Space filling curves as interconnection topology
By Jon Bringhurst
March 3rd, 2011
I haven't been able to find much information on how space filling curves, like
Hilbert curves, are used in network interconnection topology in supercomputers.
So, this paper is my attempt at putting the information in one place. We'll
start off with an introduction to gray encoding and how it relates to Hilbert
curves.
So, gray codes are the concept of the day here. Gray codes are just another
binary representation of integer counting (1, 2, 3, 4, 5, …). They were used
back in the day for mechanical counting because of a key property -- only one
bit changes for every value that a gray code is incremented or decremented.
This made it simple for a sensor to tell when the value had changed because
only one bit would be flipped every time. This is in contrast to normal binary
counting where it's common for multiple bits to be flipped for every time the
value is incremented or decremented which makes it difficult to tell if the
bits have all completely flipped into their final state.
Gray codes also have another special property. Some people like to call gray
codes "reflected binary codes". This property makes the generation of gray
codes feel very similar to the generation of a fractal. For example, start off
with the one bit gray codes -- you have 0 and 1. Now, to get two bit gray
codes, you write it forwards and backwards (0, 1, 1, 0). Then, prepend 0s to
the first half of the numbers and 1s to the second half of the numbers (00, 01,
11, 10) and you end up with the two bit gray codes. To get the n-bit gray
codes, you repeat the process n times from the base case.
Now the cool stuff. Gray codes have yet another story to tell. You can use gray
codes to reference a coordinate in a space filling curve known as a
"Hilbert Curve". Ignore the word "curve" for now -- it's just going to confuse
you. If you're like most people, you're going to think of a curve as a smooth
turn on a roller coaster or the shape of a hot air balloon. You'll think of
Hilbert curves like that eventually, but not now. Right now, you can safely
think of Hilbert curves in appearance as looking similar to the maze puzzles
on a paper placemat from a roadside greasy spoon that you solved with a wax
crayon as a child.
So, how do you use gray codes to reference locations in Hilbert curves? Lets
start off with the two bit gray codes. Just use each value like an x-y
coordinate. So, we have the coordinates of (0, 0), (0, 1), (1, 1), (1, 0) as
our first order Hilbert curve.
(0, 1) *---------* (1, 1)
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(0, 0) * * (1, 0)
Now, lets try to do the second order Hilbert curve using the four bit gray
codes. To do this, we need to utilize the fractal properties of the Hilbert
curve. Each coordinate in the first order Hilbert curve needs to be replaced
by a new sub-curve that is the same shape as the first order curve. This new
sub-curve will be translated and rotated to fit in with the design of the
first order Hilbert curve. Once each of the coordinates has been replaced, we
end up with a second order Hilbert curve.
*-------* *-------*
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* *-------* *
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*-------* *-------*
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*-------* *-------*
To address each of the coordinates in the second order Hilbert curve with gray
codes, we need to reference the original location of the coordinate that was
replaced by the sub-curve. So, for example, the lower-right corner in the
second order Hilbert curve has a gray code with bits that start with "10...".
We can then get the rest of the gray code by again picturing the single order
Hilbert curve and rotating the visualization and then referencing that same
point after rotations in the lower order. So, the remaining part of the code is
"00" -- which makes the code for the lower-right corner "1000". Using the same
concept, you can use a gray code to find the location in a Hilbert curve.
Now, how does the conversion from gray codes to Hilbert graphs and back
actually help us when it comes to network topology? Lets look at a third order
Hilbert curve for some perspective.
*---* *---* *---* *---*
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* *---* * * *---* *
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*---* *---* *---* *---*
| | | |
*---* *---*---*---* *---*
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* *---*---* *---*---* *
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*---* *---* *---* *---*
(3) | (7) |
(2) *---* *---* *---* *---*
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(1) * *---*---* *---*---* *
(4) (5) (6)
Start from the lower left corner of the curve and start ordering each
coordinate as you go along. As you pass each coordinate, calculate a rough
estimate of the physical difference in distance between each coordinate.
You'll find that as you go, coordinates that are close to each other tend to
also be close to each other in their ordering. The number of each coordinate as
you go along and give each a value is known as the "Hilbert integer".
Now that we have a bit of a base, we can try to relate this to interconnection
topology. First things first, we need to have a picture of a 3 dimensional
Hilbert curve. So, instead of filling up a 2 dimensional space through each
order of the curve, we fill up a 3 dimensional space by adding a rotation into
the third dimension to each translate and rotation in 2 dimensional space.
*
* |
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| * | *
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*-----* |
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*-----*
Now that we have a topology that closely resembles the node structure in a 3D
torus network structure, we can start thinking about how nodes are allocated.
Remember the concept of a Hilbert integer? That's going to be the node ID of
every node in the supercomputer. When the Hilbert integers of each node are
generated, they're stored by the central resource manager and stored for the
duration of the resource manager's lifecycle. After being sorted into a one
dimensional array, the node IDs can then be effectively used by a scheduler to
provide resources to jobs at a level of abstraction where the scheduler doesn't
need to worry about the actual node topology.
EOF