Is "the exchange" in the same building as the trading floor?
I don't know either way, but would be surprised if the exchange and the trading floor were in the same building.
I'd be surprised because the trading floor must be in Manhattan for social reasons (because people go there) while the exhange can be "anywhere". Given the price of space in Manhattan....
This would suggest that the NYSE datacenter is not in the same building as the trading floor. And, I would assume that there are more than one of these such datacenters.
...HFT guys pay very expensive colo rates to be in a datacenter that is milliseconds closer to wherever the time is stamped on a trade.
Yes, but that doesn't constrain the location of the exchange. The exchange (and all of the HFT colos) could be in Texas and the trading floor in NYC. (There's no reason to do HFT from the trading floor.)
EDIT: Okay, close to NYSE's datacenter, which is still in New York, I believe.
Why?
HFT needs to be close to where transactions happen, but that's not the trading floor.
The trading floor is just a place where a bunch of people with a bunch of screens connected to the "place where transactions happen" gather each trading period.
The trading floor can't be where transactions happen because it's got people in it, not machines.