Horsepower is all about getting more air into the engine. You can squirt a fire hose of gasoline.
Normally aspirated engines require a huge amount of work to improve the air flow. The only air available is what the vacuum from the piston creates, ram air effect, and really good flow through the system.
Raising the compression ratio pushes the molecules closer together and improves the amount of fuel that is ignited.
The limit is when autoignition starts raising its ugly head. At some point you have enough heat and pressure to cause the mixture to start ignition way too early in the cycle. Destructive detonation is the result.
Adding a supercharger or turbo to the equation doesn't change the mechanics of autoignition. You don't need as high a compression ratio because you are putting pressurized air into the cylinder and then compressing it further.
A high compression ratio on a supercharged engine isn't going to work on gasoline.
You can't have autoignition with direct injection because there isn't any fuel in the air at the autoignition point. But because you have the conditions, you don't have good control over the ignition point. Does the spark plug cause ignition, or the direct injection? You have to have a good mixture for the spark plug to ignite, so the injection has to come earlier. If it detonates then the direct injection hasn't gained you anything.
Boost does normally equate to power, because more air is available to put more fuel in and create more heat, which is more horsepower. You keep bumping the boost until something melts or breaks, make that part stronger, increase the boost until the next thing melts or breaks . . .
