Multidimensional algorithms and iteration

Starting with release 0.4, Julia makes it easy to write elegant and
efficient multidimensional algorithms. The new capabilities rest on
two foundations: a new type of iterator, called CartesianRange, and
sophisticated array indexing mechanisms. Before I explain, let me
emphasize that developing these capabilities was a collaborative
effort, with the bulk of the work done by Matt Bauman (@mbauman),
Jutho Haegeman (@Jutho), and myself (@timholy).

These new iterators are deceptively simple, so much so that I’ve never
been entirely convinced that this blog post is necessary: once you
learn a few principles, there’s almost nothing to it. However, like
many simple concepts, the implications can take a while to sink in.
There also seems to be some widespread confusion about the
relationship between these iterators and
Base.Cartesian,
which is a completely different (and much more painful) approach to
solving the same problem. There are still a few occasions where
Base.Cartesian is necessary, but for many problems these new
capabilities represent a vastly simplified approach.

Let’s introduce these iterators with an extension of an example taken
from the
manual.

eachindex, CartesianIndex, and CartesianRange

You may already know that, in julia 0.4, there are two recommended
ways to iterate over the elements in an AbstractArray: if you don’t
need an index associated with each element, then you can use

 for a in A    # A is an AbstractArray
# Code that does something with the element a
end


If instead you also need the index, then use

for i in eachindex(A)
# Code that does something with i and/or A[i]
end


In some cases, the first line of this loop expands to for i = 1:length(A), and i is just an integer. However, in other cases,
this will expand to the equivalent of

 for i in CartesianRange(size(A))
# i is now a CartesianIndex
# Code that does something with i and/or A[i]
end


Let’s see what these objects are:

  A = rand(3,2)

julia> for i in CartesianRange(size(A))
@show i
end
i = CartesianIndex{2}((1,1))
i = CartesianIndex{2}((2,1))
i = CartesianIndex{2}((3,1))
i = CartesianIndex{2}((1,2))
i = CartesianIndex{2}((2,2))
i = CartesianIndex{2}((3,2))


A CartesianIndex{N} represents an N-dimensional index.
CartesianIndexes are based on tuples, and indeed you can access the
underlying tuple with i.I. However, they also support certain
arithmetic operations, treating their contents like a fixed-size
Vector{Int}. Since the length is fixed, julia/LLVM can generate
very efficient code (without introducing loops) for operations with
N-dimensional CartesianIndexes.

A CartesianRange is just a pair of CartesianIndexes, encoding the
start and stop values along each dimension, respectively:

  julia> CartesianRange(size(A))
CartesianRange{CartesianIndex{2}}(CartesianIndex{2}((1,1)),CartesianIndex{2}((3,2)))


You can construct these manually: for example,

julia> CartesianRange(CartesianIndex((-7,0)), CartesianIndex((7,15)))
CartesianRange{CartesianIndex{2}}(CartesianIndex{2}((-7,0)),CartesianIndex{2}((7,15)))


constructs a range that will loop over -7:7 along the first
dimension and 0:15 along the second.

One reason that eachindex is recommended over for i = 1:length(A)
is that some AbstractArrays cannot be indexed efficiently with a
linear index; in contrast, a much wider class of objects can be
efficiently indexed with a multidimensional iterator. (SubArrays are,
generally speaking, a prime
example
.)
eachindex is designed to pick the most efficient iterator for the
given array type. You can even use

  for i in eachindex(A, B)
...


to increase the likelihood that i will be efficient for accessing
both A and B.

As we’ll see below, these iterators have another purpose: independent
of whether the underlying arrays have efficient linear indexing,
multidimensional iteration can be a powerful ally when writing
algorithms. The rest of this blog post will focus on this
latter application.

Writing multidimensional algorithms with CartesianIndex iterators

A multidimensional boxcar filter

Let’s suppose we have a multidimensional array A, and we want to
compute the “moving
average”
over a
3-by-3-by-… block around each element. From any given index position,
we’ll want to sum over a region offset by -1:1 along each dimension.
Edge positions have to be treated specially, of course, to avoid going
beyond the bounds of the array.

In many languages, writing a general (N-dimensional) implementation of
this conceptually-simple algorithm is somewhat painful, but in Julia
it’s a piece of cake:

 function boxcar3(A::AbstractArray)
out = similar(A)
R = CartesianRange(size(A))
I1, Iend = first(R), last(R)
for I in R
n, s = 0, zero(eltype(out))
for J in CartesianRange(max(I1, I-I1), min(Iend, I+I1))
s += A[J]
n += 1
end
out[I] = s/n
end
out
end


Let’s walk through this line by line:

• out = similar(A) allocates the output. In a “real” implementation,
you’d want to be a little more careful about the element type of the
output (what if the input array element type is Int?), but
we’re cutting a few corners here for simplicity.

• R = CartesianRange(size(A)) creates the iterator for the array,
ranging from CartesianIndex((1, 1, 1, ...)) to
CartesianIndex((size(A,1), size(A,2), size(A,3), ...)). We don’t
use eachindex, because we can’t be sure whether that will return a
CartesianRange iterator, and here we explicitly need one.

• I1 = first(R) and Iend = last(R) return the lower
(CartesianIndex((1, 1, 1, ...))) and upper
(CartesianIndex((size(A,1), size(A,2), size(A,3), ...))) bounds
of the iteration range, respectively. We’ll use these to ensure
that we never access out-of-bounds elements of A.

Conveniently, I1 can also be used to compute the offset range.

• for I in R: here we loop over each entry of A.

• n = 0 and s = zero(eltype(out)) initialize the accumulators. s
will hold the sum of neighboring values. n will hold the number of
neighbors used; in most cases, after the loop we’ll have n == 3^N,
but for edge points the number of valid neighbors will be smaller.

• for J in CartesianRange(max(I1, I-I1), min(Iend, I+I1)) is
probably the most “clever” line in the algorithm. I-I1 is a
CartesianIndex that is lower by 1 along each dimension, and I+I1
is higher by 1. Therefore, this constructs a range that, for
interior points, extends along each coordinate by an offset of 1 in
either direction along each dimension.

However, when I represents an edge point, either I-I1 or I+I1
(or both) might be out-of-bounds. max(I-I1, I1) ensures that each
coordinate of J is 1 or larger, while min(I+I1, Iend) ensures
that J[d] <= size(A,d).

• The inner loop accumulates the sum in s and the number of visited
neighbors in n.

• Finally, we store the average value in out[I].

Not only is this implementation simple, but it is surprisingly robust:
for edge points it computes the average of whatever nearest-neighbors
it has available. It even works if size(A, d) < 3 for some
dimension d; we don’t need any error checking on the size of A.

Computing a reduction

For a second example, consider the implementation of multidimensional
reductions. A reduction takes an input array, and returns an array
(or scalar) of smaller size. A classic example would be summing along
particular dimensions of an array: given a three-dimensional array,
you might want to compute the sum along dimension 2, leaving
dimensions 1 and 3 intact.

The core algorithm

An efficient way to write this algorithm requires that the output
array, B, is pre-allocated by the caller (later we’ll see how one
might go about allocating B programmatically). For example, if the
input A is of size (l,m,n), then when summing along just dimension
2 the output B would have size (l,1,n).

Given this setup, the implementation is shockingly simple:

function sumalongdims!(B, A)
# It's assumed that B has size 1 along any dimension that we're summing
fill!(B, 0)
Bmax = CartesianIndex(size(B))
for I in CartesianRange(size(A))
B[min(Bmax,I)] += A[I]
end
B
end


The key idea behind this algorithm is encapsulated in the single
statement B[min(Bmax,I)]. For our three-dimensional example where
A is of size (l,m,n) and B is of size (l,1,n), the inner loop
is essentially equivalent to

B[i,1,k] += A[i,j,k]


because min(1,j) = 1.

The wrapper, and handling type-instability using function barriers

As a user, you might prefer an interface more like sumalongdims(A, dims) where dims specifies the dimensions you want to sum along.
dims might be a single integer, like 2 in our example above, or
(should you want to sum along multiple dimensions at once) a tuple or
Vector{Int}. This is indeed the interface used in sum(A, dims);
here we want to write our own (somewhat simpler) implementation.

A bare-bones implementation of the wrapper is straightforward:

function sumalongdims(A, dims)
sz = [size(A)...]
sz[[dims...]] = 1
B = Array(eltype(A), sz...)
sumalongdims!(B, A)
end


Obviously, this simple implementation skips all relevant error
checking. However, here the main point I wish to explore is that the
allocation of B turns out to be
type-unstable:
sz is a Vector{Int}, the length (number of elements) of a specific
Vector{Int} is not encoded by the type itself, and therefore the
dimensionality of B cannot be inferred.

Now, we could fix that in several ways, for example by annotating the
result:

B = Array(eltype(A), sz...)::typeof(A)


However, this isn’t really necessary: in the remainder of this
function, B is not used for any performance-critical operations.
B simply gets passed to sumalongdims!, and it’s the job of the
compiler to ensure that, given the type of B, an efficient version
of sumalongdims! gets generated. In other words, the type
instability of B’s allocation is prevented from “spreading” by the
fact that B is henceforth used only as an argument in a function
call. This trick, using a function-call to separate a
performance-critical step from a potentially type-unstable
precursor
,
is sometimes referred to as introducing a function barrier.

As a general rule, when writing multidimensional code you should
ensure that the main iteration is in a separate function from
type-unstable precursors. Even when you take appropriate precautions,
there’s a potential “gotcha”: if your inner loop is small, julia’s
ability to inline code might eliminate the intended function barrier,
and you get dreadful performance. For this reason, it’s recommended
that you annotate function-barrier callees with @noinline:

@noinline function sumalongdims!(B, A)
...
end


Of course, in this example there’s a second motivation for making this
a standalone function: if this calculation is one you’re going to
repeat many times, re-using the same output array can reduce the
amount of memory allocation in your code.

Filtering along a specified dimension (exploiting multiple indexes)

One final example illustrates an important new point: when you index
an array, you can freely mix CartesianIndexes and
integers. To illustrate this, we’ll write an exponential
smoothing
filter
. An
efficient way to implement such filters is to have the smoothed output
value s[i] depend on a combination of the current input x[i] and
the previous filtered value s[i-1]; in one dimension, you can write
this as

function expfilt1!(s, x, α)
0 < α <= 1 || error("α must be between 0 and 1")
s[1] = x[1]
for i = 2:length(a)
s[i] = α*x[i] + (1-α)*s[i-1]
end
s
end


This would result in an approximately-exponential decay with timescale 1/α.

Here, we want to implement this algorithm so that it can be used to
exponentially filter an array along any chosen dimension. Once again,
the implementation is surprisingly simple:

function expfiltdim(x, dim::Integer, α)
s = similar(x)
Rpre = CartesianRange(size(x)[1:dim-1])
Rpost = CartesianRange(size(x)[dim+1:end])
_expfilt!(s, x, α, Rpre, size(x, dim), Rpost)
end

@noinline function _expfilt!(s, x, α, Rpre, n, Rpost)
for Ipost in Rpost
# Initialize the first value along the filtered dimension
for Ipre in Rpre
s[Ipre, 1, Ipost] = x[Ipre, 1, Ipost]
end
# Handle all other entries
for i = 2:n
for Ipre in Rpre
s[Ipre, i, Ipost] = α*x[Ipre, i, Ipost] + (1-α)*s[Ipre, i-1, Ipost]
end
end
end
s
end


Note once again the use of the function barrier technique. In the
core algorithm (_expfilt!), our strategy is to use two
CartesianIndex iterators, Ipre and Ipost, where the first covers
dimensions 1:dim-1 and the second dim+1:ndims(x); the filtering
dimension dim is handled separately by an integer-index i.
Because the filtering dimension is specified by an integer input,
there is no way to infer how many entries will be within each
index-tuple Ipre and Ipost. Hence, we compute the CartesianRanges in
the type-unstable portion of the algorithm, and then pass them as
arguments to the core routine _expfilt!.

What makes this implementation possible is the fact that we can index
x as x[Ipre, i, Ipost]. Note that the total number of indexes
supplied is (dim-1) + 1 + (ndims(x)-dim), which is just ndims(x).
In general, you can supply any combination of integer and
CartesianIndex indexes when indexing an AbstractArray in Julia.

The AxisAlgorithms
package makes heavy use of tricks such as these, and in turn provides
core support for high-performance packages like
Interpolations that
require multidimensional computation.

It’s worth noting one point that has thus far remained unstated: all
of the examples here are relatively cache efficient. This is a key
property to observe when writing efficient
code
. In
particular, julia arrays are stored in first-to-last dimension order
(for matrices, “column-major” order), and hence you should nest
iterations from last-to-first dimensions. For example, in the
filtering example above we were careful to iterate in the order

for Ipost ...
for i ...
for Ipre ...
x[Ipre, i, Ipost] ...


so that x would be traversed in memory-order.

Summary

As is hopefully clear by now, much of the pain of writing generic
multidimensional algorithms is eliminated by Julia’s elegant
iterators. The examples here just scratch the surface, but the
underlying principles are very simple; it is hoped that these
examples will make it easier to write your own algorithms.