Most selective companies require you to have near correct code in the most efficient algorithm, clean, correct, concise, with all edge cases in about 30-45 minutes depending on the question. They also seem to love dynamic programming algorithms and competitive programming tricks (the less common stuff you glossed over in an advanced algorithms class).
I say this as someone who worked at one and interviewed at others. You really do need to treat it like an exam.
Interviews of the sort you describe are like judging a fish by its ability to climb a tree. Indeed, whiteboards in interviews are used nothing like they are in real work. I'd hope more companies try and make interviewing closer to an actual work environment where the candidate can comfortably use their computer for solving the interview problem, using a whiteboard solely for communicating concepts and other surface level issues.
I say this as someone who worked at one and interviewed at others. You really do need to treat it like an exam.