"fast vs complete" is "SAX vs DOM"? and the IDs?

Are there a formal or more complete definition for fast-profile than this
one?
   http://dev.w3.org/csswg/selectors-4/#profiles
what is "fast"?

- - - -
Some blogs and discussions try to explain what is "slow" in the
fast-profile concept...
I learned from a  blog
<http://snook.ca/archives/html_and_css/css-parent-selectors>, and now,
personaly
I understand that
    the "fast-profile" is the hypothesis of use an "online algorithm"
    https://en.wikipedia.org/wiki/Online_algorithm
so, in XML terms, the fas-side is a good analog of the SAX-side in the "SAX
vs DOM" dicotomy.

In the fast-profile the document is traversed once, matching elements as
the online-algorithm go...
Ok, but it is strictly online, like in a streaming problem, or...
   we can suppose some pre-parse for know all IDs?
   can suppose "memory of IDs of all parsed above"? "memory of all classes
above"?
   ... other suppositions?
 what the real and complete scenario of a CSS-parser in the fast-profile?

- - - -

PS: it is a "benchmark game" with black-boxes (the future) or white-boxes
(the present)?
The *near-future* of CSS parsers is a gray-box... It's fine: let's put some
rules in this game!