Note: updated December 2018 for Julia 1.1
Julia makes it easy to write elegant and
efficient multidimensional algorithms. The new capabilities rest on
two foundations: an iterator called CartesianIndices
, 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 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.
It’s also possible to confuse these techniques with
Base.Cartesian
,
which is a completely different (and more painful) approach to
solving the same problem. There are still a few occasions where
Base.Cartesian
is helpful or 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.
You may already know that 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
If instead you also need the index, then use
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
Let’s see what these objects are:
A CartesianIndex{N}
represents an N
-dimensional index.
CartesianIndex
es are based on tuples, and indeed you can access the
underlying tuple with Tuple(i)
.
A CartesianIndices
acts like an array of CartesianIndex
values:
As a consequence iter[2,2]
and iter[5]
both return CartesianIndex(2, 2)
; indeed,
the latter is the recommended way to convert
from a linear index to a multidimensional cartesian index.
However, internally iter
is just a wrapper around the axes
range for each dimension:
As a consequence, in many applications the creation and usage of these objects has little or no overhead.
You can construct these manually: for example,
corresponds to an iterator 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 AbstractArray
s 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
to increase the likelihood that i
will be efficient for accessing
both A
and B
. (A second reason to use eachindex
is that some arrays
don’t starting indexing at 1, but that’s a topic for a separate
blog post.)
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.
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:
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 = CartesianIndices(A)
creates the iterator for the array. Assuming A
starts indexing at 1, this ranges 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
CartesianIndices
iterator, and here we explicitly need one.
Ifirst = first(R)
and Ilast = 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
.
I1 = oneunit(Ifirst)
creates an all-1s CartesianIndex
with the same
dimensionality as Ifirst
. We’ll use this in arithmetic operations to
define a region-of-interest.
for I in R
: here we loop over each entry of R
, corresponding to both
A
and out
.
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 max(Ifirst, I-I1):min(Ilast, 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.
However, when I
represents an edge point, either I-I1
or I+I1
(or both) might be out-of-bounds. max(Ifirst, I-I1)
ensures that each
coordinate of J
is 1 or larger, while min(Ilast, I+I1)
ensures
that J[d] <= size(A,d)
.
Putting these two together with a colon, Ilower:Iupper
,
creates a CartesianIndices
object that serves as an iterator.
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, it is also 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
.
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.
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:
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
.
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=dims)
;
here we want to write our own (somewhat simpler) implementation.
One possible bare-bones implementation of the wrapper looks like this:
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
non-inferrable:
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:
or by using an implementation that is inferrable:
However, here we want to emphasize that this design—having a separate
sumalongdims!
from sumalongdims
—often mitigates the worst aspects
of inference problems. 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.
It allows Julia’s compiler to generate a well-optimized version of
sumalongdims!
even if the intermediate type of B
is not known.
As a general rule, when writing multidimensional code you should
ensure that the main iteration is in a separate function from
type-unstable precursors. (In older versions of Julia, you might see
kernel functions annotated with @noinline
to prevent the
inliner from combining the two back together, but for more recent
versions of Julia this should no longer be necessary.)
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.
One final example illustrates an important new point: when you index
an array, you can freely mix CartesianIndex
es 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
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:
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 CartesianIndices
s 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
so that x
would be traversed in memory-order.
CartesianIndex
es are not broadcastable:
When you want to perform broadcast arithmetic, just extract the underlying tuple:
If desired you can package this back up in a CartesianIndex
, or just
use it directly (with splatting) for indexing.
The compiler optimizes all these operations away, so there is no actual
“cost” to constucting objects in this way.
Why is iteration disallowed? One reason is to support the following:
The underlying idea is that CartesianIndex(2, 17)
needs to act, everywhere,
like a pair of scalar indexes; consequently, a CartesianIndex
has to be
viewed as a single (scalar) entity, rather than as a container in its own right.
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.