Air gap appurtenances are required in some states. I typically install them into the inlet of a garbage disposal, or through a wye tailpiece above the trap under the kitchen sink. The reason its done this way, is so you can tell if you have a backup (it comes up in the sink basin). If it goes into a trap in the ceiling, you may have damage done without knowing it. The reason it runs up high under the sink counter before it runs into an air break, is so backup from the main soil line will come up in the sink or other fixtures, before it comes up into the dishwasher.