I would argue that a public transportation network is a requirement, maybe even a prerequisite, for high density. Manhattan simply could not work without the various public transportation methods -- if everyone commuting in from CT, LI, NJ, upstate NY, etc., had to drive in, would there even be enough space for all those cars on the island?
if everyone is driving, noone is. This is simple game theory and a system fault happens when there are too many cars. You can't widen city streets.
For example: public transportation in NY is often faster and cheaper than a car + parking.