Working. If you remember the past weeks, I always had trouble with my monitor output.

Due to some mysterious reason which escapes me currently, it's working, which is good.

Okay, without much further ado, let's start with the lecture. I think we left off here

last week, right? And after the lecture, some of you asked me, wait, wait, wait, wait, wait.

How can we build a torus network with fixed length cables? And I didn't want to answer

this because I wanted to keep it for this lecture, but since not everyone of you was

here when I actually did explain, let's repeat this. So as you can see here, this donut-shaped

object represents a two-dimensional torus, not a three-dimensional. But as you remember,

blue gene has a, at least the first instantiations had a three-dimensional torus network. So

if I draw a three-dimensional torus network, it would probably look something like this,

like a cube. Now when I'm trying to connect the opposing sides of the cube, I somehow

need longer cables, right? Now the question is, if my system gets really big, these long

cables, basically the signal's running through those, even if I'm using fiber, the signal

can at most move at light speed, right? So at some point, these long cables will limit

my network latency and thus also the size of my system. So the question is, how can

I build a torus network, especially a 3D or more dimensional torus network, without these

long cables? And the key to that construction is actually we'll first build a one-dimensional

torus, and from this one-dimensional torus with fixed length cables, we'll build a two-dimensional

and then a three-dimensional torus. So the naive construction of a one-dimensional torus

would look like this. I'll start with my nodes from zero to five, and it will always connect

nearest neighbors and the opposing sides. And again, we have this problem here, which

we don't want to have. So an alternative way of constructing this would be to split the

two node sets into two halves and say, okay, we're starting here with zero, continue with

one, two, and then have the other half interleaved with that in a reverse order. So by this construction

scheme, I always have to skip one node when connecting the first nodes and then again

when moving backwards. The drawback of this is obviously that most cables are slightly

longer. They are now not just one single node distance but two node distances, but actually

this is an upper limit. I never need a cable which is longer than the distance between

two nodes. And by interleaving those nodes, I can still build a ring or a one-dimensional

torus, which is nothing else than a ring of nodes. But if I can build a 1D torus network,

I can use the same technique to build a two-dimensional torus by simply interleaving not just on the

x-axis but also on the y-axis. I won't draw all the connections because then probably

the lecture would be over, but just a brief illustration. If we assume that these are

always one-dimensional torus, what's the plural form of torus? The connection would probably

look a bit like this in the y-direction. And as you can imagine, if I can build two-dimensional

planes of torus networks, then I can also build three-dimensional cubes. Any questions

to this? Actually, that's how I thought they would be constructing networks. So I'm not

sure if they actually build networks like this, but it's one possibility. Okay, next

in line, the hypercube network. A hypercube network has one advantage. It sounds very

cool. It has one disadvantage. If you add more nodes to the network, you typically don't

have a constant out degree. I would say let's just start with an example. A zero-dimensional

hypercube is simply one node. When I'm now constructing a hypercube of higher order,

I will just take the original hypercube or the hypercube of one lesser order, copy it,

and then connect all corresponding nodes. So this would be now a hypercube of order

one. If you continue this, again, we'll copy the hypercube and then connect all corresponding

nodes. And of course, I can now continue this. This is also the reason why it's called hypercube,

because at some point it actually looks like a cube. So this would be a three-dimensional

hypercube, but we can even go beyond that. I will now use an alternative color. I won't

go beyond a four-dimensional hypercube, because that's already complicated enough. Okay, I

think that should already be it. No. Any further edges missing? Okay. So we can already see