[cperciva]

O(n^(1/2)) really, since data centers are 2 dimensional, not 3 dimensional.

(Quite aside from the practical "we build on the surface of the earth" consideration, heat dissipation considerations limit you to a 2 dimensional circuit in 3-space.)

[mpoteat]

More fundamentally O(n^(1/2)) due to the holographic principle which states that the maximal amount of information encodable in a given region of space scales wrt its surface area, rather than its volume.

(Even more aside to your practical heat dissipation constraint)

[vlowther]

Just need to make sure all your computation is done in a volume with infinite surface area and zero volume. Encoding problem solved. Now then, how hyperbolic can we make the geometry of spacetime before things get too weird?

[frollogaston]

Hmm, I'll go with that

[frollogaston]

If you have rows of racks of machines, isn't that 3 dimensions? A machine can be on top of, behind, or next to another that it's directly connected to. And the components inside have their own non-uniform memory access.

Or if you're saying heat dissipation scales with surface area and is 2D, I don't know. Would think that water cooling makes it more about volume, but I'm not an expert on that.

[manwe150]

That example would be two dimensions still in the limit computation, since you can keep building outwards (add buildings) but not scale upwards (add floors)

[frollogaston]

You can add floors though. Some datacenters are 8 stories with cross-floor network fabrics.

[immibis]

When you get to, say, 100000 stories, you can't build more stories. At this point your computer costs more than the Earth's GDP for a century, so talking about theoretical scaling laws is irrelevant. Eventually you run out of the sun's power output so you build a Dyson sphere and eventually use all of that power, anyway.

[frollogaston]

Oh right, so the height is practically a constant. Square root for sure then.

[LPisGood]

All algorithms are O(1) in this case

[jvanderbot]

Spatial position has nothing (ok only a little) to do with topology of connections.