $N d->(J_d = J[d])
        Xs = start(X)
        @inbounds @nloops $N j d->J_d begin
            v, Xs = next(X, Xs)
            @ncall $N setindex! A v j
        end
        A
    end
end

@propagate_inbounds function _getindex{T,N}(l::LinearIndexing, A::AbstractArray{T,N}, I::Union{Real,AbstractArray,Colon,CartesianIndex}...)
    getindex(A, IteratorsMD.flatten(I)...)
end
@propagate_inbounds function _setindex!{T,N}(l::LinearIndexing, A::AbstractArray{T,N}, v, I::Union{Real,AbstractArray,Colon,CartesianIndex}...)
    setindex!(A, v, IteratorsMD.flatten(I)...)
end

##

@generated function findn{T,N}(A::AbstractArray{T,N})
    quote
        nnzA = countnz(A)
        @nexprs $N d->(I_d = Array{Int}(nnzA))
        k = 1
        @nloops $N i A begin
            @inbounds if (@nref $N A i) != zero(T)
                @nexprs $N d->(I_d[k] = i_d)
                k += 1
            end
        end
        @ntuple $N I
    end
end

for (f, fmod, op) = ((:cummin, :_cummin!, :min), (:cummax, :_cummax!, :max))
    @eval function ($f)(v::AbstractVector)
        n = length(v)
        cur_val = v[1]
        res = similar(v, n)
        res[1] = cur_val
        for i in 2:n
            cur_val = ($op)(v[i], cur_val)
            res[i] = cur_val
        end
        return res
    end

    @eval function ($f)(A::AbstractArray, axis::Integer)
        axis > 0 || throw(ArgumentError("axis must be a positive integer"))
        res = similar(A)
        axis > ndims(A) && return copy!(res, A)
        inds = indices(A)
        if isempty(inds[axis])
            return res
        end
        R1 = CartesianRange(inds[1:axis-1])
        R2 = CartesianRange(inds[axis+1:end])
        ($fmod)(res, A, R1, R2, axis)
    end

    @eval @noinline function ($fmod)(res, A::AbstractArray, R1::CartesianRange, R2::CartesianRange, axis::Integer)
        inds = indices(A, axis)
        i1 = first(inds)
        for I2 in R2
            for I1 in R1
                res[I1, i1, I2] = A[I1, i1, I2]
            end
            for i = i1+1:last(inds)
                for I1 in R1
                    res[I1, i, I2] = ($op)(A[I1, i, I2], res[I1, i-1, I2])
                end
            end
        end
        res
    end

    @eval ($f)(A::AbstractArray) = ($f)(A, 1)
end

"""
    cumsum(A, dim=1)

Cumulative sum along a dimension `dim` (defaults to 1). See also [`cumsum!`](:func:`cumsum!`)
to use a preallocated output array, both for performance and to control the precision of the
output (e.g. to avoid overflow).

```jldoctest
julia> a = [1 2 3; 4 5 6]
2×3 Array{Int64,2}:
 1  2  3
 4  5  6

julia> cumsum(a,1)
2×3 Array{Int64,2}:
 1  2  3
 5  7  9

julia> cumsum(a,2)
2×3 Array{Int64,2}:
 1  3   6
 4  9  15
```
"""
cumsum{T}(A::AbstractArray{T}, axis::Integer=1) =  cumsum!(similar(A, Base.rcum_promote_type(+, T)), A, axis)
cumsum!(B, A::AbstractArray) = cumsum!(B, A, 1)
"""
    cumprod(A, dim=1)

Cumulative product along a dimension `dim` (defaults to 1). See also
[`cumprod!`](:func:`cumprod!`) to use a preallocated output array, both for performance and
to control the precision of the output (e.g. to avoid overflow).

```jldoctest
julia> a = [1 2 3; 4 5 6]
2×3 Array{Int64,2}:
 1  2  3
 4  5  6

julia> cumprod(a,1)
2×3 Array{Int64,2}:
 1   2   3
 4  10  18

julia> cumprod(a,2)
2×3 Array{Int64,2}:
 1   2    6
 4  20  120
```
"""
cumprod(A::AbstractArray, axis::Integer=1) = cumprod!(similar(A), A, axis)
cumprod!(B, A) = cumprod!(B, A, 1)

cumsum!(B, A, axis::Integer) = cumop!(+, B, A, axis)
cumprod!(B, A, axis::Integer) = cumop!(*, B, A, axis)

function cumop!(op, B, A, axis::Integer)
    axis > 0 || throw(ArgumentError("axis must be a positive integer"))
    inds_t = indices(A)
    indices(B) == inds_t || throw(DimensionMismatch("shape of B must match A"))
    axis > ndims(A) && return copy!(B, A)
    isempty(inds_t[axis]) && return B
    if axis == 1
        # We can accumulate to a temporary variable, which allows
        # register usage and will be slightly faster
        ind1 = inds_t[1]
        @inbounds for I in CartesianRange(tail(inds_t))
            tmp = convert(eltype(B), A[first(ind1), I])
            B[first(ind1), I] = tmp
            for i_1 = first(ind1)+1:last(ind1)
                tmp = op(tmp, A[i_1, I])
                B[i_1, I] = tmp
            end
        end
    else
        R1 = CartesianRange(indices(A)[1:axis-1])   # not type-stable
        R2 = CartesianRange(indices(A)[axis+1:end])
        _cumop!(op, B, A, R1, inds_t[axis], R2) # use function barrier
    end
    return B
end

@noinline function _cumop!(op, B, A, R1, ind, R2)
    # Copy the initial element in each 1d vector along dimension `axis`
    i = first(ind)
    @inbounds for J in R2, I in R1
        B[I, i, J] = A[I, i, J]
    end
    # Accumulate
    @inbounds for J in R2, i in first(ind)+1:last(ind), I in R1
        B[I, i, J] = op(B[I, i-1, J], A[I, i, J])
    end
    B
end

### from abstractarray.jl

function fill!{T}(A::AbstractArray{T}, x)
    xT = convert(T, x)
    for I in eachindex(A)
        @inbounds A[I] = xT
    end
    A
end

function copy!{T,N}(dest::AbstractArray{T,N}, src::AbstractArray{T,N})
    @boundscheck checkbounds(dest, indices(src)...)
    for I in eachindex(linearindexing(src,dest), src)
        @inbounds dest[I] = src[I]
    end
    dest
end

function copy!(dest::AbstractArray, Rdest::CartesianRange, src::AbstractArray, Rsrc::CartesianRange)
    isempty(Rdest) && return dest
    size(Rdest) == size(Rsrc) || throw(ArgumentError("source and destination must have same size (got $(size(Rsrc)) and $(size(Rdest)))"))
    @boundscheck checkbounds(dest, Rdest.start)
    @boundscheck checkbounds(dest, Rdest.stop)
    @boundscheck checkbounds(src, Rsrc.start)
    @boundscheck checkbounds(src, Rsrc.stop)
    deltaI = Rdest.start - Rsrc.start
    for I in Rsrc
        @inbounds dest[I+deltaI] = src[I]
    end
    dest
end

# circshift!
circshift!(dest::AbstractArray, src, ::Tuple{}) = copy!(dest, src)
"""
    circshift!(dest, src, shifts)

Circularly shift the data in `src`, storing the result in
`dest`. `shifts` specifies the amount to shift in each dimension.

The `dest` array must be distinct from the `src` array (they cannot
alias each other).

See also `circshift`.
"""
@noinline function circshift!{T,N}(dest::AbstractArray{T,N}, src, shiftamt::DimsInteger)
    dest === src && throw(ArgumentError("dest and src must be separate arrays"))
    inds = indices(src)
    indices(dest) == inds || throw(ArgumentError("indices of src and dest must match (got $inds and $(indices(dest)))"))
    _circshift!(dest, (), src, (), inds, fill_to_length(shiftamt, 0, Val{N}))
end
circshift!(dest::AbstractArray, src, shiftamt) = circshift!(dest, src, (shiftamt...,))

# For each dimension, we copy the first half of src to the second half
# of dest, and the second half of src to the first half of dest. This
# uses a recursive bifurcation strategy so that these splits can be
# encoded by ranges, which means that we need only one call to `mod`
# per dimension rather than one call per index.
# `rdest` and `rsrc` are tuples-of-ranges that grow one dimension at a
# time; when all the dimensions have been filled in, you call `copy!`
# for that block. In other words, in two dimensions schematically we
# have the following call sequence (--> means a call):
#   circshift!(dest, src, shiftamt) -->
#     _circshift!(dest, src, ("first half of dim1",)) -->
#       _circshift!(dest, src, ("first half of dim1", "first half of dim2")) --> copy!
#       _circshift!(dest, src, ("first half of dim1", "second half of dim2")) --> copy!
#     _circshift!(dest, src, ("second half of dim1",)) -->
#       _circshift!(dest, src, ("second half of dim1", "first half of dim2")) --> copy!
#       _circshift!(dest, src, ("second half of dim1", "second half of dim2")) --> copy!
@inline function _circshift!(dest, rdest, src, rsrc,
                             inds::Tuple{AbstractUnitRange,Vararg{Any}},
                             shiftamt::Tuple{Integer,Vararg{Any}})
    ind1, d = inds[1], shiftamt[1]
    s = mod(d, length(ind1))
    sf, sl = first(ind1)+s, last(ind1)-s
    r1, r2 = first(ind1):sf-1, sf:last(ind1)
    r3, r4 = first(ind1):sl, sl+1:last(ind1)
    tinds, tshiftamt = tail(inds), tail(shiftamt)
    _circshift!(dest, (rdest..., r1), src, (rsrc..., r4), tinds, tshiftamt)
    _circshift!(dest, (rdest..., r2), src, (rsrc..., r3), tinds, tshiftamt)
end
# At least one of inds, shiftamt is empty
function _circshift!(dest, rdest, src, rsrc, inds, shiftamt)
    copy!(dest, CartesianRange(rdest), src, CartesianRange(rsrc))
end

# circcopy!
"""
    circcopy!(dest, src)

Copy `src` to `dest`, indexing each dimension modulo its length.
`src` and `dest` must have the same size, but can be offset in
their indices; any offset results in a (circular) wraparound. If the
arrays have overlapping indices, then on the domain of the overlap
`dest` agrees with `src`.

```julia
julia> src = reshape(collect(1:16), (4,4))
4×4 Array{Int64,2}:
 1  5   9  13
 2  6  10  14
 3  7  11  15
 4  8  12  16

julia> dest = OffsetArray{Int}((0:3,2:5))

julia> circcopy!(dest, src)
OffsetArrays.OffsetArray{Int64,2,Array{Int64,2}} with indices 0:3×2:5:
 8  12  16  4
 5   9  13  1
 6  10  14  2
 7  11  15  3

julia> dest[1:3,2:4] == src[1:3,2:4]
true
```
"""
function circcopy!(dest, src)
    dest === src && throw(ArgumentError("dest and src must be separate arrays"))
    indssrc, indsdest = indices(src), indices(dest)
    if (szsrc = map(length, indssrc)) != (szdest = map(length, indsdest))
        throw(DimensionMismatch("src and dest must have the same sizes (got $szsrc and $szdest)"))
    end
    shift = map((isrc, idest)->first(isrc)-first(idest), indssrc, indsdest)
    all(x->x==0, shift) && return copy!(dest, src)
    _circcopy!(dest, (), indsdest, src, (), indssrc)
end

# This uses the same strategy described above for _circshift!
@inline function _circcopy!(dest, rdest, indsdest::Tuple{AbstractUnitRange,Vararg{Any}},
                            src,  rsrc,  indssrc::Tuple{AbstractUnitRange,Vararg{Any}})
    indd1, inds1 = indsdest[1], indssrc[1]
    l = length(indd1)
    s = mod(first(inds1)-first(indd1), l)
    sdf = first(indd1)+s
    rd1, rd2 = first(indd1):sdf-1, sdf:last(indd1)
    ssf = last(inds1)-s
    rs1, rs2 = first(inds1):ssf, ssf+1:last(inds1)
    tindsd, tindss = tail(indsdest), tail(indssrc)
    _circcopy!(dest, (rdest..., rd1), tindsd, src, (rsrc..., rs2), tindss)
    _circcopy!(dest, (rdest..., rd2), tindsd, src, (rsrc..., rs1), tindss)
end

# At least one of indsdest, indssrc are empty (and both should be, since we've checked)
function _circcopy!(dest, rdest, indsdest, src, rsrc, indssrc)
    copy!(dest, CartesianRange(rdest), src, CartesianRange(rsrc))
end

### BitArrays

## getindex

# contiguous multidimensional indexing: if the first dimension is a range,
# we can get some performance from using copy_chunks!
@inline function _unsafe_getindex!(X::BitArray, B::BitArray, I0::Union{UnitRange{Int},Colon})
    copy_chunks!(X.chunks, 1, B.chunks, indexoffset(I0)+1, index_lengths(B, I0)[1])
    return X
end

# Optimization where the inner dimension is contiguous improves perf dramatically
@generated function _unsafe_getindex!(X::BitArray, B::BitArray, I0::Union{Colon,UnitRange{Int}}, I::Union{Int,UnitRange{Int},Colon}...)
    N = length(I)
    quote
        $(Expr(:meta, :inline))
        @nexprs $N d->(I_d = I[d])

        idxlens = @ncall $N index_lengths B I0 d->I[d]

        f0 = indexoffset(I0)+1
        l0 = idxlens[1]

        gap_lst_1 = 0
        @nexprs $N d->(gap_lst_{d+1} = idxlens[d+1])
        stride = 1
        ind = f0
        @nexprs $N d->begin
            stride *= size(B, d)
            stride_lst_d = stride
            ind += stride * indexoffset(I_d)
            gap_lst_{d+1} *= stride
        end

        storeind = 1
        Xc, Bc = X.chunks, B.chunks
        @nloops($N, i, d->(1:idxlens[d+1]),
                d->nothing, # PRE
                d->(ind += stride_lst_d - gap_lst_d), # POST
                begin # BODY
                    copy_chunks!(Xc, storeind, Bc, ind, l0)
                    storeind += l0
                end)
        return X
    end
end

# in the general multidimensional non-scalar case, can we do about 10% better
# in most cases by manually hoisting the bitarray chunks access out of the loop
# (This should really be handled by the compiler or with an immutable BitArray)
@generated function _unsafe_getindex!(X::BitArray, B::BitArray, I::Union{Int,AbstractArray{Int},Colon}...)
    N = length(I)
    quote
        $(Expr(:meta, :inline))
        stride_1 = 1
        @nexprs $N d->(stride_{d+1} = stride_d*size(B, d))
        $(Symbol(:offset_, N)) = 1
        ind = 0
        Xc, Bc = X.chunks, B.chunks
        idxlens = @ncall $N index_lengths B d->I[d]
        @nloops $N i d->(1:idxlens[d]) d->(@inbounds offset_{d-1} = offset_d + (I[d][i_d]-1)*stride_d) begin
            ind += 1
            unsafe_bitsetindex!(Xc, unsafe_bitgetindex(Bc, offset_0), ind)
        end
        return X
    end
end

## setindex!

# contiguous multidimensional indexing: if the first dimension is a range,
# we can get some performance from using copy_chunks!

@inline function setindex!(B::BitArray, X::Union{BitArray,Array}, I0::Union{Colon,UnitRange{Int}})
    @boundscheck checkbounds(B, I0)
    l0 = index_lengths(B, I0)[1]
    setindex_shape_check(X, l0)
    l0 == 0 && return B
    f0 = indexoffset(I0)+1
    copy_to_bitarray_chunks!(B.chunks, f0, X, 1, l0)
    return B
end

@inline function setindex!(B::BitArray, x, I0::Union{Colon,UnitRange{Int}})
    @boundscheck checkbounds(B, I0)
    y = Bool(x)
    l0 = index_lengths(B, I0)[1]
    l0 == 0 && return B
    f0 = indexoffset(I0)+1
    fill_chunks!(B.chunks, y, f0, l0)
    return B
end

@inline function setindex!(B::BitArray, X::Union{BitArray,Array}, I0::Union{Colon,UnitRange{Int}}, I::Union{Int,UnitRange{Int},Colon}...)
    @boundscheck checkbounds(B, I0, I...)
    _unsafe_setindex!(B, X, I0, I...)
end
@generated function _unsafe_setindex!(B::BitArray, X::Union{BitArray,Array}, I0::Union{Colon,UnitRange{Int}}, I::Union{Int,UnitRange{Int},Colon}...)
    N = length(I)
    rangeexp = [I[d] === Colon ? :(1:size(B, $(d+1))) : :(I[$d]) for d = 1:N]
    quote
        idxlens = @ncall $N index_lengths B I0 d->I[d]
        @ncall $N setindex_shape_check X idxlens[1] d->idxlens[d+1]
        isempty(X) && return B
        f0 = indexoffset(I0)+1
        l0 = idxlens[1]

        gap_lst_1 = 0
        @nexprs $N d->(gap_lst_{d+1} = idxlens[d+1])
        stride = 1
        ind = f0
        @nexprs $N d->begin
            stride *= size(B, d)
            stride_lst_d = stride
            ind += stride * indexoffset(I[d])
            gap_lst_{d+1} *= stride
        end

        refind = 1
        Bc = B.chunks
        @nloops($N, i, d->$rangeexp[d],
                d->nothing, # PRE
                d->(ind += stride_lst_d - gap_lst_d), # POST
                begin # BODY
                    copy_to_bitarray_chunks!(Bc, ind, X, refind, l0)
                    refind += l0
                end)

        return B
    end
end

@inline function setindex!(B::BitArray, x, I0::Union{Colon,UnitRange{Int}}, I::Union{Int,UnitRange{Int},Colon}...)
    @boundscheck checkbounds(B, I0, I...)
    _unsafe_setindex!(B, x, I0, I...)
end
@generated function _unsafe_setindex!(B::BitArray, x, I0::Union{Colon,UnitRange{Int}}, I::Union{Int,UnitRange{Int},Colon}...)
    N = length(I)
    rangeexp = [I[d] === Colon ? :(1:size(B, $(d+1))) : :(I[$d]) for d = 1:N]
    quote
        y = Bool(x)
        idxlens = @ncall $N index_lengths B I0 d->I[d]

        f0 = indexoffset(I0)+1
        l0 = idxlens[1]
        l0 == 0 && return B
        @nexprs $N d->(isempty(I[d]) && return B)

        gap_lst_1 = 0
        @nexprs $N d->(gap_lst_{d+1} = idxlens[d+1])
        stride = 1
        ind = f0
        @nexprs $N d->begin
            stride *= size(B, d)
            stride_lst_d = stride
            ind += stride * indexoffset(I[d])
            gap_lst_{d+1} *= stride
        end

        @nloops($N, i, d->$rangeexp[d],
                d->nothing, # PRE
                d->(ind += stride_lst_d - gap_lst_d), # POST
                fill_chunks!(B.chunks, y, ind, l0) # BODY
                )

        return B
    end
end

## findn

@generated function findn{N}(B::BitArray{N})
    quote
        nnzB = countnz(B)
        I = ntuple(x->Array{Int}(nnzB), $N)
        if nnzB > 0
            count = 1
            @nloops $N i B begin
                if (@nref $N B i) # TODO: should avoid bounds checking
                    @nexprs $N d->(I[d][count] = i_d)
                    count += 1
                end
            end
        end
        return I
    end
end

## isassigned

@generated function isassigned(B::BitArray, I_0::Int, I::Int...)
    N = length(I)
    quote
        @nexprs $N d->(I_d = I[d])
        stride = 1
        index = I_0
        @nexprs $N d->begin
            l = size(B,d)
            stride *= l
            1 <= I_{d-1} <= l || return false
            index += (I_d - 1) * stride
        end
        return isassigned(B, index)
    end
end

## permutedims

## Permute array dims ##

function permutedims(B::StridedArray, perm)
    dimsB = size(B)
    ndimsB = length(dimsB)
    (ndimsB == length(perm) && isperm(perm)) || throw(ArgumentError("no valid permutation of dimensions"))
    dimsP = ntuple(i->dimsB[perm[i]], ndimsB)::typeof(dimsB)
    P = similar(B, dimsP)
    permutedims!(P, B, perm)
end

function checkdims_perm{TP,TB,N}(P::AbstractArray{TP,N}, B::AbstractArray{TB,N}, perm)
    indsB = indices(B)
    length(perm) == N || throw(ArgumentError("expected permutation of size $N, but length(perm)=$(length(perm))"))
    isperm(perm) || throw(ArgumentError("input is not a permutation"))
    indsP = indices(P)
    for i = 1:length(perm)
        indsP[i] == indsB[perm[i]] || throw(DimensionMismatch("destination tensor of incorrect size"))
    end
    nothing
end

for (V, PT, BT) in [((:N,), BitArray, BitArray), ((:T,:N), Array, StridedArray)]
    @eval @generated function permutedims!{$(V...)}(P::$PT{$(V...)}, B::$BT{$(V...)}, perm)
        quote
            checkdims_perm(P, B, perm)

            #calculates all the strides
            strides_1 = 0
            @nexprs $N d->(strides_{d+1} = stride(B, perm[d]))

            #Creates offset, because indexing starts at 1
            offset = 1 - sum(@ntuple $N d->strides_{d+1})

            if isa(B, SubArray)
                offset += first_index(B::SubArray) - 1
                B = B.parent
            end

            ind = 1
            @nexprs 1 d->(counts_{$N+1} = strides_{$N+1}) # a trick to set counts_($N+1)
            @nloops($N, i, P,
                    d->(counts_d = strides_d), # PRE
                    d->(counts_{d+1} += strides_{d+1}), # POST
                    begin # BODY
                        sumc = sum(@ntuple $N d->counts_{d+1})
                        @inbounds P[ind] = B[sumc+offset]
                        ind += 1
                    end)

            return P
        end
    end
end

## unique across dim

# TODO: this doesn't fit into the new hashing scheme in any obvious way

immutable Prehashed
    hash::UInt
end
hash(x::Prehashed) = x.hash

"""
    unique(itr[, dim])

Returns an array containing only the unique elements of the iterable `itr`, in
the order that the first of each set of equivalent elements originally appears.
If `dim` is specified, returns unique regions of the array `itr` along `dim`.
"""
@generated function unique{T,N}(A::AbstractArray{T,N}, dim::Int)
    inds = inds -> zeros(UInt, inds)
    quote
        1 <= dim <= $N || return copy(A)
        hashes = similar($inds, indices(A, dim))

        # Compute hash for each row
        k = 0
        @nloops $N i A d->(if d == dim; k = i_d; end) begin
            @inbounds hashes[k] = hash(hashes[k], hash((@nref $N A i)))
        end

        # Collect index of first row for each hash
        uniquerow = similar(Array{Int}, indices(A, dim))
        firstrow = Dict{Prehashed,Int}()
        for k = indices(A, dim)
            uniquerow[k] = get!(firstrow, Prehashed(hashes[k]), k)
        end
        uniquerows = collect(values(firstrow))

        # Check for collisions
        collided = similar(falses, indices(A, dim))
        @inbounds begin
            @nloops $N i A d->(if d == dim
                k = i_d
                j_d = uniquerow[k]
            else
                j_d = i_d
            end) begin
                if (@nref $N A j) != (@nref $N A i)
                    collided[k] = true
                end
            end
        end

        if any(collided)
            nowcollided = similar(BitArray, indices(A, dim))
            while any(collided)
                # Collect index of first row for each collided hash
                empty!(firstrow)
                for j = indices(A, dim)
                    collided[j] || continue
                    uniquerow[j] = get!(firstrow, Prehashed(hashes[j]), j)
                end
                for v in values(firstrow)
                    push!(uniquerows, v)
                end

                # Check for collisions
                fill!(nowcollided, false)
                @nloops $N i A d->begin
                    if d == dim
                        k = i_d
                        j_d = uniquerow[k]
                        (!collided[k] || j_d == k) && continue
                    else
                        j_d = i_d
                    end
                end begin
                    if (@nref $N A j) != (@nref $N A i)
                        nowcollided[k] = true
                    end
                end
                (collided, nowcollided) = (nowcollided, collided)
            end
        end

        @nref $N A d->d == dim ? sort!(uniquerows) : (indices(A, d))
    end
end

indexoffset(i) = first(i)-1
indexoffset(::Colon) = 0


"""
    extrema(A,dims) -> Array{Tuple}

Compute the minimum and maximum elements of an array over the given dimensions.
"""
function extrema(A::AbstractArray, dims)
    sz = [size(A)...]
    sz[[dims...]] = 1
    B = Array{Tuple{eltype(A),eltype(A)}}(sz...)
    return extrema!(B, A)
end

@generated function extrema!{T,N}(B, A::AbstractArray{T,N})
    return quote
        sA = size(A)
        sB = size(B)
        @nloops $N i B begin
            AI = @nref $N A i
            (@nref $N B i) = (AI, AI)
        end
        Bmax = sB
        Istart = ones(Int,ndims(A))
        Istart[([sB...].==1) & ([sA...].!=1)] = 2
        @inbounds @nloops $N i d->(Istart[d]:size(A,d)) begin
            AI = @nref $N A i
            @nexprs $N d->(j_d = min(Bmax[d], i_{d}))
            BJ = @nref $N B j
            if AI < BJ[1]
                (@nref $N B j) = (AI, BJ[2])
            elseif AI > BJ[2]
                (@nref $N B j) = (BJ[1], AI)
            end
        end
        return B
    end
end