The controller then issues a command to the processor holding that line that requires the processors to do a write back to main memory.

Directory schemes suffer from the drawbacks of a central bottleneck and the overhead of communication between the various cache controllers and the central controller.

(b) Snoopy protocols:
Snoopy protocols distribute the responsibility for maintaining cache coherence among all of the cache controllers in a multiprocessor system.

A cache must recognize when a line that it holds is shared with other caches. 

When an update action is performed on a shared cache line, it must be announced to all other caches by a broadcast mechanism.

Each cache controller is able to "snoop" on the network to observed these broadcasted notification and react accordingly.

Snoopy protocols are ideally suited to a bus-based multiprocessor, because the shared bus provides a simple means for broadcasting and snooping.